A novel extended model with versatile shaped failure rate: Statistical inference with Covid -19 applications

Abstract

Statistical models perform an essential role in data analysis, and statisticians are constantly looking for novel or pretty recent statistical models to fit data sets across a broad variety of fields. In this study, we used modified Kies generalized transformation and the new power function to suggest an unique statistical model. We present and discuss a linear illustration of the density function. Theoretically, quantile function, characteristic function, stochastic ordering, mean, and moments are just a few of the structure properties we discuss. By defining an ideal statistical distribution for assessing the COVID-19 mortality rate, an attempt is performed to determine the model of COVID-19 spread in different nations like the United Kingdom and Italy. In some countries, the novel distribution have been shown to be more appropriate than existing competing models when fitted to COVID-19.

Introduction

Over the last three decades, scholars’ enthusiasm for developing new generalized models has grown as they seek to uncover the hidden characteristics of baseline models. Newly diversified models provide up new potential for solving real-world problems and fitting complicated and asymmetric random occurrences. As a consequence, a variety of models have been designed and investigated in the literature. The Lehmann Type I (L-I) and Type II models [1] are two of the simplest and most useful lifespan models elicited in the statistical history. The L-I model is widely addressed in support of the power function (PF) model in the literature. The L-I technique was used by Gupta et al. [2] to create a generalized form of the exponential distribution. Cordeiro et al. [3], on the other hand, suggested a dual transformation of the L-I technique and formed the Lehmann Type II (L-II) G class of models. Because of their simplistic closed CDFs, the L-I and L-II techniques have been widely employed to provide more adaptable and improved versions of classical models. A simplified version of the L-I distribution is the PF model. The PF model’s simplicity and utility have motivated numerous scholars to explore in depth its potential advancements and implementations in a variety of fields. Tavangar [4] employed dual generalized order statistics to characterize the PF distribution, while Ahsanullah et al. [5] used lower record values to explain the PF distribution. The assessment of the PF parameters using different evaluation methods was examined by Akhter [6], and the estimation of the PF parameters using the trimmed L moments was discussed by Shahzad et al. [7]. The beta-PF [8], weighted-PF [9], Weibull-PF [10], [11], odd generalized exponential PF [12], Transmuted Weibull-PF [13], and other famous extensions of the PF distribution are just a few examples. Arshad et al. [14], [15] recently formed the exponentiated-PF distribution, a finite bathtub shaped failure rate PF model using L-II class. Iqbal et al. [16] investigated a two-parameter model known as the new power function (NPF) model. The CDF and PDF of NPF model with scale $\left(\mu \right)$ and shape $\left(\lambda \right)$ are

$$\breve{G}\left(\left(t\right|\mu ,\lambda \right)=1-{\left(\frac{1-t}{1+\mu t}\right)}^{\lambda },$$ (1)

$$\breve{g}\left(\left(t\right|\mu ,\lambda \right)=\frac{\lambda \left(\mu +1\right){\left(1-t\right)}^{\lambda -1}}{{\left(1+\mu t\right)}^{\lambda +1}},-1<\mu <\infty ,\lambda >0,t\in \left(0,1\right)\text{.}$$ (2)

The primary goal of this investigation is to create a flexible extremely versatile failure rate model known as the modified Kies new power function (MKNPF) model. It may simulate hazard rates that are growing, decreasing, bathtub, and inverted-U and U-shaped. It offers some appealing features, such as a simple, closed-form PDF, CDF, and Quantile Function, all of which are straightforward to use. This model is simple to use from an application perspective. The properties of the model make it a good fit for actuarial data, biomedical life monitoring, and reliability implementations. The new model is designed to model the COVID-19 data. We have used COVID-19 real data from the United Kingdom and Italy to assess the model’s efficacy. For the United Kingdom and Italy, we used the COVID-19 daily death rate. For modeling COVID-19 data, many researchers, such as Sindhu et al. [17], [18], [19] used a new class of models.

The modified (reduced) Kies (MK) model was presented by Kumar and Dharmaja  [20] as a specific case of the Kies model. In [21] authors presented the exponentiated reduced Kies model and investigated some of its features. Dey et al. [22] used a progressive censoring strategy to develop recurrence relations for single and product moments of the MK model. Al-Babtain et al. [23] develop a novel family of generators termed as the modified Kies generalized (MK-G) on basis of the MK model. The MK-G model can be employed efficiently for analysis when given a baseline model. If $G\left(\left(t\right|\Theta \right)$ is the reference CDF for a parameter vector $\Theta$, then CDF of MK-G model is defined as

$$F\left(\left(t\right|\Psi \right)=1-exp\left\{-{\left(\frac{\breve{G}\left(\left(t\right|\Theta \right)}{1-\breve{G}\left(\left(t\right|\Theta \right)}\right)}^{\xi }\right\},\xi >0,$$ (3)

$\Psi =\left(\xi ,\Theta \right)$. The PDF of $\left(3\right)$ is

$$f\left(\left(t\right|\Psi \right)=\xi \frac{\breve{g}\left(\left(t\right|\Theta \right){\left[\breve{G}\left(\left(t\right|\Theta \right)\right]}^{\xi -1}}{{\left[1-\breve{G}\left(\left(t\right|\Theta \right)\right]}^{\xi +1}}exp\left\{-{\left(\frac{\breve{G}\left(\left(t\right|\Theta \right)}{1-\breve{G}\left(\left(t\right|\Theta \right)}\right)}^{\xi }\right\},\xi >0.$$ (4)

Parameter estimation is fundamental when studying any probability distribution. To optimize the parameters of any model, Maximum likelihood Estimation (MLE) is often used, due to its appealing properties. They are asymptotically consistent, unbiased, and normally distributed asymptotically (check [24]). Various estimating strategies for distributions have evolved over time, such as Least-squares estimation (LSE), maximum probability, Weighted least square estimation (WLSE), and L-moments approaches and minimal distance estimation (check [25], [26], [27], [28], [29]). The parameters of generalized power Weibull (GPW) model were determined using the maximum likelihood and maximum product spacing procedures in [30].

Furthermore, the MKNPF parameters are evaluated using three traditional estimation techniques MLE, LSE and WLSE. We have provided a thorough explanation for study of three approaches for guessing undetermined parameters of MKNPF model, also a study of their performance for different sample sizes in this article. We run thorough simulations to investigate behaviors of different estimators focused on bias and mean squared error because it is tough to match characteristics of different estimation strategies theoretically. The new model is an excellent competitor with certain well-known and trendy models like two-parameter modified Kies inverted Topp-Leone (MKITL) [31], modified Kies exponential (MKIEx) [32], and NPF [16] distributions, according to the results of this study. In the future, we want to investigate a new MKIF distribution implementation designed on a reduced sample (check Sindhu et al. [33], [34]) and also proposed a new two-component mixture model (see Sindhu et al. [35], [36], [37], [38], [39], [40], [41]) of three-parameter MKNPF distribution.

