Mixture modeling of hospital charge in Zimbabwe

Abstract

Objective

Zimbabwe is one of the poorest countries in the world, just emerging from a broken health care system. The objective is to figure out variables affecting hospital charge for Zimbabwe.

Material and methods

The variables used are sex, smoking status, number of children, region, age and body mass index. The first six of these are factors and the remaining are covariates. A mixture model was fitted to describe the dependence of hospital charge on these variables.

Results

A mixture model with five components each having a reversed Gumbel distribution was found to give an adequate fit. Both the covariates and all but one of the factors were found to be significant. Estimates of value at risk of hospital charge are given for all combinations of the factors.

Conclusions

The results suggest that the hospital charge could be higher for females, higher for smokers, higher if the patient had more children and higher if the patient is older. Further, estimates of value at risk given suggest, for example, that a 90 year old female not smoking and having no children and an average body mass index will have a hospital charge less than Z$49851 with probability 0.999.

1. Introduction

Zimbabwe is one of a few countries in the world with Gross National Income per capita below US$2,000. Zimbabwe's economy declined by 5% in 2000, 8% in 2001, 12% in 2002 and 18% in 2003 [1]. From 1999 to 2009, Zimbabwe saw the lowest ever economic growth with an annual Gross Domestic Product (GDP) decrease of 6.1% [2]. In the late 2000s, Zimbabwe's health care system was more or less collapsed [3], [4]. The life expectancy was 44 for men and 43 for women. In recent times, the economy of Zimbabwe has improved – according to the World bank, “the economy rebounded in 2021 driven by recovery of agriculture and industry and relative stabilization of prices and exchange rates. GDP is estimated to have grown by 5.8% in 2021 after contracting by 6.2% in 2020. An exceptionally good agriculture season, coupled with slowing inflation and higher remittances boosted domestic demand” [5]. The government has sought to provide better health care service to the public with international help. In this respect, the public needs to know and need to be informed of the cost of health care.

The hospital charge will be dependent on factors like length of stay, disease type, severity, and interventions offered. But the patients will have no idea about these factors until after they visit the hospital. It will be useful if an estimate of the average hospital charge can be made based on factors known to the patient like sex, age, region, smoking status, number of children and body mass index. For example, this can help the patient and his/her family to seek funds to cover the hospital charge especially in a poor country like Zimbabwe.

The aim of this paper is to provide statistical modeling of hospital charge in Zimbabwe. There have been no papers on modeling of hospital charge for Zimbabwe or any other part of Africa. However, there are papers using hospital charge as a variable to discriminate groups of patients having different diseases. We mention [6], [7], [8], [9]. There are also papers performing cost analysis of certain diseases, see, for example, [10], [11], [12].

The aim of this paper is to determine how hospital charge depends on various social factors. The data on hospital charge (in Zimbabwe dollars) for the year 2020 were obtained from the Ministry of Health in Harare. The data also contained the following:

• •Sex of the patient: Male or Female;
• •Age of the patient (18-64);
• •Region the patient comes from: Harare, Bulawayo, Masvingo and Gweru (the four largest cities in Zimbabwe);
• •Smoking status of the patient: yes or no;
• •Number of children the patient has (0-5);
• •Body mass index of the patient (15.96-53.13).

Age and body mass index are covariates. Sex, region, smoking status and the number of children are factors. The results of this paper could be useful to both the public and the government of Zimbabwe.

A histogram of hospital charge is shown in Fig. 1. Some summary statistics are minimum =1122, first quartile =4740, median =9382, mean =13270, third quartile =16640, maximum =63770, variance =146652372, skewness =1.51418 and kurtosis =4.595821.

Figure 1.

Histogram of the hospital charge.

We see that the histogram is multimodal. The multimodality may have arisen from social factors like sex, region and smoking status of the patient. No one statistical distribution can capture the multimodality in Fig. 1 adequately. So, we use mixtures of distributions to model hospital charge and its dependence on the social factors.

The mixture distributions applied in the paper may seem simple compared to recent advances in mixture modeling. But the paper tackles an important problem facing one of the poorest countries in the world. Hence, we have chosen to use models as simple as possible. There is no need apply advanced state of the art mixture models.

All methods in this paper were carried out in accordance with relevant guidelines and regulations. All experimental protocols with respect to the data were approved by an ethics committee of the Ministry of Health in Harare. Informed consent for the data was obtained from the patients verbally. The data are not publicly available. But they can be obtained from the corresponding author. The analysis reported in this paper is exclusively based on the data records obtained from the Ministry of Health.

The contents of this paper are organized as follows. Section 2 performs some exploratory analysis to identify features in the data and states mixture models used for hospital charge. Section 3 describes fitting of the models as well as results and discussion. Finally, some conclusions are reported in Section 4.

2. Data and models

2.1. Exploratory data analysis

The relationship of hospital charge versus age, body mass index, sex, number of children, smoking status and region is shown in Fig. 2.

Figure 2.

Scatter plot of hospital charge versus age (top left) and body mass index (top right); box plot of hospital charge versus sex (middle left), number of children (middle right), smoking status (bottom left) and region (bottom right).

We can observe the following from Fig. 2. The hospital charge generally increases with respect to age. The hospital charge generally increases with respect to body mass index too when it up to 35. Thereafter the pattern is not clear as there are not so many patients having greater body mass index than 35. The variability of the hospital charge is larger for men. The variability is largest when the number of children is 2 and smallest when the number of children is 5. The hospital charge is larger and more variable when the patient is a smoker. The variability of the hospital charge is largest when the region is Masvingo and smallest when the region is Gweru.

The correlation between age and body mass index is 0.190. The correlation between hospital charge and age is 0.299. The correlation between hospital charge and body mass index is 0.198. None of these correlations appear significant.

2.2. Models

Because of the features noted in Fig. 1, we take the probability density function of the hospital charge as

$$f(x;\mathbf{\Theta })=\sum _{i=1}^K{w}_i{f}_i(x;{\mathbf{\Theta }}_i),$$ (1)

where K is a positive integer denoting the number of densities used in the mixture, $w_i$ are non-negative weights summing to 1, $f_i(x;{\mathbf{\Theta }}_i)$ are valid probability density functions and $\mathbf{\Theta }=({\mathbf{\Theta }}_1,\dots ,{\mathbf{\Theta }}_K,w_1,\dots ,w_{K-1})$. Note that the weights themselves are treated as part of the parameters. Estimation of parameters is described later on.

We chose $f_i(x;{\mathbf{\Theta }}_i)$, $i=1,\dots ,K$ to belong to the same parametric family. It was taken to be one of the following fifty two distributions: Box-Cox Cole and Green's distribution [13]; Box-Cox power exponential distribution [14]; Box-Cox t distribution [15]; composite gamma-generalized Pareto distribution [16]; exponential distribution; ex-Gaussian distribution; exponential generalized beta type 2 distribution; gamma distribution; generalized beta type 2 distribution; generalized gamma distribution; generalized inverse Gaussian distribution [17]; generalized lambda distribution [18]; generalized t distribution; Gompertz distribution; Gumbel distribution; inverse gamma distribution; inverse Gaussian distribution; Johnson's SU distribution [19]; logistic distribution; log-logistic distribution; logit-normal distribution; log-normal distribution; Box-Cox log-normal distribution [20]; normal exponential t distribution; normal distribution; normal family distribution; normal linear quadratic distribution; Pareto type 1 distribution; Pareto type 2 distribution; power exponential type 1 distribution [21]; power exponential type 2 distribution [21]; reversed generalized extreme value distribution; reversed Gumbel distribution; skew normal distribution [22]; skew power exponential distribution of type 1 [23]; skew power exponential distribution of type 2 [23]; skew power exponential distribution of type 3 [23]; skew power exponential distribution of type 4 [23]; skew slash distribution; slash distribution; sinh-arcsinh distribution of type 1 [24]; sinh-arcsinh distribution of type 2 [25]; sinh-arcsinh distribution of type 3 [25]; skew t distribution of type 1 [26]; skew t distribution of type 2 [26]; skew t distribution of type 3 [26]; skew t distribution of type 4 [26]; skew t distribution of type 5 [27]; Student's t distribution; Weibull distribution of type 1; Weibull distribution of type 2; Weibull distribution of type 3.

We fitted (1) for each of the fifty two choices for $f_i(x;{\mathbf{\Theta }}_i)$, $i=1,\dots ,K$. The method of maximum likelihood and the Expectation Maximization (EM) algorithm were used for the fitting. All computations were performed using the packages gamlss.mx [28] and nor1mix [29] in the R software [30]. Details of the estimation methods can be found in [28], [29] and references therein. Discrimination among the fitted models was performed using the two criteria:

• •
  the Akaike information criterion due to [31] defined by

  $$\mathrm{AIC}=2(mK+K-1)-2\ln L\left(\hat{\mathbf{\Theta }}\right),$$

  where m denotes the length of each ${\mathbf{\Theta }}_i$, $\hat{\mathbf{\Theta }}$ denotes the maximum likelihood estimate of Θ, and $\ln L\left(\hat{\mathbf{\Theta }}\right)$ denotes the log-likelihood value;
• •
  the Bayesian information criterion due to [32] defined by

  $$\mathrm{BIC}=(mK+K-1)\ln n-2\ln L\left(\hat{\mathbf{\Theta }}\right),$$

  where n denotes the sample size.

The smaller the values of these criteria the better the fit.

3. Results and discussion

3.1. Mixture modeling

The model given by (1) was fitted for the fifty two choices for $f_i(x;{\mathbf{\Theta }}_i)$, $i=1,\dots ,K$ and $K=2,3,\dots ,10$. The values of AIC and BIC versus K are plotted in Figure 3, Figure 4 for ten selected choices. The values corresponding to the other forty two choices are available upon request.

Figure 3.

AIC versus K when f_i(x;Θ_i) are the inverse gamma densities (solid red), reversed Gumbel densities (solid green), inverse Gaussian densities (solid blue), log gamma densities (solid brown), Pareto type II densities (solid black), Gompertz densities (broken red), log-logistic densities (broken green), log-normal densities (broken blue), Weibull densities (broken brown) and exponential densities (broken black).

Figure 4.

BIC versus K when f_i(x;Θ_i) are the inverse gamma densities (solid red), reversed Gumbel densities (solid green), inverse Gaussian densities (solid blue), log gamma densities (solid brown), Pareto type II densities (solid black), Gompertz densities (broken red), log-logistic densities (broken green), log-normal densities (broken blue), Weibull densities (broken brown) and exponential densities (broken black).

Figure 3, Figure 4 show that AIC and BIC are smallest when $K=5$ and $f_i(x;{\mathbf{\Theta }}_i)$ are the reversed Gumbel densities. That is,

$$f_i(x;{\mathbf{\Theta }}_i)=\frac{1}{{\sigma }_i}\exp [-\frac{x-{\mu }_i}{{\sigma }_i}-\exp (-\frac{x-{\mu }_i}{{\sigma }_i})],$$ (2)

where $-\infty <x<\infty$, $-\infty <{\mu }_i<\infty$ are the location parameters, ${\sigma }_i>0$ are the scale parameters and ${\mathbf{\Theta }}_i=({\mu }_i,{\sigma }_i)$. The parameter estimates and standard errors of ${\mu }_i$, $i=1,2,\dots ,5$, ${\sigma }_i$, $i=1,2,\dots ,5$ and $w_i$, $i=1,2,\dots ,4$ (see [29] and [28] for details of the estimation method and software) are in Table 1.

Table 1.

Parameter estimates and standard errors of the mixture of reversed Gumbel densities.

Parameter	Estimate (standard error)
μ1	10.26(0.952)
σ1	0.987(0.065)
μ2	1.847(0.799)
σ2	−0.647(0.034)
μ3	38.88(3.407)
σ3	1.528(0.122)
μ4	21.84(1.967)
σ4	1.336(0.174)
μ5	4.707(1.584)
σ5	0.524(0.056)
w1	0.349(0.089)
w2	0.139(0.084)
w3	0.109(0.067)
w4	0.111(0.079)

The fitted probability density function of the hospital charge given by

$$f(x)=\sum _{i=1}^4\frac{\widehat{w_i}}{\widehat{{\sigma }_i}}\exp [-\frac{x-\widehat{{\mu }_i}}{\widehat{{\sigma }_i}}-\exp (-\frac{x-\widehat{{\mu }_i}}{\widehat{{\sigma }_i}})]+\frac{1-\widehat{w_1}-\cdots -\widehat{w_4}}{\widehat{{\sigma }_5}}\exp [-\frac{x-\widehat{{\mu }_5}}{\widehat{{\sigma }_5}}-\exp (-\frac{x-\widehat{{\mu }_5}}{\widehat{{\sigma }_5}})]$$ (3)

is shown in Fig. 5. The corresponding probability and quantile plots are shown in Fig. 6. These figures show that the fitted distribution adequately describes the data.

Figure 5.

Histogram of the hospital charge and the fitted probability density function (3).

Figure 6.

Probability plot (left) and quantile plot (right) corresponding to (3).