MKNPF distribution specifications

The CDF of MKNPF model is specified as

$$F\left(\left(t\right|\Psi \right)=1-exp\left\{-{\left(\frac{{\left(1+\mu t\right)}^{\lambda }-{\left(1-t\right)}^{\lambda }}{{\left(1-t\right)}^{\lambda }}\right)}^{\xi }\right\},t\in \left(0,1\right),\xi >0\text{,}$$ (5)

its corresponding PDF reduces to

$$f\left(\left(t\right|\Psi \right)=\frac{\xi \lambda \left(\mu +1\right){\left\{{\left(1+\mu t\right)}^{\lambda }-{\left(1-t\right)}^{\lambda }\right\}}^{\xi -1}}{{\left(1+\mu t\right)}^{1-\lambda }{\left(1-t\right)}^{\lambda \xi +1}}exp\left\{-{\left(\frac{{\left(1+\mu t\right)}^{\lambda }-{\left(1-t\right)}^{\lambda }}{{\left(1-t\right)}^{\lambda }}\right)}^{\xi }\right\},$$ (6)

where $\Psi =\left(\lambda ,\mu ,\xi \right)$, where $t\in \left(0,1\right)$, $\mu$ $>-1$, is a scale and $\lambda$, $\xi$ $>0$ are the two shape parameters, respectively. In the literature, terms like ”failure rate function”, is widely mentioned. This term is employed to indicate an element’s failure rate over a given time period ($t$) and mathematically formulated as $h\text{(}\left(t\right|$ $\Psi \text{)}=f\text{(}\left(t\right|$ $\Psi \text{)}/\text{[}1-F\text{(}\left(t\right|$ $\Psi \text{)]}$. The failure rate function is

$$h\left(\left(t\right|\Psi \right)=\frac{\xi \lambda \left(\mu +1\right){\left\{{\left(1+\mu t\right)}^{\lambda }-{\left(1-t\right)}^{\lambda }\right\}}^{\xi -1}}{{\left(1+\mu t\right)}^{1-\lambda }{\left(1-t\right)}^{\lambda \xi +1}},$$ (7)

an ideal mechanism in reliability study. The chance that a component will survive at time $t$ can be described as the reliability function $S\left(\left(t\right|\Psi \right)$. Analytically, it is characterized as $S\left(\left(t\right|\Psi \right)=1-F\text{(}\left(t\right|$ $\Psi \text{)}$, here, $S\left(\left(y\right|\Psi \right)$ functions of MKNPF$\left(\Psi \right)$ model is

$$S\left(\left(t\right|\Psi \right)=exp\left\{-{\left(\frac{{\left(1+\mu t\right)}^{\lambda }-{\left(1-t\right)}^{\lambda }}{{\left(1-t\right)}^{\lambda }}\right)}^{\xi }\right\}.$$ (8)

One of valuable reliability indicators is the CHRF. The CHRF is a measure of risk: higher the $H\left(\left(t\right|\Psi \right)$, elevate the risk of collapse by $t$-time.

$$H\left(\left(t\right|\Psi \right)={\int }_0^{t}h\left(\left(y\right|\Psi \right)dy=-log\left[S\left(\left(t\right|\Psi \right)\right].$$ (9)

$$H\left(\left(t\right|\Psi \right)={\left(\frac{{\left(1+\mu t\right)}^{\lambda }-{\left(1-t\right)}^{\lambda }}{{\left(1-t\right)}^{\lambda }}\right)}^{\xi }.$$ (10)

Mills ratio is defined by $M\text{(}\left(t\right|$ $\Psi \text{)}=S\text{(}\left(t\right|$ $\Psi \text{)}/f\text{(}\left(t\right|$ $\Psi \text{)}$. Mills ratio of $T$ is given by

$$M\left(\left(t\right|\Psi \right)=\frac{{\left(1+\mu t\right)}^{1-\lambda }{\left(1-t\right)}^{\lambda \xi +1}}{\xi \lambda \left(\mu +1\right){\left\{{\left(1+\mu t\right)}^{\lambda }-{\left(1-t\right)}^{\lambda }\right\}}^{\xi -1}}.$$ (11)

The odd function is defined by $O\text{(}\left(t\right|$ $\Psi \text{)}=F\text{(}\left(t\right|$ $\Psi \text{)}/S\text{(}\left(t\right|$ $\Psi \text{)}$. The odd function of $T$ is

$$O\left(\left(t\right|\Psi \right)=exp\left\{{\left(\frac{{\left(1+\mu t\right)}^{\lambda }-{\left(1-t\right)}^{\lambda }}{{\left(1-t\right)}^{\lambda }}\right)}^{\xi }\right\}-1.$$ (12)

The reverse hazard rate function (RHRF) is defined by $=f\text{(}\left(t\right|$ $\Psi \text{)}/F\text{(}\left(t\right|$ $\Psi \text{)}$. The RHRF of $T$ is given by

$$RHRF\left(\left(t\right|\Psi \right)=\frac{\xi \lambda \left(\mu +1\right){\left\{{\left(1+\mu t\right)}^{\lambda }-{\left(1-t\right)}^{\lambda }\right\}}^{\xi -1}}{{\left(1-t\right)}^{\lambda \xi +1}{\left(1+\mu t\right)}^{1-\lambda }exp\left[\left\{{\left(\frac{{\left(1+\mu t\right)}^{\lambda }-{\left(1-t\right)}^{\lambda }}{{\left(1-t\right)}^{\lambda }}\right)}^{\xi }\right\}-1\right]}.$$ (13)

Useful expansion of PDF of MKNPF

The power series and exponential function are

$${\left(1-t\right)}^{-\left(\varsigma +1\right)}=\sum _{k=0}^{\infty }\left(\genfrac{}{}{0ex}{}{\varsigma +k}{k}\right)t^k,\text{for }\left|t\right|<1.$$ (14)

and

$$e^{-t}=\sum _{\gamma =0}^{\infty }{\left(-1\right)}^{\gamma }\frac{t^{\gamma }}{\gamma !}.$$ (15)

Then, using (14), (15) to expand ${\left[1-\breve{G}\left(\left(t\right|\Theta \right)\right]}^{-\left(\xi +1\right)}$ and $exp\left\{-{\left(\frac{\breve{G}\left(\left(t\right|\Theta \right)}{1-\breve{G}\left(\left(t\right|\Theta \right)}\right)}^{\xi }\right\}$ respectively, it (4) follows as,

$$f\left(\left(t\right|\Psi \right)=\sum _{l.,m=0}^{+\infty }\frac{{\left(-1\right)}^l}{l!}\left(\genfrac{}{}{0ex}{}{\left(1+l\right)\xi +m}{m}\right)\xi \check{g}\left(\left(t\right|\mu ,\lambda \right){\left\{\breve{G}\left(\left(t\right|\mu ,\lambda \right)\right\}}^{\left(1+l\right)\xi +m-1}.$$ (16)

If $s\in {\mathbb{R}}^{+}$, and $\left|t\right|<1$, then it holds

$${\left(1-t\right)}^{\upsilon -1}=\sum _{i=0}^{+\infty }{\left(-1\right)}^p\left(\genfrac{}{}{0ex}{}{\upsilon -1}{p}\right)t^p.$$ (17)

Applying (17) to expand ${\left[1-{\left(\frac{1-t}{1+\mu t}\right)}^{\lambda }\right]}^{\left(1+l\right)\xi +m-1}$, we get PDF of MKNPF in (16) is with

$${\kappa }_{l,m,p}=\frac{{\left(-1\right)}^{l+p}\xi }{l!\left(p+1\right)}\left(\genfrac{}{}{0ex}{}{\left(1+l\right)\xi +m}{m}\right)\left(\genfrac{}{}{0ex}{}{\left(1+l\right)\xi +m-1}{p}\right),$$

$$f\left(\left(t\right|\Psi \right)=\sum _{l,m,p=0}^{+\infty }{\kappa }_{l,m,p}\frac{\lambda \left(p+1\right)\left(\mu +1\right){\left(1-t\right)}^{\lambda \left(p+1\right)-1}}{{\left(1+\mu t\right)}^{\lambda \left(p+1\right)+1}},$$ (18)

As seen in (18) MKNPF density can be represented as a linear conjunction of NPF densities. As a result, the features of MKNPF model can be extrapolated from those of NPF model. In the hereafter, the result in (18) will be used to calculate numerous mathematical features of the MKNPF distribution.

Shape

Fig. 1, Fig. 2, Fig. 3 show possible shapes for the MKNPF density, CDF and FR functions based on different parameter values. The potential shapes of the PDF corresponding to the parameter $\lambda$, that regulates the distribution’s scale, as well as the two shape parameters $\mu$ and $\xi$, which govern the distribution’s shapes, include growing, bathtub, symmetric, asymmetric, inverted U, decreasing, and J forms. Fig. 1(a–h) demonstrate such shapes. Fig. 2(a–h) also shows CDF shapes for the MKNPF model. The failure rate function (FRF) forms, which include rising, U, and bathtub shapes, are shown in Fig. 3(a–h). These adaptable FRF shapes are appropriate for both monotonic and non-monotonic hazard rate behaviors, which are most common in real-time scenarios. Non-stationary lifespan phenomena frequently exhibit these types of forms.

Fig. 1.

Variations of PDF of MKNPF along with $\mu$, $\lambda$ and $\xi$.

Fig. 2.

Fluctuations of CDF of MKNPF along with $\mu$, $\lambda$ and $\xi$.

Fig. 3.

Fluctuations of CDF of MKNPF with $\mu$, $\lambda$ and $\xi$.

Simulation

Hyndman and Fan [42] first proposed the notion of a quantile function (QF). The $qth$ QF of MKNPF is obtained by inverting the CDF (5). The MKNPF model can be easily simulated from (19). The generated variate having PDF (6) is

$$t\left(\left(q\right|\Psi \right)=\frac{{\left\{1+{\left[-log(1-q)\right]}^{\frac{1}{\xi }}\right\}}^{\frac{1}{\lambda }}-1}{\mu +{\left\{1+{\left[-log(1-q)\right]}^{\frac{1}{\xi }}\right\}}^{\frac{1}{\lambda }}},0<q<1.$$ (19)

As a consequence, the median as well as lower and upper quantiles, are computed as follows:

$$\tilde{T}=\frac{{\left\{1+{\left[log\left(2\right)\right]}^{\frac{1}{\xi }}\right\}}^{\frac{1}{\lambda }}-1}{\mu +{\left\{1+{\left[log\left(2\right)\right]}^{\frac{1}{\xi }}\right\}}^{\frac{1}{\lambda }}},$$ (20)

$$t\left(\left(0.25\right|\Psi \right)=\frac{{\left\{1+{\left[-log(0.75)\right]}^{\frac{1}{\xi }}\right\}}^{\frac{1}{\lambda }}-1}{\mu +{\left\{1+{\left[-log(0.75)\right]}^{\frac{1}{\xi }}\right\}}^{\frac{1}{\lambda }}},$$ (21)

$$t\left(\left(0.75\right|\Psi \right)=\frac{{\left\{1+{\left[-log(0.25)\right]}^{\frac{1}{\xi }}\right\}}^{\frac{1}{\lambda }}-1}{\mu +{\left\{1+{\left[-log(0.25)\right]}^{\frac{1}{\xi }}\right\}}^{\frac{1}{\lambda }}}.$$ (22)

The differentiation of (19) provides the corresponding quantile density function

$$t^{\prime }(q;\Psi )=\frac{\left(1+\mu \right){\left\{1+{\left[-log(1-q)\right]}^{\frac{1}{\xi }}\right\}}^{\frac{1}{\lambda }-1}{\left[-log(1-q)\right]}^{\frac{1}{\xi }-1}}{\lambda \xi (1-q){\left[\mu +{\left\{1+{\left[-log(1-q)\right]}^{\frac{1}{\xi }}\right\}}^{\frac{1}{\lambda }}\right]}^2}.$$ (23)

Skewness and Kurtosis

Eq. (19) with the following two formulas can be used to compute the Galton skewness coefficient, say $S_{\Psi }$, and Moors kurtosis, say $K_{\Psi }$, of the MKNPF distribution:

$$S_{\Psi }=\frac{Q\left(\left(0.75\right|\Psi \right)-2Q\left(\left(0.5\right|\Psi \right)+Q\left(\left(0.25\right|\Psi \right)}{IQR}.$$ (24)

and

$$K_{\Psi }=\frac{Q\left(\left(0.875\right|\Psi \right)-Q\left(\left(0.625\right|\Psi \right)+Q\left(\left(0.375\right|\Psi \right)-Q\left(\left(0.125\right|\Psi \right)}{IQR}.$$ (25)

These descriptive indicators, which are developed through quartiles and octiles, can offer more robust estimates than classical skewness and kurtosis metrics. Furthermore, $S_{\Psi }$ and $K_{\Psi }$ are less responsive to exceptions and perform better with inadequate moment models. Fig. 4 shows three-dimensional plots of possible $S_{\Psi }$ and $K_{\Psi }$ shapes for various values of $\mu$, $\lambda$ and $\xi$. At different levels of $\lambda$ The QF and quantile density function layouts are shown in Fig. 4. The model is noticed to be positively skewed shaped. Fig. 5 shows three-dimensional visualizations of the median function, skewness and kurtosis at various levels of $\lambda$. The stronger the change in the median curve, the lower the inputs of the parameters $\mu$ and all potential values of $\xi$. Also, when the $\mu$ approaches 6, the median function provides decreasing values. On the other hand there is a noticeable shift in the skewness trend along $\mu$ at lesser characteristics of $\xi$, but as $\xi$ rises, it comes up to nearly 0.1. As $\xi$ increases, the extent of peakedness of the model increases and may also be platykurtic.