3.2. Regression modeling

We now incorporate the factors (sex, region, smoking status and the number of children) and covariates (age and body mass index) into modeling to see how they affect hospital charge. We let the location parameters in (2) depend on the factors and covariates as follows:

$${\mu }_i={\mu }_{\mathrm{intercept},i}+{\mu }_{\mathrm{sex},i}\cdot I\{\mathrm{sex}=1\}+\sum _{j=1}^3{\mu }_{\mathrm{region},i,j}\cdot I\{\mathrm{region}=j\}+{\mu }_{\mathrm{smoking},i}\cdot I\{\mathrm{smoking}=1\}+\sum _{j=1}^5{\mu }_{\mathrm{child},i,j}\cdot I\{\mathrm{child}=j\}+{\mu }_{\mathrm{age},i}\cdot \mathrm{Age}+{\mu }_{\mathrm{BMI},i}\cdot \mathrm{BMI}$$

for $i=1,2,\dots ,5$, where $I\{\cdot \}$ denotes the indicator function. We begin by fitting the constant model ${\mu }_i={\mu }_{\mathrm{intercept},i}$ for $i=1,2,\dots ,5$. Then, we fit the term that is most significant among the factors and covariates. Thereafter, we fit the term that is most significant among the remaining factors and covariates. We continue this process until the model cannot be improved further. In the final model, all of the factors and covariates turned out to be significant except for region. That is,

$$\widehat{{\mu }_i}=\widehat{{\mu }_{\mathrm{intercept},i}}+\widehat{{\mu }_{\mathrm{sex},i}}\cdot I\{\mathrm{sex}=1\}+\widehat{{\mu }_{\mathrm{smoking},i}}\cdot I\{\mathrm{smoking}=1\}+\sum _{j=1}^5\widehat{{\mu }_{\mathrm{child},i,j}}\cdot I\{\mathrm{child}=j\}+\widehat{{\mu }_{\mathrm{age},i}}\cdot \mathrm{Age}+\widehat{{\mu }_{\mathrm{BMI},i}}\cdot \mathrm{BMI}$$ (4)

for $i=1,2,\dots ,5$.

The parameter estimates in (4) (see [29] and [28] for details of the estimation method and software) are in Table 2. The parameter estimates related to sex are mostly negative, meaning that the hospital charge could be higher for females. The parameter estimates related to smoking status are all positive, meaning that the hospital charge could be higher for smokers. The parameter estimates related to the number of children are mostly positive, meaning that the hospital charge could be higher if the patient had more children. The parameter estimates related to age are all positive, meaning that the hospital charge could be higher if the patient is older. Three of the parameter estimates related to the body mass index are negative, but all of them are close to zero. The estimates of ${\mu }_{\mathrm{intercept},1}$, ${\mu }_{\mathrm{intercept},2}$, ${\mu }_{\mathrm{intercept},3}$, ${\mu }_{\mathrm{intercept},4}$, ${\mu }_{\mathrm{intercept},5}$, ${\mu }_{\mathrm{sex},1}$, ${\mu }_{\mathrm{sex},2}$, ${\mu }_{\mathrm{sex},4}$, ${\mu }_{\mathrm{sex},5}$, ${\mu }_{\mathrm{child},1,1}$, ${\mu }_{\mathrm{child},1,2}$, ${\mu }_{\mathrm{child},2,5}$, ${\mu }_{\mathrm{child},3,4}$, ${\mu }_{\mathrm{child},3,5}$, ${\mu }_{\mathrm{child},5,5}$, ${\mu }_{\mathrm{BMI},1}$, ${\mu }_{\mathrm{BMI},4}$ and ${\mu }_{\mathrm{BMI},5}$ were significantly positive. The estimates of the remaining parameters were significantly negative. The significance was tested using the likelihood ratio test ([33], page 331) and the software due to [29], [28].

Table 2.

Estimates of the parameters in (4).

Parameter	Estimate	Parameter	Estimate
μintercept,1	−3.281	μchild,3,1	0.065
μintercept,2	−10.662	μchild,3,2	0.506
μintercept,3	−18.715	μchild,3,3	1.452
μintercept,4	−2.201	μchild,3,4	−4.861
μintercept,5	−5.840	μchild,3,5	−5.308
μsex,1	−0.361	μchild,4,1	0.514
μsex,2	−7.384	μchild,4,2	1.110
μsex,3	14.153	μchild,4,3	1.732
μsex,4	−0.447	μchild,4,4	2.155
μsex,5	−0.511	μchild,4,5	2.792
μsmoking,1	34.863	μchild,5,1	0.523
μsmoking,2	15.718	μchild,5,2	1.152
μsmoking,3	16.520	μchild,5,3	1.755
μsmoking,4	15.179	μchild,5,4	2.276
μsmoking,5	13.514	μchild,5,5	−0.584
μchild,1,1	−0.681	μage,1	0.270
μchild,1,2	−0.072	μage,2	0.247
μchild,1,3	0.749	μage,3	0.270
μchild,1,4	1.279	μage,4	0.218
μchild,1,5	1.549	μage,5	0.310
μchild,2,1	2.293	μBMI,1	−0.002
μchild,2,2	3.813	μBMI,2	0.662
μchild,2,3	1.063	μBMI,3	0.491
μchild,2,4	2.513	μBMI,4	−0.012
μchild,2,5	−0.813	μBMI,5	−0.009

According to the fitted probability density function in (3), the first four moments of hospital charge are

$$E\left(\text{hospital charge}\right)=\sum _{i=1}^4\widehat{w_i}(\widehat{{\mu }_i}+\gamma \widehat{{\sigma }_i})+(1-\widehat{w_1}-\cdots -\widehat{w_4})(\widehat{{\mu }_5}+\gamma \widehat{{\sigma }_5}),$$ (5)

$$E\left({\text{hospital charge}}^2\right)=\sum _{i=1}^4\widehat{w_i}[{\widehat{{\mu }_i}}^2+2\gamma \widehat{{\mu }_i}\widehat{{\sigma }_i}+(\frac{{\pi }^2}{6}+{\gamma }^2){\widehat{{\sigma }_i}}^2]+(1-\widehat{w_1}-\cdots -\widehat{w_4})[{\widehat{{\mu }_5}}^2+2\gamma \widehat{{\mu }_5}\widehat{{\sigma }_5}+(\frac{{\pi }^2}{6}+{\gamma }^2){\widehat{{\sigma }_5}}^2],$$ (6)

$$E\left({\text{hospital charge}}^3\right)=\sum _{i=1}^4\widehat{w_i}[{\widehat{{\mu }_i}}^3+3\gamma {\widehat{{\mu }_i}}^2\widehat{{\sigma }_i}+3(\frac{{\pi }^2}{6}+{\gamma }^2)\widehat{{\mu }_i}{\widehat{{\sigma }_i}}^2+(2\eta (3)+\frac{{\pi }^2\gamma }{2}+{\gamma }^3){\widehat{{\sigma }_i}}^3]+(1-\widehat{w_1}-\cdots -\widehat{w_4})[{\widehat{{\mu }_5}}^3+3\gamma {\widehat{{\mu }_5}}^2\widehat{{\sigma }_5}+3(\frac{{\pi }^2}{6}+{\gamma }^2)\widehat{{\mu }_5}{\widehat{{\sigma }_5}}^2+(2\eta (3)+\frac{{\pi }^2\gamma }{2}+{\gamma }^3){\widehat{{\sigma }_5}}^3]$$ (7)

and

$$E\left({\text{hospital charge}}^4\right)=\sum _{i=1}^4\widehat{w_i}[{\widehat{{\mu }_i}}^4+4\gamma {\widehat{{\mu }_i}}^3\widehat{{\sigma }_i}+6(\frac{{\pi }^2}{6}+{\gamma }^2){\widehat{{\mu }_i}}^2{\widehat{{\sigma }_i}}^2]+\sum _{i=1}^4\widehat{w_i}[4(2\eta (3)+\frac{{\pi }^2\gamma }{2}+{\gamma }^3)\widehat{{\mu }_i}{\widehat{{\sigma }_i}}^3+(\frac{3{\pi }^4}{20}+8\eta (3)\gamma +{\pi }^2{\gamma }^2+{\gamma }^4){\widehat{{\sigma }_i}}^4]+(1-\widehat{w_1}-\cdots -\widehat{w_4})[{\widehat{{\mu }_5}}^4+4\gamma {\widehat{{\mu }_5}}^3\widehat{{\sigma }_5}+6(\frac{{\pi }^2}{6}+{\gamma }^2){\widehat{{\mu }_5}}^2{\widehat{{\sigma }_5}}^2]+(1-\widehat{w_1}-\cdots -\widehat{w_4})[4(2\eta (3)+\frac{{\pi }^2\gamma }{2}+{\gamma }^3)\widehat{{\mu }_5}{\widehat{{\sigma }_5}}^3+(\frac{3{\pi }^4}{20}+8\eta (3)\gamma +{\pi }^2{\gamma }^2+{\gamma }^4){\widehat{{\sigma }_5}}^4],$$ (8)

where γ denotes Euler's constant and $\eta (\cdot )$ denotes Riemann's zeta function. (5)-(8) together can be used to calculate the variance, skewness and kurtosis of hospital charge. Fig. 7 plots the expected hospital charge versus age, body mass index, sex, number of children and smoking status. Fig. 8 plots the variance of hospital charge versus age, body mass index, sex, number of children and smoking status. Fig. 9 plots the skewness of hospital charge versus age, body mass index, sex, number of children and smoking status. Fig. 10 plots the kurtosis of hospital charge versus age, body mass index, sex, number of children and smoking status.

Figure 7.

Expected hospital charge versus age for all combinations of the levels of sex, number of children, smoking status and when body mass index takes its average value (top left); expected hospital charge versus body mass index for all combinations of the levels of sex, number of children, smoking status and when age takes its average value (top right); expected hospital charge versus sex for all combinations of the levels of number of children, smoking status and when age and body mass index take their average values (middle left); expected hospital charge versus number of children for all combinations of the levels of sex, smoking status and when age and body mass index take their average values (middle right); expected hospital charge versus smoking status for all combinations of the levels of sex, number of children and when age and body mass index take their average values (bottom).

Figure 8.

Variance of hospital charge versus age for all combinations of the levels of sex, number of children, smoking status and when body mass index takes its average value (top left); variance of hospital charge versus body mass index for all combinations of the levels of sex, number of children, smoking status and when age takes its average value (top right); variance of hospital charge versus sex for all combinations of the levels of number of children, smoking status and when age and body mass index take their average values (middle left); variance of hospital charge versus number of children for all combinations of the levels of sex, smoking status and when age and body mass index take their average values (middle right); variance of hospital charge versus smoking status for all combinations of the levels of sex, number of children and when age and body mass index take their average values (bottom).

Figure 9.

Skewness of hospital charge versus age for all combinations of the levels of sex, number of children, smoking status and when body mass index takes its average value (top left); skewness of hospital charge versus body mass index for all combinations of the levels of sex, number of children, smoking status and when age takes its average value (top right); skewness of hospital charge versus sex for all combinations of the levels of number of children, smoking status and when age and body mass index take their average values (middle left); skewness of hospital charge versus number of children for all combinations of the levels of sex, smoking status and when age and body mass index take their average values (middle right); skewness of hospital charge versus smoking status for all combinations of the levels of sex, number of children and when age and body mass index take their average values (bottom).

Figure 10.

Kurtosis of hospital charge versus age for all combinations of the levels of sex, number of children, smoking status and when body mass index takes its average value (top left); kurtosis of hospital charge versus body mass index for all combinations of the levels of sex, number of children, smoking status and when age takes its average value (top right); kurtosis of hospital charge versus sex for all combinations of the levels of number of children, smoking status and when age and body mass index take their average values (middle left); kurtosis of hospital charge versus number of children for all combinations of the levels of sex, smoking status and when age and body mass index take their average values (middle right); kurtosis of hospital charge versus smoking status for all combinations of the levels of sex, number of children and when age and body mass index take their average values (bottom).

Fig. 7 shows that the expected hospital charge increases with increasing age. The expected hospital charge also increases with increasing body mass index. The expected hospital charge is slightly higher for females. The expected hospital charge increases with the number of children except when the number of children is five. The expected hospital charge is higher for smokers. There are two clusters of lines in the first four plots in Fig. 7. The lower cluster corresponds to non-smokers while the upper cluster corresponds to smokers.

Fig. 8 shows that the variance of hospital charge either increases with increasing age or initially decreases before increasing with increasing age. The minimum variance appears to occur around the age of 40. The variance of hospital charge also increases with increasing body mass index. The variance of hospital charge does not appear to change much with respect to sex or the number of children. The variance of hospital charge is higher for smokers. There are two clusters of lines in the second, third and fourth plots in Fig. 8. The lower cluster corresponds to non-smokers while the upper cluster corresponds to smokers.

Fig. 9 shows that the skewness of hospital charge either decreases with increasing body mass index or initially increases before decreasing with increasing body mass index. The maximum skewness appears to occur around the body mass index of 20. The skewness of hospital charge is smaller for smokers. The skewness of hospital charge does not appear to change much with respect to age, sex or the number of children. There are two clusters of lines in the first four plots in Fig. 9. The lower cluster corresponds to smokers while the upper cluster corresponds to non-smokers.

Fig. 10 shows that the kurtosis of hospital charge either increases with increasing body mass index or initially increases before decreasing with increasing body mass index. The maximum kurtosis appears to occur around the body mass index of 30. The kurtosis of hospital charge is smaller for smokers. The kurtosis of hospital charge does not appear to change much with respect to age, sex or the number of children. There are two clusters of lines in the first four plots in Fig. 10. The lower cluster corresponds to smokers while the upper cluster corresponds to non-smokers.