Fig. 4.

Variations of QF and quantile density function of MKNPF with $\xi$ and $q$ at different extents of $\lambda$ with $\mu =0.7$.

Fig. 5.

Fluctuation of $\tilde{T}$, skewness and kurtosis of MKNPF with $\mu$ and $\xi$ at different extents of $\lambda$.

Moments

Moments are utilized in statistics to explain the different features of a model. The central tendency, skewness, dispersion, and kurtosis of the model can all be examined using moments.

Theorem 1

If $\tilde{T}$ MKNPF $\left(\Psi \right)$ , then $r\text{-}th$ moment ${\acute{\mu }}_r$ of $T$ is

$${\acute{\mu }}_r=\lambda \left(p+1\right)\left(\mu +1\right)\sum _{l,m,p=0}^{+\infty }\sum _{i=0}^{+\infty }{\kappa }_i{\kappa }_{l,m,p}B\left\{r+i+1,\lambda \left(p+1\right)\right\}.$$ (26)

where $B(\rho ,\sigma )={\int }_0^{\infty }{z}^{\rho -1}\left(1-z^{\sigma -1}\right)dz$ , and $B\left(\rho ,\sigma \right)=\frac{\Gamma \rho \Gamma \sigma }{\Gamma \left(\rho +\sigma \right)}$ .

Proof

(18) can be used to write ${\acute{\mu }}_r$ as

$${\acute{\mu }}_r={\int }_0^1{t}^{r}d{F}_t\left(\left(t\right|\Psi \right);r=1,2,\dots$$ (27)

$${\acute{\mu }}_r={\int }_0^1\sum _{l,m,p=0}^{+\infty }{t}^r{\kappa }_{l,m,p}\frac{\lambda \left(p+1\right)\left(\mu +1\right){\left(1-t\right)}^{\lambda \left(p+1\right)-1}}{{\left(1+\mu t\right)}^{\lambda \left(p+1\right)+1}}dt,$$ (28)

After some algebra, we get

$${\acute{\mu }}_r=\lambda \left(p+1\right)\times \left(\mu +1\right)\sum _{i=0}^{+\infty }\sum _{l,m,p=0}^{+\infty }{\left(-1\right)}^i{\mu }^i\left(\genfrac{}{}{0ex}{}{\lambda \left(p+1\right)+i}{i}\right){\kappa }_{l,m,p}{\int }_0^1{t}^{r+i}{\left(1-t\right)}^{\lambda \left(p+1\right)-1}dt,$$

The ultimate expression of ${\acute{\mu }}_r$ is as follows, based on quick arithmetic on the last expression.

$${\acute{\mu }}_r=\lambda \left(p+1\right)\left(\mu +1\right)\sum _{l,m,p=0}^{+\infty }\sum _{i=0}^{+\infty }{\kappa }_i{\kappa }_{l,m,p}B\left\{r+i+1,\lambda \left(p+1\right)\right\},$$ (29)

where ${\kappa }_i={\left(-1\right)}^i{\mu }^i\left(\genfrac{}{}{0ex}{}{\lambda \left(p+1\right)+i}{i}\right)$.

The moment formula (29) can help you come up with some valuable statistical metrics. In (29), for example, the mean of $T$ follows with $r=1$. The negative moment of $T$ can be simply determined by substituting $r$ with $-\tau$ in (29).

Remark 1

The Moment Generating Function (MGF) is commonly employed in model characterization. The MGF of MKNPF model using the Maclaurin series expansion of an exponential function is mentioned as

$$M_t\left(\left(d\right|\Psi \right)=E\left(e^{dt}\right)=\sum _{r=0}^{+\infty }\frac{d^r}{r!}{\acute{\mu }}_r$$ (30)

Substituting by (29) into (30), we get

$$M_t\left(\left(d\right|\Psi \right)=\lambda \left(p+1\right)\left(\mu +1\right)\sum _{r=0}^{+\infty }\frac{d^r}{r!}\sum _{l,m,p=0}^{+\infty }\sum _{i=0}^{+\infty }{\kappa }_i{\kappa }_{l,m,p}B\left\{r+i+1,\lambda \left(p+1\right)\right\}.$$ (31)

Proposition 1

Suppose $T$ be a random variable following MKNPF model, then central moments is

$${\mu }_s\left(\left(t\right|\Psi \right)=\lambda \left(p+1\right)\left(\mu +1\right)\left(\sum _{j=0}^s\sum _{l,m,p=0}^{+\infty }\sum _{i=0}^{+\infty }{\kappa }_i{\kappa }_j{\kappa }_{l,m,p}B\left\{s-j+i+1,\lambda \left(p+1\right)\right\}\right),$$ (32)

${\kappa }_j={\left(-1\right)}^j{\mu }^j\left(\genfrac{}{}{0ex}{}{s}{j}\right),{\kappa }_i={\left(-1\right)}^i{\mu }^i\left(\genfrac{}{}{0ex}{}{\lambda \left(p+1\right)+i}{i}\right),B(\rho ,\sigma )={\int }_0^{\infty }{z}^{\rho -1}\left(1-z^{\sigma -1}\right)dz=B\left(\rho ,\sigma \right)$ , where $B\left(\rho ,\sigma \right)=\frac{\Gamma \rho \Gamma \sigma }{\Gamma \left(\rho +\sigma \right)}$ .

$${\mu }_s\left(\left(t\right|\Psi \right)={\int }_0^{\infty }{\left(t-{\mu }_t\right)}^{s}f\left(\left(t\right|\Psi \right)dt$$ (33)

Substituting (18) into (33) and after simple arithmetic, we get

$${\mu }_s\left(\left(t\right|\Psi \right)=\lambda \left(p+1\right)\left(\mu +1\right)\left(\sum _{j=0}^s\sum _{l,m,p=0}^{+\infty }\sum _{i=0}^{+\infty }{\kappa }_i{\kappa }_j{\kappa }_{l,m,p}B\left\{s-j+i+1,\lambda \left(p+1\right)\right\}.\right)$$ (34)

Remark 2

The moment formula (34) facilitates in the formulation of statistical measures that are helpful. In (34), for example, the variance of $T$ follows with $s=2$. Furthermore, by putting $s=2,3,4$ in (34), the skewness (${\vartheta }_1={\mu }_3^2/{\mu }_2^3$) and kurtosis (${\vartheta }_2={\mu }_4^2/{\mu }_2^2$) of $T$ can well be calculated using (34).

Characteristic function

The CF of random variable $T$ is

$$\breve{\Delta }\left(\left(t\right|\Psi \right)={\int }_0^1{e}^{idt}dF\left(\left(t\right|\Psi \right).$$ (35)

After using exponential series, we have

$$\breve{\Delta }\left(\left(t\right|\Psi \right)=\sum _{r=0}^{\infty }\frac{{\left(id\right)}^r}{r!}{\int }_0^1{t}^{r}dF\left(\left(t\right|\Psi \right).$$ (36)

Hence, we obtain

$$\breve{\Delta }\left(\left(t\right|\Psi \right)=\lambda \left(p+1\right)\left(\mu +1\right)\sum _{r=0}^{\infty }\sum _{l,m,p=0}^{+\infty }\sum _{i=0}^{+\infty }\frac{{\left(id\right)}^r}{r!}{\kappa }_i{\kappa }_{l,m,p}B\left\{r+i+1,\lambda \left(p+1\right)\right\}.$$ (37)

Factorial generating function

The FGF of MKNPF is

$$F_t\left(\left(\delta t\right|\Psi \right)={\int }_0^1{e}^{log{\left(1+\delta \right)}^t}dF\left(\left(t\right|\Psi \right),=\sum _{r=0}^{\infty }\frac{{\left(log\left(1+\delta \right)\right)}^r}{r!}{\int }_0^1{t}^{r}dF\left(\left(t\right|\Psi \right),$$ (38)

so, we can put together the integral as follows:

$$F_t\left(\left(\delta t\right|\left(t\right|\Psi \right)=\lambda \left(p+1\right)\left(\mu +1\right)\sum _{r=0}^{\infty }\frac{{\left(log\left(1+\delta \right)\right)}^r}{r!}\times \sum _{l,m,p=0}^{+\infty }\sum _{i=0}^{+\infty }{\kappa }_i{\kappa }_{l,m,p}B\left\{r+i+1,\lambda \left(p+1\right)\right\}.$$ (39)

Stochastic ordering

In this subsection, we compare the MKNP${\mathrm{F}}_1\left(\left(t\right|{\Psi }_1\right)$ and the MKNPF₂ $\left(\left(t\right|{\Psi }_2\right)$ with respect to stochastic ordering information. Assume that $T_1$ and $T_2$ be two random variables with reliability functions, cdfs, and pdfs $S_1\left(\left(t\right|{\Psi }_1\right)$ and $S_2\left(\left(t\right|{\Psi }_2\right)$; $F_1\left(\left(t\right|{\Psi }_1\right)$ and $F_2\left(\left(t\right|{\Psi }_2\right)$; and $f_1\left(\left(t\right|{\Psi }_1\right)$ and $f_2\left(\left(t\right|{\Psi }_2\right)$, respectively, where ${\Psi }_1=\left({\mu }_1;{\xi }_1;{\lambda }_1\right)$ and ${\Psi }_2=\left({\mu }_2;{\xi }_2;{\lambda }_2\right)$ respectively. A random variable $T_1\le T_2$ in the following ordering (see [43]), if: (i) Stochastic order $\left(T_1{\preccurlyeq }_{st}{T}_2\right)$ if $S_1\left(\left(t\right|{\Psi }_1\right)\preccurlyeq S_2\left(\left(t\right|{\Psi }_2\right)$ $\forall$ $t$; (ii) Hazard rate order $\left(T_1{\preccurlyeq }_{hr}{T}_2\right)$ if $h_1\left(\left(t\right|{\Psi }_1\right)\succcurlyeq h_2\left(\left(t\right|{\Psi }_2\right)$ $\forall$ $t$; (iii) likelihood ratio order $X{\preccurlyeq }_{lr}T$ if $\frac{f_{T_1}\left(t\right)}{f_{T_2}\left(t\right)}$ decreases in $t$. Among the various partial orderings discussed above, the following chain of implications follows.

$$T_1{\preccurlyeq }_{lr}{T}_2\Rightarrow T_1{\preccurlyeq }_{hr}{T}_2\Rightarrow T_1{\preccurlyeq }_{st}{T}_2.$$ (40)

As stated in the following theorem, MKNPF$\left(\left(t\right|\Psi \right)$ models are ranked according to the strongest ”likelihood ratio” ordering.

Theorem 2

Let $T_1\sim MKNP{F}_1\left({\mu }_1;{\xi }_1;{\lambda }_1\right)$ , and $T_2\sim MKNP{F}_2\left({\mu }_2;{\xi }_2;{\lambda }_2\right)$ , if ${\mu }_1={\mu }_2$ , ${\lambda }_1={\lambda }_2$ and ${\xi }_1\le {\xi }_2$ , then $T_1{\preccurlyeq }_{lr}{T}_2$ $\left(T_1{\preccurlyeq }_{hr}{T}_2,T_1{\preccurlyeq }_{st}{T}_2\right)$ in all three cases exist.

Proof

It is sufficient to show $\frac{f_{T_1}\left(t\right)}{f_{T_2}\left(t\right)}$ is a decreasing function of $t$; the likelihood ratio is

$$\frac{f_{MKNP{F}_1}\left(t\right)}{f_{MKNP{F}_2}\left(t\right)}=\frac{{\xi }_1}{{\xi }_2}{\left(1-t\right)}^{\lambda \left({\xi }_2-{\xi }_1\right)}{\left[{\left(1+\mu t\right)}^{\lambda }-{\left(1-t\right)}^{\lambda }\right]}^{{\xi }_1-{\xi }_2}\times e^{-{\left[-1+{\left(1+\mu t\right)}^{\lambda }{\left(1-t\right)}^{-\lambda }\right]}^{{\xi }_1}+{\left[-1+{\left(1+\mu t\right)}^{\lambda }{\left(1-t\right)}^{-\lambda }\right]}^{{\xi }_2}}.$$ (41)

Thus if ${\mu }_1={\mu }_2$, ${\lambda }_1={\lambda }_2$, and ${\xi }_1\le {\xi }_2$, then

$$\frac{d}{dt}\left[\frac{f_{MKNEP1}\left(t\right)}{f_{MKNEP2}\left(t\right)}\right]=\frac{1}{{\xi }_2}{e}^{-{\left[-1+{\left(1+\mu t\right)}^{\lambda }{\left(1-t\right)}^{-\lambda }\right]}^{{\xi }_1}+{\left[-1+{\left(1+\mu t\right)}^{\lambda }{\left(1-t\right)}^{-\lambda }\right]}^{{\xi }_2}}{\left(1-t\right)}^{-1-\lambda \left({\xi }_1-{\xi }_2\right)}\lambda \left(1+\mu \right){\left(1+\mu t\right)}^{-1+\lambda }{\left[{\left(1+\mu t\right)}^{\lambda }-{\left(1-t\right)}^{\lambda }\right]}^{-1+{\xi }_1-{\xi }_2}\times {\xi }_1\left\{{\xi }_1-{\left[-1+{\left(1+\mu t\right)}^{\lambda }{\left(1-t\right)}^{-\lambda }\right]}^{{\xi }_1}{\xi }_1\right)\left(\left(+{\left[-1+{\left(1+\mu t\right)}^{\lambda }{\left(1-t\right)}^{-\lambda }\right]}^{{\xi }_2}{\xi }_2\right\}\le 0.\right)$$ (42)

Hence it shows that $T_1{\preccurlyeq }_{lr}{T}_2$, and according to $\left(40\right)$ these both are $T_1{\preccurlyeq }_{hr}{T}_2$, $T_1{\preccurlyeq }_{st}{T}_2$ also hold. □

The certain estimation techniques with simulation

In this part, we focus on the three techniques for estimating MKNPF model parameters: MLE ordinary LSE, and WLSE techniques. Simulation studies are used to investigate the effectiveness of certain techniques. From now, $t_1,t_2,\dots ,t_n$ indicate $n$ observed characteristics from $T$ and their ascending ordering values $t_{\left(1\right)}\le t_{\left(2\right)}\le \cdots \le t_{\left(n\right)}$.