The probability and quantile plots corresponding to (4) are shown in Fig. 11, compare with Fig. 6. These plots show that the fitted regression adequately describes the data.

Figure 11.

Probability plot (left) and quantile plot (right) corresponding to (4).

3.3. Risk estimation

We now give risk estimates associated with the hospital charge. The most popular risk measure is the value at risk [34]. The value at risk with probability p corresponding to (3) is the root of

$$\sum _{i=1}^4\widehat{w_i}\exp [-\exp (-\frac{{\widehat{\mathrm{VaR}}}_p-\widehat{{\mu }_i}}{\widehat{{\sigma }_i}})]+(1-\widehat{w_1}-\cdots -\widehat{w_4})\exp [-\exp (-\frac{{\widehat{\mathrm{VaR}}}_p-\widehat{{\mu }_5}}{\widehat{{\sigma }_5}})]=p.$$ (9)

(9) was solved using the uniroot function in R. The estimates of value at risk versus age, body mass index, sex, number of children and smoking status are shown in Table 3, Table 4, Table 5, Table 6, Table 7, Table 8, Table 9, Table 10, Table 11, Table 12, Table 13, Table 14, Table 15, Table 16, Table 17, Table 18, Table 19, Table 20, Table 21, Table 22, Table 23, Table 24, Table 25, Table 26, Table 27 of Appendix A.

The numbers in Table 3, Table 4, Table 5, Table 6, Table 7, Table 8, Table 9, Table 10, Table 11, Table 12, Table 13, Table 14, Table 15, Table 16, Table 17, Table 18, Table 19, Table 20, Table 21, Table 22, Table 23, Table 24, Table 25, Table 26, Table 27 can be useful for the public and the government of Zimbabwe. For example, a 90 year old female not smoking and having no children and an average body mass index will have a hospital charge less than Z$49851 with probability 0.999. A 90 year old male smoking and having five children and an average body mass index will have a hospital charge less than Z$65843 with probability 0.999. A female not smoking with a body mass index of 90 and having no children and an average age will have a hospital charge less than Z$54553 with probability 0.999. A male smoking with a body mass index of 90 and having five children and an average age will have a hospital charge less than Z$69656 with probability 0.999. A female not smoking and having no children, an average age and an average body mass index will have a hospital charge less than Z$35804 with probability 0.999. A male smoking and having five children, an average age and an average body mass index will have a hospital charge less than Z$50907 with probability 0.999.

4. Conclusions

We have provided the first statistical modeling of hospital charge for any country in Africa. The modeling was based on the fitting of over fifty mixture distributions each with the number of components ranging from two to ten. A five component mixture of the reversed Gumbel distribution turned out to give the best fit. The effect of factors (sex, smoking status, number of children and region) and covariates (age and body mass index) was examined by parameterizing this distribution appropriately. Age, body mass index, sex, smoking status and number of children turned out to be significant. Estimates of the value at risk of the hospital charge were given for all combinations of sex, smoking status and number of children and for certain ranges of age and body mass index. The results of this paper (especially estimates of value at risk) can be useful for the public and the government of Zimbabwe.

A future work is to perform similar analysis for other poorest countries in the world, including Burundi, Somalia, Mozambique, Madagascar, Sierra Leone, Afghanistan, Eritrea, Central African Republic, Liberia and Niger. In Section 3, we fitted mixtures of distributions belonging to the same parametric family. Another future work is to consider fitting mixtures of distributions belonging to different parametric families. Also Section 3 does not take into account the heterogeneity over regions. There are more advanced mixture modeling under the existence of cluster information (see, for example, [35], [36]). These advanced mixture methods would extend the scope of the paper.

Funding statement

This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.

CRediT authorship contribution statement

Nyasha Mangava, Saralees Nadarajah: Conceived and designed the experiments; Performed the experiments; Analyzed and interpreted the data; Contributed reagents, materials, analysis tools or data; Wrote the paper.

Declaration of Competing Interest

The authors declare no competing interests.

Appendix A. Table 3, Table 4, Table 5, Table 6, Table 7, Table 8, Table 9, Table 10, Table 11, Table 12, Table 13, Table 14, Table 15, Table 16, Table 17, Table 18, Table 19, Table 20, Table 21, Table 22, Table 23, Table 24, Table 25, Table 26, Table 27

Table 3.

Estimates of (9) versus p = 0.9,0.95,0.99,0.999, smoking status, and age when the patient is a female, has no children and an average body mass index.

	Not smoker				Smoker
Age	${\widehat{\mathrm{VaR}}}_{0.9}$	${\widehat{\mathrm{VaR}}}_{0.95}$	${\widehat{\mathrm{VaR}}}_{0.99}$	${\widehat{\mathrm{VaR}}}_{0.999}$	${\widehat{\mathrm{VaR}}}_{0.9}$	${\widehat{\mathrm{VaR}}}_{0.95}$	${\widehat{\mathrm{VaR}}}_{0.99}$	${\widehat{\mathrm{VaR}}}_{0.999}$
18	12070	15619	21810	29939	34826	35943	41982	50111
20	12624	16172	22363	30492	35301	36471	42535	50664
22	13177	16725	22917	31045	35777	37002	43088	51217
24	13730	17278	23470	31598	36253	37536	43642	51770
26	14283	17831	24023	32151	36729	38074	44195	52323
28	14836	18385	24576	32704	37206	38613	44748	52876
30	15389	18938	25129	33257	37683	39155	45301	53429
32	15942	19491	25682	33811	38161	39699	45854	53982
34	16496	20044	26235	34364	38639	40245	46407	54536
36	17049	20597	26789	34917	39117	40792	46960	55089
38	17602	21150	27342	35470	39596	41340	47513	55642
40	18155	21703	27895	36023	40075	41890	48067	56195
42	18708	22256	28448	36576	40556	42440	48620	56748
44	19261	22810	29001	37129	41036	42990	49173	57301
46	19814	23363	29554	37682	41517	43542	49726	57854
48	20367	23916	30107	38236	41999	44093	50279	58408
50	20921	24469	30660	38789	42482	44645	50832	58961
52	21474	25022	31214	39342	42965	45197	51385	59514
54	22027	25575	31767	39895	43449	45750	51939	60067
56	22580	26128	32320	40448	43934	46302	52492	60620
58	23133	26682	32873	41001	44420	46855	53045	61173
60	23686	27235	33426	41554	44907	47408	53598	61726
62	24239	27788	33979	42108	45395	47961	54151	62279
64	24792	28341	34532	42661	45884	48514	54704	62833
66	25346	28894	35085	43214	46375	49067	55257	63386
68	25899	29447	35639	43767	46867	49620	55810	63939
70	26452	30000	36192	44320	47360	50173	56364	64492
72	27005	30553	36745	44873	47854	50726	56917	65045
74	27558	31107	37298	45426	48351	51279	57470	65598
76	28111	31660	37851	45980	48849	51832	58023	66151
78	28664	32213	38404	46533	49349	52385	58576	66704
80	29218	32766	38957	47086	49852	52938	59129	67258
82	29771	33319	39511	47639	50356	53491	59682	67811
84	30324	33872	40064	48192	50864	54044	60236	68364
86	30877	34425	40617	48745	51373	54597	60789	68917
88	31430	34978	41170	49298	51886	55150	61342	69470
90	31983	35532	41723	49851	52402	55704	61895	70023

Table 4.

Estimates of (9) versus p = 0.9,0.95,0.99,0.999, smoking status, and age when the patient is a female, has one child and an average body mass index.

	Not smoker				Smoker
Age	${\widehat{\mathrm{VaR}}}_{0.9}$	${\widehat{\mathrm{VaR}}}_{0.95}$	${\widehat{\mathrm{VaR}}}_{0.99}$	${\widehat{\mathrm{VaR}}}_{0.999}$	${\widehat{\mathrm{VaR}}}_{0.9}$	${\widehat{\mathrm{VaR}}}_{0.95}$	${\widehat{\mathrm{VaR}}}_{0.99}$	${\widehat{\mathrm{VaR}}}_{0.999}$
18	14336	17884	24076	32204	35630	38058	44248	52376
20	14889	18438	24629	32757	36117	38611	44801	52929
22	15442	18991	25182	33310	36605	39164	45354	53482
24	15995	19544	25735	33864	37094	39716	45907	54035
26	16548	20097	26288	34417	37584	40269	46460	54589
28	17102	20650	26841	34970	38076	40822	47013	55142
30	17655	21203	27395	35523	38569	41375	47566	55695
32	18208	21756	27948	36076	39063	41928	48120	56248
34	18761	22309	28501	36629	39560	42482	48673	56801
36	19314	22863	29054	37182	40058	43035	49226	57354
38	19867	23416	29607	37735	40558	43588	49779	57907
40	20420	23969	30160	38289	41060	44141	50332	58460
42	20974	24522	30713	38842	41564	44694	50885	59014
44	21527	25075	31267	39395	42071	45247	51438	59567
46	22080	25628	31820	39948	42581	45800	51992	60120
48	22633	26181	32373	40501	43093	46353	52545	60673
50	23186	26734	32926	41054	43608	46906	53098	61226
52	23739	27288	33479	41607	44127	47460	53651	61779
54	24292	27841	34032	42161	44648	48013	54204	62332
56	24845	28394	34585	42714	45173	48566	54757	62886
58	25399	28947	35138	43267	45701	49119	55310	63439
60	25952	29500	35692	43820	46232	49672	55863	63992
62	26505	30053	36245	44373	46766	50225	56417	64545
64	27058	30606	36798	44926	47303	50778	56970	65098
66	27611	31160	37351	45479	47842	51331	57523	65651
68	28164	31713	37904	46032	48384	51885	58076	66204
70	28717	32266	38457	46586	48927	52438	58629	66757
72	29271	32819	39010	47139	49473	52991	59182	67311
74	29824	33372	39564	47692	50020	53544	59735	67864
76	30377	33925	40117	48245	50568	54097	60288	68417
78	30930	34478	40670	48798	51117	54650	60842	68970
80	31483	35031	41223	49351	51667	55203	61395	69523
82	32036	35585	41776	49904	52217	55756	61948	70076
84	32589	36138	42329	50458	52768	56310	62501	70629
86	33142	36691	42882	51011	53320	56863	63054	71182
88	33696	37244	43435	51564	53872	57416	63607	71736
90	34249	37797	43989	52117	54424	57969	64160	72289

Table 5.

Estimates of (9) versus p = 0.9,0.95,0.99,0.999, smoking status, and age when the patient is a female, has two children and an average body mass index.

	Not smoker				Smoker
Age	${\widehat{\mathrm{VaR}}}_{0.9}$	${\widehat{\mathrm{VaR}}}_{0.95}$	${\widehat{\mathrm{VaR}}}_{0.99}$	${\widehat{\mathrm{VaR}}}_{0.999}$	${\widehat{\mathrm{VaR}}}_{0.9}$	${\widehat{\mathrm{VaR}}}_{0.95}$	${\widehat{\mathrm{VaR}}}_{0.99}$	${\widehat{\mathrm{VaR}}}_{0.999}$
18	14600	18149	24340	32468	36103	38324	44512	52640
20	15153	18702	24893	33022	36587	38876	45065	53193
22	15706	19255	25446	33575	37072	39429	45618	53747
24	16260	19808	25999	34128	37558	39982	46171	54300
26	16813	20361	26553	34681	38044	40534	46724	54853
28	17366	20914	27106	35234	38532	41087	47278	55406
30	17919	21467	27659	35787	39021	41640	47831	55959
32	18472	22021	28212	36340	39511	42193	48384	56512
34	19025	22574	28765	36893	40003	42746	48937	57065
36	19578	23127	29318	37447	40496	43299	49490	57618
38	20132	23680	29871	38000	40990	43852	50043	58172
40	20685	24233	30425	38553	41486	44405	50596	58725
42	21238	24786	30978	39106	41984	44958	51150	59278
44	21791	25339	31531	39659	42484	45511	51703	59831
46	22344	25892	32084	40212	42986	46064	52256	60384
48	22897	26446	32637	40765	43491	46618	52809	60937
50	23450	26999	33190	41319	43997	47171	53362	61490
52	24003	27552	33743	41872	44507	47724	53915	62043
54	24557	28105	34296	42425	45019	48277	54468	62597
56	25110	28658	34850	42978	45534	48830	55021	63150
58	25663	29211	35403	43531	46052	49383	55575	63703
60	26216	29764	35956	44084	46574	49936	56128	64256
62	26769	30318	36509	44637	47098	50489	56681	64809
64	27322	30871	37062	45190	47626	51043	57234	65362
66	27875	31424	37615	45744	48157	51596	57787	65915
68	28429	31977	38168	46297	48691	52149	58340	66469
70	28982	32530	38721	46850	49227	52702	58893	67022
72	29535	33083	39275	47403	49766	53255	59446	67575
74	30088	33636	39828	47956	50308	53808	60000	68128
76	30641	34189	40381	48509	50852	54361	60553	68681
78	31194	34743	40934	49062	51397	54914	61106	69234
80	31747	35296	41487	49616	51944	55468	61659	69787
82	32300	35849	42040	50169	52492	56021	62212	70340
84	32854	36402	42593	50722	53041	56574	62765	70894
86	33407	36955	43147	51275	53591	57127	63318	71447
88	33960	37508	43700	51828	54141	57680	63872	72000
90	34513	38061	44253	52381	54692	58233	64425	72553

Table 6.

Estimates of (9) versus p = 0.9,0.95,0.99,0.999, smoking status, and age when the patient is a female, has three children and an average body mass index.

	Not smoker				Smoker
Age	${\widehat{\mathrm{VaR}}}_{0.9}$	${\widehat{\mathrm{VaR}}}_{0.95}$	${\widehat{\mathrm{VaR}}}_{0.99}$	${\widehat{\mathrm{VaR}}}_{0.999}$	${\widehat{\mathrm{VaR}}}_{0.9}$	${\widehat{\mathrm{VaR}}}_{0.95}$	${\widehat{\mathrm{VaR}}}_{0.99}$	${\widehat{\mathrm{VaR}}}_{0.999}$
18	12960	16508	22700	30828	36378	37166	42872	51000
20	13513	17062	23253	31381	36851	37669	43425	51553
22	14066	17615	23806	31934	37324	38174	43978	52106
24	14619	18168	24359	32488	37798	38682	44531	52659
26	15173	18721	24912	33041	38272	39192	45084	53213
28	15726	19274	25466	33594	38745	39706	45637	53766
30	16279	19827	26019	34147	39220	40223	46191	54319
32	16832	20380	26572	34700	39694	40743	46744	54872
34	17385	20933	27125	35253	40169	41267	47297	55425
36	17938	21487	27678	35806	40644	41794	47850	55978
38	18491	22040	28231	36360	41120	42324	48403	56531
40	19044	22593	28784	36913	41595	42857	48956	57084
42	19598	23146	29337	37466	42072	43394	49509	57638
44	20151	23699	29891	38019	42548	43932	50062	58191
46	20704	24252	30444	38572	43025	44474	50616	58744
48	21257	24805	30997	39125	43503	45017	51169	59297
50	21810	25359	31550	39678	43980	45562	51722	59850
52	22363	25912	32103	40231	44459	46109	52275	60403
54	22916	26465	32656	40785	44937	46657	52828	60956
56	23470	27018	33209	41338	45417	47205	53381	61510
58	24023	27571	33762	41891	45896	47755	53934	62063
60	24576	28124	34316	42444	46377	48306	54487	62616
62	25129	28677	34869	42997	46858	48857	55041	63169
64	25682	29230	35422	43550	47340	49408	55594	63722
66	26235	29784	35975	44103	47822	49960	56147	64275
68	26788	30337	36528	44656	48305	50512	56700	64828
70	27341	30890	37081	45210	48789	51065	57253	65381
72	27895	31443	37634	45763	49274	51617	57806	65935
74	28448	31996	38188	46316	49759	52170	58359	66488
76	29001	32549	38741	46869	50246	52722	58913	67041
78	29554	33102	39294	47422	50733	53275	59466	67594
80	30107	33656	39847	47975	51222	53828	60019	68147
82	30660	34209	40400	48528	51712	54381	60572	68700
84	31213	34762	40953	49082	52203	54934	61125	69253
86	31766	35315	41506	49635	52696	55487	61678	69807
88	32320	35868	42059	50188	53190	56040	62231	70360
90	32873	36421	42613	50741	53686	56593	62784	70913

Table 7.

Estimates of (9) versus p = 0.9,0.95,0.99,0.999, smoking status, and age when the patient is a female, has four children and an average body mass index.

	Not smoker				Smoker
Age	${\widehat{\mathrm{VaR}}}_{0.9}$	${\widehat{\mathrm{VaR}}}_{0.95}$	${\widehat{\mathrm{VaR}}}_{0.99}$	${\widehat{\mathrm{VaR}}}_{0.999}$	${\widehat{\mathrm{VaR}}}_{0.9}$	${\widehat{\mathrm{VaR}}}_{0.95}$	${\widehat{\mathrm{VaR}}}_{0.99}$	${\widehat{\mathrm{VaR}}}_{0.999}$
18	13293	16841	23032	31161	36967	37674	43204	51333
20	13846	17394	23586	31714	37439	38170	43757	51886
22	14399	17947	24139	32267	37912	38667	44311	52439
24	14952	18500	24692	32820	38384	39167	44864	52992
26	15505	19054	25245	33373	38857	39669	45417	53545
28	16058	19607	25798	33926	39330	40174	45970	54098
30	16611	20160	26351	34480	39804	40681	46523	54651
32	17165	20713	26904	35033	40277	41192	47076	55205
34	17718	21266	27458	35586	40751	41705	47629	55758
36	18271	21819	28011	36139	41226	42221	48182	56311
38	18824	22372	28564	36692	41700	42741	48736	56864
40	19377	22926	29117	37245	42175	43264	49289	57417
42	19930	23479	29670	37798	42650	43790	49842	57970
44	20483	24032	30223	38352	43125	44320	50395	58523
46	21036	24585	30776	38905	43601	44853	50948	59076
48	21590	25138	31329	39458	44077	45388	51501	59630
50	22143	25691	31883	40011	44554	45927	52054	60183
52	22696	26244	32436	40564	45031	46468	52608	60736
54	23249	26797	32989	41117	45508	47011	53161	61289
56	23802	27351	33542	41670	45986	47555	53714	61842
58	24355	27904	34095	42223	46464	48102	54267	62395
60	24908	28457	34648	42777	46942	48649	54820	62948
62	25462	29010	35201	43330	47422	49198	55373	63502
64	26015	29563	35754	43883	47901	49748	55926	64055
66	26568	30116	36308	44436	48382	50298	56479	64608
68	27121	30669	36861	44989	48863	50849	57033	65161
70	27674	31222	37414	45542	49344	51401	57586	65714
72	28227	31776	37967	46095	49826	51952	58139	66267
74	28780	32329	38520	46649	50309	52504	58692	66820
76	29333	32882	39073	47202	50793	53057	59245	67373
78	29887	33435	39626	47755	51278	53609	59798	67927
80	30440	33988	40180	48308	51763	54162	60351	68480
82	30993	34541	40733	48861	52249	54715	60905	69033
84	31546	35094	41286	49414	52737	55267	61458	69586
86	32099	35648	41839	49967	53225	55820	62011	70139
88	32652	36201	42392	50520	53715	56373	62564	70692
90	33205	36754	42945	51074	54206	56926	63117	71245

Table 8.

Estimates of (9) versus p = 0.9,0.95,0.99,0.999, smoking status, and age when the patient is a female, has five children and an average body mass index.

	Not smoker				Smoker
Age	${\widehat{\mathrm{VaR}}}_{0.9}$	${\widehat{\mathrm{VaR}}}_{0.95}$	${\widehat{\mathrm{VaR}}}_{0.99}$	${\widehat{\mathrm{VaR}}}_{0.999}$	${\widehat{\mathrm{VaR}}}_{0.9}$	${\widehat{\mathrm{VaR}}}_{0.95}$	${\widehat{\mathrm{VaR}}}_{0.99}$	${\widehat{\mathrm{VaR}}}_{0.999}$
18	7897	11439	17630	25758	37224	37541	38614	45930
20	8449	11992	18183	26312	37690	38009	39105	46483
22	9001	12545	18736	26865	38156	38476	39597	47037
24	9553	13098	19290	27418	38622	38943	40091	47590
26	10106	13651	19843	27971	39089	39411	40587	48143
28	10658	14204	20396	28524	39555	39879	41085	48696
30	11211	14758	20949	29077	40021	40346	41586	49249
32	11764	15311	21502	29630	40488	40814	42089	49802
34	12317	15864	22055	30184	40954	41282	42594	50355
36	12870	16417	22608	30737	41420	41750	43103	50909
38	13422	16970	23161	31290	41887	42218	43615	51462
40	13975	17523	23715	31843	42353	42686	44130	52015
42	14528	18076	24268	32396	42820	43154	44648	52568
44	15081	18629	24821	32949	43287	43623	45169	53121
46	15635	19183	25374	33502	43753	44091	45694	53674
48	16188	19736	25927	34055	44220	44560	46223	54227
50	16741	20289	26480	34609	44687	45028	46754	54780
52	17294	20842	27033	35162	45154	45497	47289	55334
54	17847	21395	27587	35715	45620	45966	47827	55887
56	18400	21948	28140	36268	46087	46435	48367	56440
58	18953	22501	28693	36821	46554	46904	48909	56993
60	19506	23055	29246	37374	47021	47373	49453	57546
62	20059	23608	29799	37927	47488	47843	49999	58099
64	20612	24161	30352	38481	47956	48312	50546	58652
66	21166	24714	30905	39034	48423	48782	51095	59205
68	21719	25267	31458	39587	48890	49252	51644	59759
70	22272	25820	32012	40140	49357	49722	52194	60312
72	22825	26373	32565	40693	49825	50192	52745	60865
74	23378	26926	33118	41246	50292	50663	53296	61418
76	23931	27480	33671	41799	50760	51133	53848	61971
78	24484	28033	34224	42352	51227	51604	54400	62524
80	25037	28586	34777	42906	51695	52075	54952	63077
82	25591	29139	35330	43459	52163	52546	55505	63631
84	26144	29692	35884	44012	52631	53017	56057	64184
86	26697	30245	36437	44565	53098	53489	56610	64737
88	27250	30798	36990	45118	53566	53961	57163	65290
90	27803	31351	37543	45671	54034	54433	57716	65843

Table 9.

Estimates of (9) versus p = 0.9,0.95,0.99,0.999, smoking status, and age when the patient is a male, has no children and an average body mass index.

	Not smoker				Smoker
Age	${\widehat{\mathrm{VaR}}}_{0.9}$	${\widehat{\mathrm{VaR}}}_{0.95}$	${\widehat{\mathrm{VaR}}}_{0.99}$	${\widehat{\mathrm{VaR}}}_{0.999}$	${\widehat{\mathrm{VaR}}}_{0.9}$	${\widehat{\mathrm{VaR}}}_{0.95}$	${\widehat{\mathrm{VaR}}}_{0.99}$	${\widehat{\mathrm{VaR}}}_{0.999}$
18	11182	14730	20922	29050	34341	35234	41093	49222
20	11735	15283	21475	29603	34815	35745	41647	49775
22	12288	15836	22028	30156	35289	36260	42200	50328
24	12841	16390	22581	30709	35764	36778	42753	50881
26	13394	16943	23134	31263	36238	37299	43306	51434
28	13947	17496	23687	31816	36713	37823	43859	51987
30	14501	18049	24240	32369	37188	38351	44412	52541
32	15054	18602	24794	32922	37664	38882	44965	53094
34	15607	19155	25347	33475	38140	39416	45519	53647
36	16160	19708	25900	34028	38616	39952	46072	54200
38	16713	20262	26453	34581	39093	40492	46625	54753
40	17266	20815	27006	35134	39570	41034	47178	55306
42	17819	21368	27559	35688	40047	41577	47731	55859
44	18372	21921	28112	36241	40525	42123	48284	56413
46	18926	22474	28665	36794	41003	42670	48837	56966
48	19479	23027	29219	37347	41482	43218	49390	57519
50	20032	23580	29772	37900	41962	43767	49944	58072
52	20585	24133	30325	38453	42442	44317	50497	58625
54	21138	24687	30878	39006	42922	44868	51050	59178
56	21691	25240	31431	39559	43403	45419	51603	59731
58	22244	25793	31984	40113	43885	45970	52156	60284
60	22798	26346	32537	40666	44368	46522	52709	60838
62	23351	26899	33090	41219	44851	47074	53262	61391
64	23904	27452	33644	41772	45335	47627	53816	61944
66	24457	28005	34197	42325	45820	48179	54369	62497
68	25010	28558	34750	42878	46306	48732	54922	63050
70	25563	29112	35303	43431	46793	49285	55475	63603
72	26116	29665	35856	43985	47280	49838	56028	64156
74	26669	30218	36409	44538	47769	50391	56581	64710
76	27223	30771	36962	45091	48260	50943	57134	65263
78	27776	31324	37516	45644	48751	51496	57687	65816
80	28329	31877	38069	46197	49244	52050	58241	66369
82	28882	32430	38622	46750	49739	52603	58794	66922
84	29435	32984	39175	47303	50235	53156	59347	67475
86	29988	33537	39728	47856	50733	53709	59900	68028
88	30541	34090	40281	48410	51233	54262	60453	68581
90	31095	34643	40834	48963	51735	54815	61006	69135

Table 10.

Estimates of (9) versus p = 0.9,0.95,0.99,0.999, smoking status, and age when the patient is a male, has one child and an average body mass index.