MLE approach

The maximum likelihood strategy is the most extensively used methodology for estimating parameters. Let $T_1,T_2,\dots ,T_n$ be a random sample and the corresponding observed values, $t_1,t_2,\dots ,t_n$ from MKNPF model with parameter vector $\Psi =\left(\mu ,\lambda ,\xi \right)$. Then the joint probability function $L\left(\left(\mathbf{t}\right|\Psi \right)={\prod }_{i=1}^{n}f\text{(}\left(t\right|$ $\Psi \text{)}$ of $T_1,T_2,\dots ,T_n$ as a log-likelihood function is

$$l\left(\left(\mathbf{t}\right|\Psi \right)=ln\prod _{i=1}^{n}f(t_i;\Psi ),=nlog\left(\xi \right)+nlog\left(\lambda \right)+nlog\left(\mu +1\right)+\sum _{i=1}^{n}log\left[{\left\{{\left(1+\mu t_i\right)}^{\lambda }-{\left(1-t_i\right)}^{\lambda }\right\}}^{\xi -1}\right]+\sum _{i=1}^{n}log{\left(1+\mu t_i\right)}^{-\left(1-\lambda \right)}+\sum _{i=1}^{n}log{\left(1-t_i\right)}^{-\left(\lambda \xi +1\right)}-\sum _{i=1}^n{\left(\frac{{\left(1+\mu t_i\right)}^{\lambda }-{\left(1-t_i\right)}^{\lambda }}{{\left(1-t_i\right)}^{\lambda }}\right)}^{\xi },$$ (43)

$$\frac{\partial l\left(\left(\mathbf{t}\right|\Psi \right)}{\partial \xi }=\frac{n}{\xi }-\lambda \sum _{i=1}^{n}log\left(1-t_i\right)+\sum _{i=1}^{n}log\left(1+\mu t_i\right)+\sum _{i=1}^{n}log\left\{{\left(1+\mu t_i\right)}^{\lambda }-{\left(1-t_i\right)}^{\lambda }\right\}\left(-\sum _{i=1}^n{\left(\frac{{\left(1+\mu t_i\right)}^{\lambda }-{\left(1-t_i\right)}^{\lambda }}{{\left(1-t_i\right)}^{\lambda }}\right)}^{\xi }log\left(\frac{{\left(1+\mu t_i\right)}^{\lambda }-{\left(1-t_i\right)}^{\lambda }}{{\left(1-t_i\right)}^{\lambda }}\right)\right),$$ (44)

$$\frac{\partial l\left(\left(\mathbf{t}\right|\Psi \right)}{\partial \lambda }=\frac{n}{\lambda }-\xi \sum _{i=1}^{n}log\left(1-t_i\right)+\sum _{i=1}^{n}log\left(1+\mu t_i\right)-\xi \sum _{i=1}^n{\left(\frac{{\left(1+\mu t_i\right)}^{\lambda }-{\left(1-t_i\right)}^{\lambda }}{{\left(1-t_i\right)}^{\lambda }}\right)}^{\xi -1}\left(+\left(\xi -1\right)\sum _{i=1}^n\frac{{\left(1+\mu t_i\right)}^{\lambda }log\left(1+\mu t_i\right)-{\left(1-t_i\right)}^{\lambda }log\left(1-t_i\right)}{\left\{{\left(1+\mu t_i\right)}^{\lambda }-{\left(1-t_i\right)}^{\lambda }\right\}}\right)\times \left[\left\{{\left(1-t_i\right)}^{-\lambda }log\left(1-t_i\right)\right\}{\left(1+\mu t_i\right)}^{\lambda }-\xi {\left(1-t_i\right)}^{-\lambda }\right)\left(\times {\left(1+\mu t_i\right)}^{\lambda }log\left(1+\mu t_i\right)\right],$$ (45)

$$\frac{\partial l\left(\left(\mathbf{t}\right|\Psi \right)}{\partial \mu }=\frac{n}{\left(\mu +1\right)}+\left(\lambda -1\right)\sum _{i=1}^n\frac{t_i}{1+\mu t_i}+\left(\xi -1\right)\sum _{i=1}^n\frac{\lambda t_i{\left(1+\mu t_i\right)}^{\lambda -1}}{{\left(1+\mu t_i\right)}^{\lambda }-{\left(1-t_i\right)}^{\lambda }}-\lambda \xi \sum _{i=1}^n{t}_i{\left(1-t_i\right)}^{-\lambda }{\left(1+\mu t_i\right)}^{\lambda -1}{\left(\frac{{\left(1+\mu t_i\right)}^{\lambda }-{\left(1-t_i\right)}^{\lambda }}{{\left(1-t_i\right)}^{\lambda }}\right)}^{\xi -1}.$$ (46)

The MLEs of the model parameters are achieved by solving the equations above simultaneously. The bias and mean square error of the MLEs, with few exceptions, decrease as sample sizes rise, which fits the common criteria of asymptotic properties of MLEs, according to the simulation study (Section ‘Numerical and Graphical analysis’).

Method of ordinary and weighted least squares

For estimating undetermined parameters, the LS and WLS techniques are extensively used [44]. The two methods for evaluating the parameters of MKNPF model are covered here. The LSEs of parameter vector $\Psi =\left(\mu ,\lambda ,\xi \right)$, can be achieved by minimizing the following

$$LS\left(\Psi \right)=\sum _{i=1}^n{\left[F(t_{\left(i\right)};\Psi )-\frac{i}{n+1}\right]}^2,$$ (47)

$$LS\left(\Psi \right)=\sum _{i=1}^n{\left[\left\{1-exp\left[-{\left(\frac{{\left(1+\mu t_{\left(i\right)}\right)}^{\lambda }-{\left(1-t_{\left(i\right)}\right)}^{\lambda }}{{\left(1-t_{\left(i\right)}\right)}^{\lambda }}\right)}^{\xi }\right]\right\}-\frac{i}{n+1}\right]}^2,$$ (48)

with respect to unknown parameters of the model . These can be extracted equivalently by solving: $\partial LS\left(\Psi \right)/\partial \mu =0$, $\partial LS\left(\Psi \right)/\partial \lambda =0$, and $LS\left(\Psi \right)/\partial \xi =0$ where