	Not smoker				Smoker
Age	${\widehat{\mathrm{VaR}}}_{0.9}$	${\widehat{\mathrm{VaR}}}_{0.95}$	${\widehat{\mathrm{VaR}}}_{0.99}$	${\widehat{\mathrm{VaR}}}_{0.999}$	${\widehat{\mathrm{VaR}}}_{0.9}$	${\widehat{\mathrm{VaR}}}_{0.95}$	${\widehat{\mathrm{VaR}}}_{0.99}$	${\widehat{\mathrm{VaR}}}_{0.999}$
18	13447	16996	23187	31315	35095	37173	43359	51487
20	14000	17549	23740	31869	35578	37725	43912	52040
22	14553	18102	24293	32422	36061	38277	44465	52594
24	15107	18655	24847	32975	36545	38830	45018	53147
26	15660	19208	25400	33528	37030	39382	45571	53700
28	16213	19761	25953	34081	37516	39935	46125	54253
30	16766	20314	26506	34634	38002	40488	46678	54806
32	17319	20868	27059	35187	38490	41040	47231	55359
34	17872	21421	27612	35741	38979	41593	47784	55912
36	18425	21974	28165	36294	39469	42146	48337	56466
38	18979	22527	28718	36847	39960	42699	48890	57019
40	19532	23080	29272	37400	40453	43252	49443	57572
42	20085	23633	29825	37953	40948	43805	49997	58125
44	20638	24186	30378	38506	41444	44359	50550	58678
46	21191	24740	30931	39059	41941	44912	51103	59231
48	21744	25293	31484	39612	42441	45465	51656	59784
50	22297	25846	32037	40166	42943	46018	52209	60337
52	22850	26399	32590	40719	43447	46571	52762	60891
54	23404	26952	33143	41272	43954	47124	53315	61444
56	23957	27505	33697	41825	44463	47677	53868	61997
58	24510	28058	34250	42378	44975	48230	54422	62550
60	25063	28611	34803	42931	45490	48783	54975	63103
62	25616	29165	35356	43484	46008	49336	55528	63656
64	26169	29718	35909	44037	46529	49890	56081	64209
66	26722	30271	36462	44591	47053	50443	56634	64762
68	27276	30824	37015	45144	47581	50996	57187	65316
70	27829	31377	37569	45697	48112	51549	57740	65869
72	28382	31930	38122	46250	48645	52102	58293	66422
74	28935	32483	38675	46803	49182	52655	58847	66975
76	29488	33037	39228	47356	49721	53208	59400	67528
78	30041	33590	39781	47909	50262	53762	59953	68081
80	30594	34143	40334	48463	50805	54315	60506	68634
82	31147	34696	40887	49016	51351	54868	61059	69188
84	31701	35249	41440	49569	51897	55421	61612	69741
86	32254	35802	41994	50122	52445	55974	62165	70294
88	32807	36355	42547	50675	52994	56527	62719	70847
90	33360	36908	43100	51228	53544	57080	63272	71400

Table 11.

Estimates of (9) versus p = 0.9,0.95,0.99,0.999, smoking status, and age when the patient is a male, has two children and an average body mass index.

	Not smoker				Smoker
Age	${\widehat{\mathrm{VaR}}}_{0.9}$	${\widehat{\mathrm{VaR}}}_{0.95}$	${\widehat{\mathrm{VaR}}}_{0.99}$	${\widehat{\mathrm{VaR}}}_{0.999}$	${\widehat{\mathrm{VaR}}}_{0.9}$	${\widehat{\mathrm{VaR}}}_{0.95}$	${\widehat{\mathrm{VaR}}}_{0.99}$	${\widehat{\mathrm{VaR}}}_{0.999}$
18	13712	17260	23451	31580	35580	37444	43623	51752
20	14265	17813	24004	32133	36060	37994	44176	52305
22	14818	18366	24558	32686	36541	38546	44729	52858
24	15371	18919	25111	33239	37023	39097	45283	53411
26	15924	19472	25664	33792	37506	39649	45836	53964
28	16477	20026	26217	34345	37989	40201	46389	54517
30	17030	20579	26770	34899	38473	40753	46942	55070
32	17583	21132	27323	35452	38957	41306	47495	55623
34	18137	21685	27876	36005	39443	41859	48048	56177
36	18690	22238	28430	36558	39930	42411	48601	56730
38	19243	22791	28983	37111	40417	42964	49155	57283
40	19796	23344	29536	37664	40906	43517	49708	57836
42	20349	23898	30089	38217	41396	44070	50261	58389
44	20902	24451	30642	38770	41888	44623	50814	58942
46	21455	25004	31195	39324	42380	45176	51367	59495
48	22008	25557	31748	39877	42875	45729	51920	60049
50	22562	26110	32301	40430	43370	46282	52473	60602
52	23115	26663	32855	40983	43868	46835	53026	61155
54	23668	27216	33408	41536	44368	47388	53580	61708
56	24221	27769	33961	42089	44870	47941	54133	62261
58	24774	28323	34514	42642	45374	48495	54686	62814
60	25327	28876	35067	43196	45880	49048	55239	63367
62	25880	29429	35620	43749	46389	49601	55792	63920
64	26434	29982	36173	44302	46901	50154	56345	64474
66	26987	30535	36727	44855	47416	50707	56898	65027
68	27540	31088	37280	45408	47933	51260	57452	65580
70	28093	31641	37833	45961	48454	51813	58005	66133
72	28646	32195	38386	46514	48979	52366	58558	66686
74	29199	32748	38939	47067	49506	52920	59111	67239
76	29752	33301	39492	47621	50036	53473	59664	67792
78	30305	33854	40045	48174	50570	54026	60217	68346
80	30859	34407	40598	48727	51106	54579	60770	68899
82	31412	34960	41152	49280	51645	55132	61323	69452
84	31965	35513	41705	49833	52186	55685	61877	70005
86	32518	36066	42258	50386	52730	56238	62430	70558
88	33071	36620	42811	50939	53275	56791	62983	71111
90	33624	37173	43364	51492	53821	57345	63536	71664

Table 12.

Estimates of (9) versus p = 0.9,0.95,0.99,0.999, smoking status, and age when the patient is a male, has three children and an average body mass index.

	Not smoker				Smoker
Age	${\widehat{\mathrm{VaR}}}_{0.9}$	${\widehat{\mathrm{VaR}}}_{0.95}$	${\widehat{\mathrm{VaR}}}_{0.99}$	${\widehat{\mathrm{VaR}}}_{0.999}$	${\widehat{\mathrm{VaR}}}_{0.9}$	${\widehat{\mathrm{VaR}}}_{0.95}$	${\widehat{\mathrm{VaR}}}_{0.99}$	${\widehat{\mathrm{VaR}}}_{0.999}$
18	12071	15620	21811	29940	35904	36570	41983	50111
20	12624	16173	22364	30493	36376	37062	42536	50664
22	13178	16726	22917	31046	36848	37556	43089	51218
24	13731	17279	23471	31599	37320	38052	43642	51771
26	14284	17832	24024	32152	37793	38550	44196	52324
28	14837	18385	24577	32705	38265	39050	44749	52877
30	15390	18939	25130	33258	38738	39552	45302	53430
32	15943	19492	25683	33811	39211	40057	45855	53983
34	16496	20045	26236	34365	39685	40564	46408	54536
36	17049	20598	26789	34918	40159	41075	46961	55090
38	17603	21151	27342	35471	40632	41588	47514	55643
40	18156	21704	27896	36024	41107	42105	48067	56196
42	18709	22257	28449	36577	41581	42624	48621	56749
44	19262	22810	29002	37130	42056	43148	49174	57302
46	19815	23364	29555	37683	42531	43674	49727	57855
48	20368	23917	30108	38237	43006	44204	50280	58408
50	20921	24470	30661	38790	43482	44737	50833	58961
52	21475	25023	31214	39343	43958	45273	51386	59515
54	22028	25576	31768	39896	44435	45811	51939	60068
56	22581	26129	32321	40449	44912	46352	52492	60621
58	23134	26682	32874	41002	45389	46895	53046	61174
60	23687	27236	33427	41555	45867	47440	53599	61727
62	24240	27789	33980	42108	46345	47986	54152	62280
64	24793	28342	34533	42662	46824	48534	54705	62833
66	25346	28895	35086	43215	47303	49083	55258	63387
68	25900	29448	35639	43768	47783	49633	55811	63940
70	26453	30001	36193	44321	48263	50183	56364	64493
72	27006	30554	36746	44874	48744	50734	56918	65046
74	27559	31107	37299	45427	49226	51285	57471	65599
76	28112	31661	37852	45980	49708	51837	58024	66152
78	28665	32214	38405	46533	50191	52389	58577	66705
80	29218	32767	38958	47087	50675	52942	59130	67258
82	29772	33320	39511	47640	51159	53494	59683	67812
84	30325	33873	40064	48193	51645	54047	60236	68365
86	30878	34426	40618	48746	52131	54599	60789	68918
88	31431	34979	41171	49299	52619	55152	61343	69471
90	31984	35532	41724	49852	53107	55705	61896	70024

Table 13.

Estimates of (9) versus p = 0.9,0.95,0.99,0.999, smoking status, and age when the patient is a male, has four children and an average body mass index.

	Not smoker				Smoker
Age	${\widehat{\mathrm{VaR}}}_{0.9}$	${\widehat{\mathrm{VaR}}}_{0.95}$	${\widehat{\mathrm{VaR}}}_{0.99}$	${\widehat{\mathrm{VaR}}}_{0.999}$	${\widehat{\mathrm{VaR}}}_{0.9}$	${\widehat{\mathrm{VaR}}}_{0.95}$	${\widehat{\mathrm{VaR}}}_{0.99}$	${\widehat{\mathrm{VaR}}}_{0.999}$
18	12404	15952	22144	30272	36496	37106	42316	50444
20	12957	16505	22697	30825	36967	37594	42869	50997
22	13510	17059	23250	31378	37439	38083	43422	51550
24	14063	17612	23803	31932	37910	38573	43975	52103
26	14616	18165	24356	32485	38382	39065	44528	52656
28	15170	18718	24909	33038	38854	39559	45081	53210
30	15723	19271	25463	33591	39326	40054	45634	53763
32	16276	19824	26016	34144	39799	40551	46188	54316
34	16829	20377	26569	34697	40272	41051	46741	54869
36	17382	20931	27122	35250	40744	41553	47294	55422
38	17935	21484	27675	35803	41218	42057	47847	55975
40	18488	22037	28228	36357	41691	42564	48400	56528
42	19041	22590	28781	36910	42164	43074	48953	57082
44	19595	23143	29334	37463	42638	43587	49506	57635
46	20148	23696	29888	38016	43112	44103	50059	58188
48	20701	24249	30441	38569	43587	44622	50613	58741
50	21254	24802	30994	39122	44062	45145	51166	59294
52	21807	25356	31547	39675	44537	45671	51719	59847
54	22360	25909	32100	40228	45012	46200	52272	60400
56	22913	26462	32653	40782	45488	46732	52825	60953
58	23467	27015	33206	41335	45964	47267	53378	61507
60	24020	27568	33760	41888	46440	47805	53931	62060
62	24573	28121	34313	42441	46917	48346	54484	62613
64	25126	28674	34866	42994	47394	48889	55038	63166
66	25679	29228	35419	43547	47872	49433	55591	63719
68	26232	29781	35972	44100	48350	49979	56144	64272
70	26785	30334	36525	44654	48829	50527	56697	64825
72	27338	30887	37078	45207	49308	51076	57250	65379
74	27892	31440	37631	45760	49788	51625	57803	65932
76	28445	31993	38185	46313	50268	52175	58356	66485
78	28998	32546	38738	46866	50749	52726	58910	67038
80	29551	33099	39291	47419	51230	53278	59463	67591
82	30104	33653	39844	47972	51712	53829	60016	68144
84	30657	34206	40397	48525	52195	54381	60569	68697
86	31210	34759	40950	49079	52679	54934	61122	69250
88	31764	35312	41503	49632	53163	55486	61675	69804
90	32317	35865	42056	50185	53648	56039	62228	70357

Table 14.

Estimates of (9) versus p = 0.9,0.95,0.99,0.999, smoking status, and age when the patient is a male, has five children and an average body mass index.

	Not smoker				Smoker
Age	${\widehat{\mathrm{VaR}}}_{0.9}$	${\widehat{\mathrm{VaR}}}_{0.95}$	${\widehat{\mathrm{VaR}}}_{0.99}$	${\widehat{\mathrm{VaR}}}_{0.999}$	${\widehat{\mathrm{VaR}}}_{0.9}$	${\widehat{\mathrm{VaR}}}_{0.95}$	${\widehat{\mathrm{VaR}}}_{0.99}$	${\widehat{\mathrm{VaR}}}_{0.999}$
18	7012	10550	16741	24870	36781	37094	38068	45042
20	7564	11103	17295	25423	37247	37560	38552	45595
22	8115	11656	17848	25976	37713	38028	39037	46148
24	8667	12209	18401	26529	38179	38495	39523	46701
26	9219	12763	18954	27082	38646	38962	40010	47254
28	9771	13316	19507	27635	39112	39429	40499	47807
30	10324	13869	20060	28189	39578	39896	40989	48360
32	10876	14422	20613	28742	40044	40364	41481	48914
34	11429	14975	21166	29295	40510	40831	41975	49467
36	11981	15528	21720	29848	40976	41299	42471	50020
38	12534	16081	22273	30401	41443	41766	42969	50573
40	13087	16634	22826	30954	41909	42234	43469	51126
42	13640	17188	23379	31507	42375	42702	43972	51679
44	14193	17741	23932	32061	42842	43170	44477	52232
46	14746	18294	24485	32614	43308	43638	44985	52785
48	15299	18847	25038	33167	43775	44106	45497	53339
50	15852	19400	25592	33720	44241	44574	46011	53892
52	16405	19953	26145	34273	44708	45042	46529	54445
54	16958	20506	26698	34826	45175	45510	47050	54998
56	17511	21060	27251	35379	45641	45979	47575	55551
58	18064	21613	27804	35932	46108	46447	48103	56104
60	18617	22166	28357	36486	46575	46916	48634	56657
62	19171	22719	28910	37039	47042	47385	49168	57211
64	19724	23272	29463	37592	47508	47853	49705	57764
66	20277	23825	30017	38145	47975	48322	50245	58317
68	20830	24378	30570	38698	48442	48792	50787	58870
70	21383	24931	31123	39251	48909	49261	51331	59423
72	21936	25485	31676	39804	49376	49730	51877	59976
74	22489	26038	32229	40357	49843	50200	52424	60529
76	23042	26591	32782	40911	50311	50670	52972	61082
78	23596	27144	33335	41464	50778	51139	53522	61636
80	24149	27697	33889	42017	51245	51609	54072	62189
82	24702	28250	34442	42570	51713	52080	54622	62742
84	25255	28803	34995	43123	52180	52550	55174	63295
86	25808	29357	35548	43676	52648	53020	55725	63848
88	26361	29910	36101	44229	53115	53491	56277	64401
90	26914	30463	36654	44783	53583	53962	56829	64954

Table 15.