$$\frac{\partial LS\left(\Psi \right)}{\partial \mu }=2\sum _{i=1}^n{\varpi }_i^{\left(1\right)}\left(\Psi \right)\left[\left\{1-exp\left[-{\left(\frac{{\left(1+\mu t_{\left(i\right)}\right)}^{\lambda }-{\left(1-t_{\left(i\right)}\right)}^{\lambda }}{{\left(1-t_{\left(i\right)}\right)}^{\lambda }}\right)}^{\xi }\right]\right\}-\frac{i}{n+1}\right],$$ (49)

$$\frac{\partial LS\left(\Psi \right)}{\partial \lambda }=2\sum _{i=1}^n{\varpi }_i^2\left(\Psi \right)\left[\left\{1-exp\left[-{\left(\frac{{\left(1+\mu t_{\left(i\right)}\right)}^{\lambda }-{\left(1-t_{\left(i\right)}\right)}^{\lambda }}{{\left(1-t_{\left(i\right)}\right)}^{\lambda }}\right)}^{\xi }\right]\right\}-\frac{i}{n+1}\right],$$ (50)

$$\frac{\partial LS\left(\Psi \right)}{\partial \xi }=2\sum _{i=1}^n{\varpi }_i^3\left(\Psi \right)\left[\left\{1-exp\left[-{\left(\frac{{\left(1+\mu t_{\left(i\right)}\right)}^{\lambda }-{\left(1-t_{\left(i\right)}\right)}^{\lambda }}{{\left(1-t_{\left(i\right)}\right)}^{\lambda }}\right)}^{\xi }\right]\right\}-\frac{i}{n+1}\right],$$ (51)

$W\left(t_{\left(i\right)};\Psi \right)=exp\left[-{\left(\frac{{\left(1+\mu t_{\left(i\right)}\right)}^{\lambda }-{\left(1-t_{\left(i\right)}\right)}^{\lambda }}{{\left(1-t_{\left(i\right)}\right)}^{\lambda }}\right)}^{\xi }\right]$ and

$${\varpi }_i^{\left(1\right)}\left(\Psi \right)=W\left(t_{\left(i\right)};\Psi \right)\frac{t_{\left(i\right)}\lambda \xi {\left(1+\mu t_{\left(i\right)}\right)}^{\lambda -1}{\left\{-1+{\left(1-t_{\left(i\right)}\right)}^{-\lambda }{\left(1+\mu t_{\left(i\right)}\right)}^{\lambda }\right\}}^{\xi }}{{\left(1-t_{\left(i\right)}\right)}^{\lambda }-{\left(1+\mu t_{\left(i\right)}\right)}^{\lambda }},$$ (52)

$${\varpi }_i^{\left(2\right)}\left(\Psi \right)=W\left(t_{\left(i\right)};\Psi \right)\times \frac{{\left(1+\mu t_{\left(i\right)}\right)}^{\lambda }{\left\{-1+{\left(1-t_{\left(i\right)}\right)}^{-\lambda }{\left(1+\mu t_{\left(i\right)}\right)}^{\lambda }\right\}}^{\xi }\xi \left(log\left(1-t_{\left(i\right)}\right)-log{\left(1+\mu t_{\left(i\right)}\right)}^{\lambda }\right)}{{\left(1-t_{\left(i\right)}\right)}^{\lambda }-{\left(1+\mu t_{\left(i\right)}\right)}^{\lambda }}$$ (53)

$${\varpi }_i^{\left(3\right)}\left(\Psi \right)=W\left(t_{\left(i\right)};\Psi \right)\times {\left\{-1+{\left(1-t_{\left(i\right)}\right)}^{-\lambda }{\left(1+\mu t_{\left(i\right)}\right)}^{\lambda }\right\}}^{\xi }log(-1+{\left(1-t_{\left(i\right)}\right)}^{-\lambda }{\left(1+\mu t_{\left(i\right)}\right)}^{\lambda }).$$ (54)

The WLSE of MKNPF model parameters, on the other hand, can be got by minimizing

$$WLS\left(\Psi \right)=\sum _{i=1}^n\frac{{(n+1)}^2(n+2)}{i(n-i+1)}{\left[F(t_{\left(i\right)};\Psi )-\frac{i}{n+1}\right]}^2,$$ (55)

$$=\sum _{i=1}^n\frac{{(n+1)}^2(n+2)}{i(n-i+1)}\times \sum _{i=1}^n{\left[\left\{1-exp\left[-{\left(\frac{{\left(1+\mu t_{\left(i\right)}\right)}^{\lambda }-{\left(1-t_{\left(i\right)}\right)}^{\lambda }}{{\left(1-t_{\left(i\right)}\right)}^{\lambda }}\right)}^{\xi }\right]\right\}-\frac{i}{n+1}\right]}^2.$$ (56)

The estimates can also be obtained by solving: $\partial WLS\left(\Psi \right)/\partial \mu =0$, $\partial WLS\left(\Psi \right)/\partial \lambda =0$ and $\partial WLS\left(\Psi \right)/\partial \xi =0$ where

$$\frac{\partial WLS\left(\Psi \right)}{\partial \mu }=2\sum _{i=1}^n\frac{{(n+1)}^2(n+2)}{i(n-i+1)}{\varpi }_i^{\left(1\right)}\left(\Psi \right)\times \left[\left\{1-exp\left[-{\left(\frac{{\left(1+\mu t_{\left(i\right)}\right)}^{\lambda }-{\left(1-t_{\left(i\right)}\right)}^{\lambda }}{{\left(1-t_{\left(i\right)}\right)}^{\lambda }}\right)}^{\xi }\right]\right\}-\frac{i}{n+1}\right],$$ (57)

$$\frac{\partial WLS\left(\Theta \right)}{\partial \lambda }=2\sum _{i=1}^n\frac{{(n+1)}^2(n+2)}{i(n-i+1)}{\varpi }_i^{\left(2\right)}\left(\Psi \right)\times \left[\left\{1-exp\left[-{\left(\frac{{\left(1+\mu t_{\left(i\right)}\right)}^{\lambda }-{\left(1-t_{\left(i\right)}\right)}^{\lambda }}{{\left(1-t_{\left(i\right)}\right)}^{\lambda }}\right)}^{\xi }\right]\right\}-\frac{i}{n+1}\right],$$ (58)

$$\frac{\partial WLS\left(\Theta \right)}{\partial \xi }=2\sum _{i=1}^n\frac{{(n+1)}^2(n+2)}{i(n-i+1)}{\varpi }_i^{\left(3\right)}\left(\Psi \right)\times \left[\left\{1-exp\left[-{\left(\frac{{\left(1+\mu t_{\left(i\right)}\right)}^{\lambda }-{\left(1-t_{\left(i\right)}\right)}^{\lambda }}{{\left(1-t_{\left(i\right)}\right)}^{\lambda }}\right)}^{\xi }\right]\right\}-\frac{i}{n+1}\right],$$ (59)

where ${\varpi }_i^{\left(k\right)}\left(\Psi \right)$, $k=1,2,3$. are given in (52–54). A simulation study (see the next section) reveals that all parameter combinations have a downward bias under LSE and WLSE. In particular, variation in bias and mean square error is observed in all parameter combinations that approaches to zero. Such reflections can be found in Table 2, Table 3.