Estimates of (9) versus p = 0.9,0.95,0.99,0.999, smoking status, and body mass index when the patient is a female, has no children and an average age.

	Not smoker				Smoker
BMI	${\widehat{\mathrm{VaR}}}_{0.9}$	${\widehat{\mathrm{VaR}}}_{0.95}$	${\widehat{\mathrm{VaR}}}_{0.99}$	${\widehat{\mathrm{VaR}}}_{0.999}$	${\widehat{\mathrm{VaR}}}_{0.9}$	${\widehat{\mathrm{VaR}}}_{0.95}$	${\widehat{\mathrm{VaR}}}_{0.99}$	${\widehat{\mathrm{VaR}}}_{0.999}$
15	12901	16450	22641	30770	39597	39975	42817	50942
17	13544	17093	23284	31412	39606	40013	43457	51584
19	14187	17736	23927	32055	39620	40067	44099	52227
21	14830	18378	24570	32698	39640	40144	44742	52870
23	15473	19021	25213	33341	39667	40258	45384	53513
25	16115	19664	25855	33984	39703	40435	46027	54155
27	16758	20307	26498	34626	39751	40726	46670	54798
29	17401	20949	27141	35269	39815	41186	47313	55441
31	18044	21592	27784	35912	39902	41776	47955	56084
33	18687	22235	28426	36555	40021	42409	48598	56727
35	19329	22878	29069	37198	40194	43050	49241	57369
37	19972	23521	29712	37840	40461	43692	49884	58012
39	20615	24163	30355	38483	40881	44335	50527	58655
41	21258	24806	30998	39126	41448	44978	51169	59298
43	21901	25449	31640	39769	42075	45621	51812	59941
45	22543	26092	32283	40411	42716	46264	52455	60583
47	23186	26734	32926	41054	43358	46906	53098	61226
49	23829	27377	33569	41697	44001	47549	53741	61869
51	24472	28020	34211	42340	44644	48192	54383	62512
53	25114	28663	34854	42983	45286	48835	55026	63154
55	25757	29306	35497	43625	45929	49477	55669	63797
57	26400	29948	36140	44268	46572	50120	56312	64440
59	27043	30591	36783	44911	47215	50763	56955	65083
61	27686	31234	37425	45554	47857	51406	57597	65726
63	28328	31877	38068	46197	48500	52049	58240	66368
65	28971	32520	38711	46839	49143	52691	58883	67011
67	29614	33162	39354	47482	49786	53334	59526	67654
69	30257	33805	39997	48125	50429	53977	60168	68297
71	30900	34448	40639	48768	51071	54620	60811	68940
73	31542	35091	41282	49411	51714	55263	61454	69582
75	32185	35734	41925	50053	52357	55905	62097	70225
77	32828	36376	42568	50696	53000	56548	62740	70868
79	33471	37019	43211	51339	53642	57191	63382	71511
81	34114	37662	43853	51982	54285	57834	64025	72153
83	34758	38305	44496	52624	54928	58477	64668	72796
85	35403	38947	45139	53267	55571	59119	65311	73439
87	36051	39590	45782	53910	56214	59762	65953	74082
89	36702	40233	46424	54553	56856	60405	66596	74725

Table 16.

Estimates of (9) versus p = 0.9,0.95,0.99,0.999, smoking status, and body mass index when the patient is a female, has one child and an average age.

	Not smoker				Smoker
BMI	${\widehat{\mathrm{VaR}}}_{0.9}$	${\widehat{\mathrm{VaR}}}_{0.95}$	${\widehat{\mathrm{VaR}}}_{0.99}$	${\widehat{\mathrm{VaR}}}_{0.999}$	${\widehat{\mathrm{VaR}}}_{0.9}$	${\widehat{\mathrm{VaR}}}_{0.95}$	${\widehat{\mathrm{VaR}}}_{0.99}$	${\widehat{\mathrm{VaR}}}_{0.999}$
15	15167	18715	24907	33035	40179	40662	45079	53207
17	15810	19358	25550	33678	40203	40762	45721	53850
19	16453	20001	26192	34321	40237	40915	46364	54493
21	17095	20644	26835	34964	40281	41161	47007	55135
23	17738	21287	27478	35606	40339	41564	47650	55778
25	18381	21929	28121	36249	40417	42122	48293	56421
27	19024	22572	28764	36892	40524	42747	48935	57064
29	19666	23215	29406	37535	40676	43387	49578	57707
31	20309	23858	30049	38177	40907	44030	50221	58349
33	20952	24501	30692	38820	41272	44672	50864	58992
35	21595	25143	31335	39463	41799	45315	51507	59635
37	22238	25786	31977	40106	42415	45958	52149	60278
39	22880	26429	32620	40749	43053	46601	52792	60920
41	23523	27072	33263	41391	43695	47244	53435	61563
43	24166	27714	33906	42034	44338	47886	54078	62206
45	24809	28357	34549	42677	44981	48529	54720	62849
47	25452	29000	35191	43320	45623	49172	55363	63492
49	26094	29643	35834	43963	46266	49815	56006	64134
51	26737	30286	36477	44605	46909	50457	56649	64777
53	27380	30928	37120	45248	47552	51100	57292	65420
55	28023	31571	37763	45891	48195	51743	57934	66063
57	28665	32214	38405	46534	48837	52386	58577	66706
59	29308	32857	39048	47176	49480	53029	59220	67348
61	29951	33500	39691	47819	50123	53671	59863	67991
63	30594	34142	40334	48462	50766	54314	60506	68634
65	31237	34785	40976	49105	51408	54957	61148	69277
67	31879	35428	41619	49748	52051	55600	61791	69919
69	32522	36071	42262	50390	52694	56243	62434	70562
71	33165	36713	42905	51033	53337	56885	63077	71205
73	33808	37356	43548	51676	53980	57528	63719	71848
75	34451	37999	44190	52319	54622	58171	64362	72491
77	35093	38642	44833	52962	55265	58814	65005	73133
79	35736	39285	45476	53604	55908	59456	65648	73776
81	36379	39927	46119	54247	56551	60099	66291	74419
83	37022	40570	46762	54890	57194	60742	66933	75062
85	37665	41213	47404	55533	57836	61385	67576	75705
87	38307	41856	48047	56176	58479	62028	68219	76347
89	38950	42499	48690	56818	59122	62670	68862	76990

Table 17.

Estimates of (9) versus p = 0.9,0.95,0.99,0.999, smoking status, and body mass index when the patient is a female, has two children and an average age.

	Not smoker				Smoker
BMI	${\widehat{\mathrm{VaR}}}_{0.9}$	${\widehat{\mathrm{VaR}}}_{0.95}$	${\widehat{\mathrm{VaR}}}_{0.99}$	${\widehat{\mathrm{VaR}}}_{0.999}$	${\widehat{\mathrm{VaR}}}_{0.9}$	${\widehat{\mathrm{VaR}}}_{0.95}$	${\widehat{\mathrm{VaR}}}_{0.99}$	${\widehat{\mathrm{VaR}}}_{0.999}$
15	15431	18980	25171	33299	40699	41159	45343	53471
17	16074	19622	25814	33942	40721	41244	45986	54114
19	16717	20265	26457	34585	40750	41372	46628	54757
21	17360	20908	27099	35228	40789	41573	47271	55400
23	18002	21551	27742	35871	40841	41905	47914	56042
25	18645	22194	28385	36513	40911	42408	48557	56685
27	19288	22836	29028	37156	41004	43016	49200	57328
29	19931	23479	29671	37799	41135	43652	49842	57971
31	20574	24122	30313	38442	41328	44294	50485	58614
33	21216	24765	30956	39085	41630	44937	51128	59256
35	21859	25408	31599	39727	42094	45579	51771	59899
37	22502	26050	32242	40370	42685	46222	52414	60542
39	23145	26693	32885	41013	43318	46865	53056	61185
41	23787	27336	33527	41656	43960	47508	53699	61828
43	24430	27979	34170	42298	44602	48151	54342	62470
45	25073	28621	34813	42941	45245	48793	54985	63113
47	25716	29264	35456	43584	45888	49436	55628	63756
49	26359	29907	36098	44227	46530	50079	56270	64399
51	27001	30550	36741	44870	47173	50722	56913	65041
53	27644	31193	37384	45512	47816	51365	57556	65684
55	28287	31835	38027	46155	48459	52007	58199	66327
57	28930	32478	38670	46798	49102	52650	58841	66970
59	29573	33121	39312	47441	49744	53293	59484	67613
61	30215	33764	39955	48084	50387	53936	60127	68255
63	30858	34407	40598	48726	51030	54578	60770	68898
65	31501	35049	41241	49369	51673	55221	61413	69541
67	32144	35692	41884	50012	52316	55864	62055	70184
69	32786	36335	42526	50655	52958	56507	62698	70827
71	33429	36978	43169	51297	53601	57150	63341	71469
73	34072	37620	43812	51940	54244	57792	63984	72112
75	34715	38263	44455	52583	54887	58435	64627	72755
77	35358	38906	45097	53226	55529	59078	65269	73398
79	36000	39549	45740	53869	56172	59721	65912	74040
81	36643	40192	46383	54511	56815	60363	66555	74683
83	37286	40834	47026	55154	57458	61006	67198	75326
85	37929	41477	47669	55797	58101	61649	67840	75969
87	38572	42120	48311	56440	58743	62292	68483	76612
89	39215	42763	48954	57083	59386	62935	69126	77254

Table 18.

Estimates of (9) versus p = 0.9,0.95,0.99,0.999, smoking status, and body mass index when the patient is a female, has three children and an average age.

	Not smoker				Smoker
BMI	${\widehat{\mathrm{VaR}}}_{0.9}$	${\widehat{\mathrm{VaR}}}_{0.95}$	${\widehat{\mathrm{VaR}}}_{0.99}$	${\widehat{\mathrm{VaR}}}_{0.999}$	${\widehat{\mathrm{VaR}}}_{0.9}$	${\widehat{\mathrm{VaR}}}_{0.95}$	${\widehat{\mathrm{VaR}}}_{0.99}$	${\widehat{\mathrm{VaR}}}_{0.999}$
15	13791	17339	23531	31659	41203	41558	43730	51831
17	14434	17982	24174	32302	41208	41583	44350	52474
19	15077	18625	24816	32945	41217	41620	44989	53117
21	15719	19268	25459	33588	41230	41671	45631	53759
23	16362	19911	26102	34230	41249	41745	46274	54402
25	17005	20553	26745	34873	41275	41853	46917	55045
27	17648	21196	27388	35516	41310	42021	47559	55688
29	18291	21839	28030	36159	41357	42294	48202	56331
31	18933	22482	28673	36802	41418	42732	48845	56973
33	19576	23125	29316	37444	41502	43311	49488	57616
35	20219	23767	29959	38087	41616	43942	50131	58259
37	20862	24410	30602	38730	41781	44582	50773	58902
39	21504	25053	31244	39373	42033	45225	51416	59544
41	22147	25696	31887	40015	42432	45868	52059	60187
43	22790	26338	32530	40658	42985	46510	52702	60830
45	23433	26981	33173	41301	43608	47153	53345	61473
47	24076	27624	33815	41944	44248	47796	53987	62116
49	24718	28267	34458	42587	44890	48439	54630	62758
51	25361	28910	35101	43229	45533	49081	55273	63401
53	26004	29552	35744	43872	46176	49724	55916	64044
55	26647	30195	36387	44515	46819	50367	56558	64687
57	27290	30838	37029	45158	47461	51010	57201	65330
59	27932	31481	37672	45801	48104	51653	57844	65972
61	28575	32124	38315	46443	48747	52295	58487	66615
63	29218	32766	38958	47086	49390	52938	59130	67258
65	29861	33409	39601	47729	50033	53581	59772	67901
67	30504	34052	40243	48372	50675	54224	60415	68544
69	31146	34695	40886	49014	51318	54867	61058	69186
71	31789	35337	41529	49657	51961	55509	61701	69829
73	32432	35980	42172	50300	52604	56152	62344	70472
75	33075	36623	42814	50943	53246	56795	62986	71115
77	33719	37266	43457	51586	53889	57438	63629	71757
79	34364	37909	44100	52228	54532	58080	64272	72400
81	35010	38551	44743	52871	55175	58723	64915	73043
83	35660	39194	45386	53514	55818	59366	65557	73686
85	36316	39837	46028	54157	56460	60009	66200	74329
87	36985	40480	46671	54800	57103	60652	66843	74971
89	37671	41123	47314	55442	57746	61294	67486	75614

Table 19.

Estimates of (9) versus p = 0.9,0.95,0.99,0.999 and body mass index when the patient is a female, has four children and an average age.

	Not smoker				Smoker
BMI	${\widehat{\mathrm{VaR}}}_{0.9}$	${\widehat{\mathrm{VaR}}}_{0.95}$	${\widehat{\mathrm{VaR}}}_{0.99}$	${\widehat{\mathrm{VaR}}}_{0.999}$	${\widehat{\mathrm{VaR}}}_{0.9}$	${\widehat{\mathrm{VaR}}}_{0.95}$	${\widehat{\mathrm{VaR}}}_{0.99}$	${\widehat{\mathrm{VaR}}}_{0.999}$
15	14124	17672	23863	31992	41810	42157	44090	52164
17	14766	18315	24506	32635	41813	42178	44688	52806
19	15409	18958	25149	33277	41819	42210	45322	53449
21	16052	19600	25792	33920	41830	42254	45964	54092
23	16695	20243	26435	34563	41847	42317	46607	54735
25	17338	20886	27077	35206	41870	42408	47249	55378
27	17980	21529	27720	35849	41901	42546	47892	56020
29	18623	22172	28363	36491	41942	42767	48535	56663
31	19266	22814	29006	37134	41997	43130	49178	57306
33	19909	23457	29649	37777	42070	43660	49820	57949
35	20552	24100	30291	38420	42169	44277	50463	58592
37	21194	24743	30934	39062	42309	44915	51106	59234
39	21837	25386	31577	39705	42518	45557	51749	59877
41	22480	26028	32220	40348	42847	46200	52392	60520
43	23123	26671	32863	40991	43342	46843	53034	61163
45	23765	27314	33505	41634	43946	47486	53677	61805
47	24408	27957	34148	42276	44581	48129	54320	62448
49	25051	28599	34791	42919	45223	48771	54963	63091
51	25694	29242	35434	43562	45866	49414	55605	63734
53	26337	29885	36076	44205	46508	50057	56248	64377
55	26979	30528	36719	44848	47151	50700	56891	65019
57	27622	31171	37362	45490	47794	51342	57534	65662
59	28265	31813	38005	46133	48437	51985	58177	66305
61	28908	32456	38648	46776	49080	52628	58819	66948
63	29551	33099	39290	47419	49722	53271	59462	67591
65	30193	33742	39933	48061	50365	53914	60105	68233
67	30836	34385	40576	48704	51008	54556	60748	68876
69	31479	35027	41219	49347	51651	55199	61391	69519
71	32122	35670	41862	49990	52294	55842	62033	70162
73	32765	36313	42504	50633	52936	56485	62676	70804
75	33408	36956	43147	51275	53579	57128	63319	71447
77	34051	37598	43790	51918	54222	57770	63962	72090
79	34694	38241	44433	52561	54865	58413	64605	72733
81	35339	38884	45075	53204	55507	59056	65247	73376
83	35985	39527	45718	53847	56150	59699	65890	74018
85	36635	40170	46361	54489	56793	60341	66533	74661
87	37291	40812	47004	55132	57436	60984	67176	75304
89	37959	41455	47647	55775	58079	61627	67818	75947

Table 20.

Estimates of (9) versus p = 0.9,0.95,0.99,0.999, smoking status, and body mass index when the patient is a female, has five children and an average age.

	Not smoker				Smoker
BMI	${\widehat{\mathrm{VaR}}}_{0.9}$	${\widehat{\mathrm{VaR}}}_{0.95}$	${\widehat{\mathrm{VaR}}}_{0.99}$	${\widehat{\mathrm{VaR}}}_{0.999}$	${\widehat{\mathrm{VaR}}}_{0.9}$	${\widehat{\mathrm{VaR}}}_{0.95}$	${\widehat{\mathrm{VaR}}}_{0.99}$	${\widehat{\mathrm{VaR}}}_{0.999}$
15	10042	12273	18461	26590	42227	42516	43199	46763
17	10192	12913	19104	27232	42216	42507	43213	47404
19	10421	13555	19747	27875	42206	42500	43234	48047
21	10789	14198	20390	28518	42196	42494	43267	48690
23	11322	14841	21032	29161	42188	42490	43315	49333
25	11940	15484	21675	29803	42181	42488	43388	49975
27	12579	16127	22318	30446	42175	42489	43500	50618
29	13221	16769	22961	31089	42170	42493	43682	51261
31	13864	17412	23603	31732	42168	42502	43988	51904
33	14508	18055	24246	32375	42169	42517	44470	52546
35	15153	18698	24889	33017	42172	42538	45070	53189
37	15799	19340	25532	33660	42179	42570	45705	53832
39	16449	19983	26175	34303	42190	42615	46347	54475
41	17105	20626	26817	34946	42206	42679	46989	55118
43	17773	21269	27460	35589	42230	42771	47632	55760
45	18459	21912	28103	36231	42261	42912	48275	56403
47	19170	22554	28746	36874	42303	43136	48918	57046
49	19908	23197	29389	37517	42358	43506	49560	57689
51	20673	23840	30031	38160	42432	44041	50203	58332
53	21462	24483	30674	38802	42533	44659	50846	58974
55	22270	25126	31317	39445	42674	45298	51489	59617
57	23092	25770	31960	40088	42887	45940	52132	60260
59	23927	26414	32602	40731	43222	46583	52774	60903
61	24771	27060	33245	41374	43722	47226	53417	61545
63	25622	27710	33888	42016	44328	47868	54060	62188
65	26478	28365	34531	42659	44965	48511	54703	62831
67	27339	29031	35174	43302	45607	49154	55345	63474
69	28204	29715	35816	43945	46251	49797	55988	64117
71	29070	30424	36459	44588	46896	50440	56631	64759
73	29938	31162	37102	45230	47543	51082	57274	65402
75	30807	31928	37745	45873	48195	51725	57917	66045
77	31675	32720	38388	46516	48854	52368	58559	66688
79	32541	33534	39030	47159	49528	53011	59202	67331
81	33402	34364	39673	47801	50222	53654	59845	67973
83	34256	35208	40316	48444	50942	54296	60488	68616
85	35094	36063	40959	49087	51690	54939	61131	69259
87	35896	36926	41602	49730	52463	55582	61773	69902
89	36612	37797	42244	50373	53259	56225	62416	70544

Table 21.

Estimates of (9) versus p = 0.9,0.95,0.99,0.999, smoking status, and body mass index when the patient is a male, has no children and an average age.

	Not smoker				Smoker
BMI	${\widehat{\mathrm{VaR}}}_{0.9}$	${\widehat{\mathrm{VaR}}}_{0.95}$	${\widehat{\mathrm{VaR}}}_{0.99}$	${\widehat{\mathrm{VaR}}}_{0.999}$	${\widehat{\mathrm{VaR}}}_{0.9}$	${\widehat{\mathrm{VaR}}}_{0.95}$	${\widehat{\mathrm{VaR}}}_{0.99}$	${\widehat{\mathrm{VaR}}}_{0.999}$
15	12013	15561	21753	29881	39147	39510	41937	50053
17	12656	16204	22395	30524	39153	39540	42569	50696
19	13298	16847	23038	31167	39164	39582	43210	51338
21	13941	17490	23681	31809	39179	39642	43853	51981
23	14584	18132	24324	32452	39201	39729	44496	52624
25	15227	18775	24967	33095	39231	39859	45138	53267
27	15870	19418	25609	33738	39271	40065	45781	53910
29	16512	20061	26252	34381	39324	40404	46424	54552
31	17155	20704	26895	35023	39394	40914	47067	55195
33	17798	21346	27538	35666	39489	41525	47710	55838
35	18441	21989	28181	36309	39622	42162	48352	56481
37	19083	22632	28823	36952	39819	42804	48995	57123
39	19726	23275	29466	37594	40128	43447	49638	57766
41	20369	23917	30109	38237	40600	44089	50281	58409
43	21012	24560	30752	38880	41194	44732	50924	59052
45	21655	25203	31394	39523	41828	45375	51566	59695
47	22297	25846	32037	40166	42469	46018	52209	60337
49	22940	26489	32680	40808	43112	46660	52852	60980
51	23583	27131	33323	41451	43755	47303	53495	61623
53	24226	27774	33966	42094	44398	47946	54137	62266
55	24869	28417	34608	42737	45040	48589	54780	62909
57	25511	29060	35251	43380	45683	49232	55423	63551
59	26154	29703	35894	44022	46326	49874	56066	64194
61	26797	30345	36537	44665	46969	50517	56709	64837
63	27440	30988	37180	45308	47612	51160	57351	65480
65	28082	31631	37822	45951	48254	51803	57994	66122
67	28725	32274	38465	46593	48897	52446	58637	66765
69	29368	32916	39108	47236	49540	53088	59280	67408
71	30011	33559	39751	47879	50183	53731	59923	68051
73	30654	34202	40393	48522	50825	54374	60565	68694
75	31297	34845	41036	49165	51468	55017	61208	69336
77	31940	35488	41679	49807	52111	55659	61851	69979
79	32583	36130	42322	50450	52754	56302	62494	70622
81	33226	36773	42965	51093	53397	56945	63136	71265
83	33871	37416	43607	51736	54039	57588	63779	71908
85	34517	38059	44250	52379	54682	58231	64422	72550
87	35166	38702	44893	53021	55325	58873	65065	73193
89	35821	39344	45536	53664	55968	59516	65708	73836

Table 22.

Estimates of (9) versus p = 0.9,0.95,0.99,0.999, smoking status, and body mass index when the patient is a male, has one child and an average age.

	Not smoker				Smoker
BMI	${\widehat{\mathrm{VaR}}}_{0.9}$	${\widehat{\mathrm{VaR}}}_{0.95}$	${\widehat{\mathrm{VaR}}}_{0.99}$	${\widehat{\mathrm{VaR}}}_{0.999}$	${\widehat{\mathrm{VaR}}}_{0.9}$	${\widehat{\mathrm{VaR}}}_{0.95}$	${\widehat{\mathrm{VaR}}}_{0.99}$	${\widehat{\mathrm{VaR}}}_{0.999}$
15	14278	17827	24018	32146	39720	40167	44190	52318
17	14921	18470	24661	32789	39740	40243	44833	52961
19	15564	19112	25304	33432	39767	40357	45476	53604
21	16207	19755	25947	34075	39803	40533	46118	54247
23	16849	20398	26589	34718	39851	40822	46761	54889
25	17492	21041	27232	35360	39915	41278	47404	55532
27	18135	21683	27875	36003	40001	41867	48047	56175
29	18778	22326	28518	36646	40120	42500	48689	56818
31	19421	22969	29160	37289	40292	43141	49332	57461
33	20063	23612	29803	37932	40557	43784	49975	58103
35	20706	24255	30446	38574	40974	44426	50618	58746
37	21349	24897	31089	39217	41540	45069	51261	59389
39	21992	25540	31732	39860	42167	45712	51903	60032
41	22635	26183	32374	40503	42807	46355	52546	60675
43	23277	26826	33017	41146	43449	46998	53189	61317
45	23920	27469	33660	41788	44092	47640	53832	61960
47	24563	28111	34303	42431	44735	48283	54475	62603
49	25206	28754	34946	43074	45378	48926	55117	63246
51	25848	29397	35588	43717	46020	49569	55760	63888
53	26491	30040	36231	44359	46663	50212	56403	64531
55	27134	30682	36874	45002	47306	50854	57046	65174
57	27777	31325	37517	45645	47949	51497	57689	65817
59	28420	31968	38159	46288	48591	52140	58331	66460
61	29062	32611	38802	46931	49234	52783	58974	67102
63	29705	33254	39445	47573	49877	53425	59617	67745
65	30348	33896	40088	48216	50520	54068	60260	68388
67	30991	34539	40731	48859	51163	54711	60902	69031
69	31634	35182	41373	49502	51805	55354	61545	69674
71	32276	35825	42016	50145	52448	55997	62188	70316
73	32919	36468	42659	50787	53091	56639	62831	70959
75	33562	37110	43302	51430	53734	57282	63474	71602
77	34205	37753	43945	52073	54377	57925	64116	72245
79	34847	38396	44587	52716	55019	58568	64759	72887
81	35490	39039	45230	53358	55662	59211	65402	73530
83	36133	39681	45873	54001	56305	59853	66045	74173
85	36776	40324	46516	54644	56948	60496	66688	74816
87	37419	40967	47158	55287	57590	61139	67330	75459
89	38062	41610	47801	55930	58233	61782	67973	76101

Table 23.

Estimates of (9) versus p = 0.9,0.95,0.99,0.999, smoking status, and body mass index when the patient is a male, has two children and an average age.

	Not smoker				Smoker
BMI	${\widehat{\mathrm{VaR}}}_{0.9}$	${\widehat{\mathrm{VaR}}}_{0.95}$	${\widehat{\mathrm{VaR}}}_{0.99}$	${\widehat{\mathrm{VaR}}}_{0.999}$	${\widehat{\mathrm{VaR}}}_{0.9}$	${\widehat{\mathrm{VaR}}}_{0.95}$	${\widehat{\mathrm{VaR}}}_{0.99}$	${\widehat{\mathrm{VaR}}}_{0.999}$
15	14543	18091	24282	32411	40242	40671	44454	52583
17	15185	18734	24925	33054	40259	40737	45097	53225
19	15828	19377	25568	33696	40283	40833	45740	53868
21	16471	20019	26211	34339	40315	40979	46383	54511
23	17114	20662	26854	34982	40358	41213	47025	55154
25	17756	21305	27496	35625	40415	41598	47668	55797
27	18399	21948	28139	36267	40491	42144	48311	56439
29	19042	22591	28782	36910	40594	42766	48954	57082
31	19685	23233	29425	37553	40741	43406	49597	57725
33	20328	23876	30067	38196	40961	44048	50239	58368
35	20970	24519	30710	38839	41310	44691	50882	59010
37	21613	25162	31353	39481	41823	45333	51525	59653
39	22256	25804	31996	40124	42434	45976	52168	60296
41	22899	26447	32639	40767	43072	46619	52810	60939
43	23542	27090	33281	41410	43714	47262	53453	61582
45	24184	27733	33924	42053	44356	47905	54096	62224
47	24827	28376	34567	42695	44999	48547	54739	62867
49	25470	29018	35210	43338	45642	49190	55382	63510
51	26113	29661	35853	43981	46285	49833	56024	64153
53	26755	30304	36495	44624	46927	50476	56667	64796
55	27398	30947	37138	45266	47570	51119	57310	65438
57	28041	31590	37781	45909	48213	51761	57953	66081
59	28684	32232	38424	46552	48856	52404	58596	66724
61	29327	32875	39066	47195	49498	53047	59238	67367
63	29969	33518	39709	47838	50141	53690	59881	68009
65	30612	34161	40352	48480	50784	54332	60524	68652
67	31255	34803	40995	49123	51427	54975	61167	69295
69	31898	35446	41638	49766	52070	55618	61809	69938
71	32541	36089	42280	50409	52712	56261	62452	70581
73	33183	36732	42923	51052	53355	56904	63095	71223
75	33826	37375	43566	51694	53998	57546	63738	71866
77	34469	38017	44209	52337	54641	58189	64381	72509
79	35112	38660	44852	52980	55284	58832	65023	73152
81	35755	39303	45494	53623	55926	59475	65666	73795
83	36397	39946	46137	54265	56569	60118	66309	74437
85	37040	40589	46780	54908	57212	60760	66952	75080
87	37683	41231	47423	55551	57855	61403	67595	75723
89	38326	41874	48065	56194	58497	62046	68237	76366

Table 24.

Estimates of (9) versus p = 0.9,0.95,0.99,0.999, smoking status, and body mass index when the patient is a male, has three children and an average age.

	Not smoker				Smoker
BMI	${\widehat{\mathrm{VaR}}}_{0.9}$	${\widehat{\mathrm{VaR}}}_{0.95}$	${\widehat{\mathrm{VaR}}}_{0.99}$	${\widehat{\mathrm{VaR}}}_{0.999}$	${\widehat{\mathrm{VaR}}}_{0.9}$	${\widehat{\mathrm{VaR}}}_{0.95}$	${\widehat{\mathrm{VaR}}}_{0.99}$	${\widehat{\mathrm{VaR}}}_{0.999}$
15	12902	16451	22642	30771	40756	41100	42897	50942
17	13545	17094	23285	31413	40759	41119	43472	51585
19	14188	17736	23928	32056	40764	41148	44102	52228
21	14831	18379	24571	32699	40774	41188	44743	52871
23	15473	19022	25213	33342	40789	41245	45385	53514
25	16116	19665	25856	33984	40810	41328	46028	54156
27	16759	20307	26499	34627	40839	41451	46671	54799
29	17402	20950	27142	35270	40877	41645	47314	55442
31	18045	21593	27784	35913	40928	41965	47956	56085
33	18687	22236	28427	36556	40996	42455	48599	56727
35	19330	22879	29070	37198	41087	43059	49242	57370
37	19973	23521	29713	37841	41215	43695	49885	58013
39	20616	24164	30356	38484	41401	44336	50527	58656
41	21259	24807	30998	39127	41693	44979	51170	59299
43	21901	25450	31641	39770	42144	45622	51813	59941
45	22544	26093	32284	40412	42729	46264	52456	60584
47	23187	26735	32927	41055	43361	46907	53099	61227
49	23830	27378	33570	41698	44002	47550	53741	61870
51	24472	28021	34212	42341	44644	48193	54384	62513
53	25115	28664	34855	42983	45287	48836	55027	63155
55	25758	29306	35498	43626	45930	49478	55670	63798
57	26401	29949	36141	44269	46573	50121	56313	64441
59	27044	30592	36783	44912	47215	50764	56955	65084
61	27686	31235	37426	45555	47858	51407	57598	65726
63	28329	31878	38069	46197	48501	52049	58241	66369
65	28972	32520	38712	46840	49144	52692	58884	67012
67	29615	33163	39355	47483	49787	53335	59526	67655
69	30258	33806	39997	48126	50429	53978	60169	68298
71	30901	34449	40640	48769	51072	54621	60812	68940
73	31544	35092	41283	49411	51715	55263	61455	69583
75	32187	35734	41926	50054	52358	55906	62098	70226
77	32831	36377	42569	50697	53001	56549	62740	70869
79	33477	37020	43211	51340	53643	57192	63383	71512
81	34125	37663	43854	51982	54286	57835	64026	72154
83	34778	38306	44497	52625	54929	58477	64669	72797
85	35440	38948	45140	53268	55572	59120	65312	73440
87	36117	39591	45782	53911	56214	59763	65954	74083
89	36817	40234	46425	54554	56857	60406	66597	74725

Table 25.

Estimates of (9) versus p = 0.9,0.95,0.99,0.999, smoking status, and body mass index when the patient is a male, has four children and an average age.

	Not smoker				Smoker
BMI	${\widehat{\mathrm{VaR}}}_{0.9}$	${\widehat{\mathrm{VaR}}}_{0.95}$	${\widehat{\mathrm{VaR}}}_{0.99}$	${\widehat{\mathrm{VaR}}}_{0.999}$	${\widehat{\mathrm{VaR}}}_{0.9}$	${\widehat{\mathrm{VaR}}}_{0.95}$	${\widehat{\mathrm{VaR}}}_{0.99}$	${\widehat{\mathrm{VaR}}}_{0.999}$
15	13235	16783	22975	31103	41363	41701	43298	51275
17	13878	17426	23618	31746	41364	41717	43822	51918
19	14521	18069	24260	32389	41369	41741	44438	52561
21	15163	18712	24903	33032	41377	41776	45076	53203
23	15806	19355	25546	33674	41389	41825	45718	53846
25	16449	19997	26189	34317	41407	41896	46361	54489
27	17092	20640	26832	34960	41433	41999	47003	55132
29	17734	21283	27474	35603	41467	42158	47646	55774
31	18377	21926	28117	36245	41512	42416	48289	56417
33	19020	22568	28760	36888	41572	42833	48932	57060
35	19663	23211	29403	37531	41652	43401	49575	57703
37	20306	23854	30045	38174	41762	44029	50217	58346
39	20948	24497	30688	38817	41919	44669	50860	58988
41	21591	25140	31331	39459	42159	45312	51503	59631
43	22234	25782	31974	40102	42538	45954	52146	60274
45	22877	26425	32617	40745	43077	46597	52788	60917
47	23520	27068	33259	41388	43696	47240	53431	61560
49	24162	27711	33902	42031	44335	47883	54074	62202
51	24805	28354	34545	42673	44977	48525	54717	62845
53	25448	28996	35188	43316	45620	49168	55360	63488
55	26091	29639	35831	43959	46262	49811	56002	64131
57	26733	30282	36473	44602	46905	50454	56645	64774
59	27376	30925	37116	45244	47548	51097	57288	65416
61	28019	31567	37759	45887	48191	51739	57931	66059
63	28662	32210	38402	46530	48834	52382	58574	66702
65	29305	32853	39044	47173	49476	53025	59216	67345
67	29947	33496	39687	47816	50119	53668	59859	67987
69	30590	34139	40330	48458	50762	54310	60502	68630
71	31233	34781	40973	49101	51405	54953	61145	69273
73	31876	35424	41616	49744	52048	55596	61787	69916
75	32519	36067	42258	50387	52690	56239	62430	70559
77	33163	36710	42901	51030	53333	56882	63073	71201
79	33807	37353	43544	51672	53976	57524	63716	71844
81	34452	37995	44187	52315	54619	58167	64359	72487
83	35100	38638	44830	52958	55262	58810	65001	73130
85	35753	39281	45472	53601	55904	59453	65644	73773
87	36415	39924	46115	54243	56547	60096	66287	74415
89	37092	40566	46758	54886	57190	60738	66930	75058

Table 26.

Estimates of (9) versus p = 0.9,0.95,0.99,0.999, smoking status, and body mass index when the patient is a male, has five children and an average age.

	Not smoker				Smoker
BMI	${\widehat{\mathrm{VaR}}}_{0.9}$	${\widehat{\mathrm{VaR}}}_{0.95}$	${\widehat{\mathrm{VaR}}}_{0.99}$	${\widehat{\mathrm{VaR}}}_{0.999}$	${\widehat{\mathrm{VaR}}}_{0.9}$	${\widehat{\mathrm{VaR}}}_{0.95}$	${\widehat{\mathrm{VaR}}}_{0.99}$	${\widehat{\mathrm{VaR}}}_{0.999}$
15	9520	11392	17572	25701	41789	42076	42747	45877
17	9636	12026	18215	26344	41778	42067	42757	46516
19	9806	12667	18858	26986	41767	42059	42773	47158
21	10071	13309	19501	27629	41757	42052	42797	47801
23	10493	13952	20144	28272	41748	42047	42834	48444
25	11063	14595	20786	28915	41740	42043	42889	49087
27	11692	15238	21429	29558	41733	42042	42972	49729
29	12333	15881	22072	30200	41727	42044	43102	50372
31	12976	16523	22715	30843	41724	42050	43315	51015
33	13620	17166	23358	31486	41722	42060	43674	51658
35	14266	17809	24000	32129	41723	42077	44203	52301
37	14914	18452	24643	32771	41728	42101	44820	52943
39	15567	19095	25286	33414	41736	42136	45459	53586
41	16229	19737	25929	34057	41749	42186	46101	54229
43	16906	20380	26571	34700	41767	42258	46743	54872
45	17605	21023	27214	35343	41793	42363	47386	55514
47	18331	21666	27857	35985	41827	42524	48029	56157
49	19084	22309	28500	36628	41873	42787	48672	56800
51	19863	22951	29143	37271	41933	43211	49314	57443
53	20663	23595	29785	37914	42014	43783	49957	58086
55	21479	24238	30428	38557	42126	44411	50600	58728
57	22309	24882	31071	39199	42285	45052	51243	59371
59	23149	25527	31714	39842	42529	45694	51886	60014
61	23997	26175	32357	40485	42914	46337	52528	60657
63	24851	26827	32999	41128	43458	46980	53171	61300
65	25710	27488	33642	41770	44079	47623	53814	61942
67	26573	28163	34285	42413	44719	48265	54457	62585
69	27439	28860	34928	43056	45363	48908	55100	63228
71	28307	29584	35571	43699	46009	49551	55742	63871
73	29175	30338	36213	44342	46658	50194	56385	64513
75	30043	31119	36856	44984	47314	50837	57028	65156
77	30910	31924	37499	45627	47981	51479	57671	65799
79	31774	32747	38142	46270	48665	52122	58314	66442
81	32632	33586	38784	46913	49373	52765	58956	67085
83	33478	34436	39427	47556	50108	53408	59599	67727
85	34301	35296	40070	48198	50871	54051	60242	68370
87	35062	36163	40713	48841	51657	54694	60885	69013
89	35727	37037	41356	49484	52463	55337	61527	69656

Table 27.

Estimates of (9) versus p = 0.9,0.95,0.99,0.999, smoking status, sex and number of children when the patient is of average age and average body mass index.

		Not smoker				Smoker
Sex	No of children	${\widehat{\mathrm{VaR}}}_{0.9}$	${\widehat{\mathrm{VaR}}}_{0.95}$	${\widehat{\mathrm{VaR}}}_{0.99}$	${\widehat{\mathrm{VaR}}}_{0.999}$	${\widehat{\mathrm{VaR}}}_{0.9}$	${\widehat{\mathrm{VaR}}}_{0.95}$	${\widehat{\mathrm{VaR}}}_{0.99}$	${\widehat{\mathrm{VaR}}}_{0.999}$
Female	0	17936	21484	27675	35804	39885	41672	47847	55976
Female	1	20201	23750	29941	38069	40860	43922	50113	58241
Female	2	20465	24014	30205	38334	41289	44186	50377	58505
Female	3	18825	22374	28565	36693	41407	42646	48737	56865
Female	4	19158	22706	28898	37026	41986	43056	49069	57198
Female	5	13756	17304	23495	31624	42168	42500	43925	51796
Male	0	17047	20595	26787	34915	39380	40818	46959	55087
Male	1	19312	22861	29052	37181	40258	43033	49224	57352
Male	2	19577	23125	29317	37445	40712	43298	49488	57617
Male	3	17936	21485	27676	35805	40919	41899	47848	55976
Male	4	18269	21817	28009	36137	41503	42363	48181	56309
Male	5	12868	16415	22607	30735	41724	42049	43270	50907

Data availability

Data will be made available on request.