Table 2.

Average values of Biases and MSEs values of LSEs from Simulation of the MKNPF distribution for $\left(\mu =0.95,\lambda =1.15,\xi =2.5\right)$.

$n$	Bias $\left(\hat{\mu }\right)$	Bias $\left(\hat{\lambda }\right)$	Bias $\left(\hat{\xi }\right)$	MSE $\left(\hat{\mu }\right)$	MSE $\left(\hat{\lambda }\right)$	MSE $\left(\hat{\xi }\right)$
30	1.0238	0.3900	0.0860	8.5295	1.0509	0.3548
65	0.0874	0.2665	0.0886	4.8184	0.7614	0.2571
100	0.7270	0.2946	0.0457	2.9078	0.5108	0.1845
135	0.6304	0.1963	0.0625	2.4030	0.4818	0.1524
170	0.5532	0.2289	0.0478	1.7104	0.4140	0.1450
205	0.4032	0.1767	0.0475	1.6661	0.3590	0.1204
310	0.4010	0.1184	0.0486	1.5938	0.2895	0.0891
400	028588	0.1628	0.0148	1.5104	0.2689	0.0746

Table 3.

Average values of Biases and MSEs values of WLSEs from Simulation of the MKNPF distribution for $\left(\mu =0.95,\lambda =1.15,\xi =2.5\right)$.

$n$	Bias $\left(\hat{\mu }\right)$	Bias $\left(\hat{\lambda }\right)$	Bias $\left(\hat{\xi }\right)$	MSE $\left(\hat{\mu }\right)$	MSE $\left(\hat{\lambda }\right)$	MSE $\left(\hat{\xi }\right)$
30	1.3919	0.5147	0.1429	14.0451	1.5953	0.4569
65	1.1370	0.4113	0.1538	9.1774	1.1553	0.3538
100	1.0068	0.2739	0.0866	4.2017	0.7628	0.2411
135	0.7836	0.2892	0.0955	3.3592	0.7398	0.2091
170	0.7462	0.2262	0.0733	2.6377	0.5784	0.1894
205	0.7233	0.2033	0.0645	2.3582	0.5227	0.1566
310	0.5538	0.1594	0.0539	2.1982	0.3887	0.1226
400	0.3443	0.1493	0.0243	2.1334	0.3687	0.0950

Numerical and graphical analysis

The ML, LS, and WLS estimators of the MKNPF model are extremely difficult to obtain, as shown in the prior section. As a result, a simulation experiment is conducted to assess the trend of estimates utilizing various metrics such as mean square errors (MSEs) and average bias values, and also their asymptotic behavior for finite samples.

To evaluate the finite sample behavior of MLEs, LSEs, and WLSEs, we can perform simulation experiments numerically and graphically. The decision has been made using the given algorithm:

1. Generate a thousand samples of size $n$ from (6). QF accomplished all of the work and gleaned the data from a uniform distribution.

2. The Set $:\left(0.95,1.15,2.5\right)$ of true parameter values $\mu ,\lambda$ and $\xi$ is employed . The simulated and theoretical model for true parameter values of $\mu ,\lambda$ and $\xi$ is shown in Fig. 6.

Fig. 6.

Simulated model for true parameter values of $\mu ,\lambda$ and $\xi$.

3. Compute the values for 5000 samples, say $\left({\check{\mu }}_k,{\check{\lambda }}_k,{\check{\xi }}_k\right)$ for $k=1,2,\dots ,5000$.

4. Appraise average bias values and MSEs. These targets are acquired with following formulas:

$$Bia{s}_{\Psi }\left(n\right)=\frac{1}{5000}\sum _{i=1}^{5000}\left({\check{\Psi }}_i-\Psi \right),$$

$$MS{E}_{\Psi }\left(n\right)=\frac{1}{5000}\sum _{i=1}^{5000}{\left({\check{\Psi }}_i-\Psi \right)}^2,$$

where $\Psi =\left(\lambda ,\mu ,\xi \right)$.

5. These processes have been replicated with the defined parameters for MLEs, LSEs, and WLSEs for $n=30,35,\dots ,400$. The bias ${}_{\Psi }$ ($n$) and MSE${}_{\Psi }$ ($n$) have both been computed. We utilized optim function of R to assess the quality of estimates. Table 1, Table 2, Table 3 and Fig. 7 illustrate the findings of the simulations. These biases and MSEs fluctuate with respect to $n$ in Fig. 7 (left panels and right panels).

Table 1.

Average values of Biases and MSEs values of MLEs from Simulation of the MKNPF distribution for $\left(\mu =0.95,\lambda =1.15,\xi =2.5\right)$.

$n$	Bias $\left(\hat{\mu }\right)$	Bias $\left(\hat{\lambda }\right)$	Bias $\left(\hat{\xi }\right)$	MSE $\left(\hat{\mu }\right)$	MSE $\left(\hat{\lambda }\right)$	MSE $\left(\hat{\xi }\right)$
30	0.4407	0.99860	−0.1381	0.3440	1.5420	0.3524
65	0.2801	1.03936	−0.1451	0.1895	1.2529	0.2076
100	0.1540	0.94022	−0.1187	0.1327	0.9886	0.0852
135	0.1506	0.82407	−0.0867	0.0852	0.7575	0.0316
170	0.1695	0.73511	−0.0613	0.0734	0.5918	0.0263
205	0.1838	0.64434	−0.0571	0.0719	0.4400	0.0262
310	0.2123	0.60378	−0.0695	0.0596	0.3827	0.0188
400	0.2793	0.56925	−0.1170	0.0414	0.3425	0.0312

Fig. 7.

Variation of Bias and MSE of $\check{\mu }$, $\check{\lambda }$ and $\check{\xi }$.

Because as $n$ increases, the bias approaches zero, we may infer that estimators exhibit the attribute of asymptotic unbiasedness. Meanwhile, the trend in the MSE indicates consistency because the error approaches zero as $n$ rises.

Conclusions on the Simulation Results

The outcomes of the study are interpreted through graphs and tables as described in the results and discussion. The main findings of study can be stated as follows:

• •The biases of $\check{\mu },\check{\lambda }$ and $\check{\xi }$ decrease as $n$ rises in all estimating methods.
• •The biases of and are relatively positive for MLEs, however there exist negative biases for $\check{\xi }$.
• •The MLEs, LSEs and WLSEs are overestimated however, MLEs of $\check{\xi }$ are underestimated (see left panel of Fig. 7).
• •The LSE and WLSE have analogous results, while the MLE has a slightly different result.
• •It is noticed that Bias of ${\check{\mu }}_{MLE}<$ Bias of ${\check{\mu }}_{LSE}<$ Bias of ${\check{\mu }}_{WLSE}$, Bias of ${\check{\lambda }}_{MLE}>$ Bias of ${\check{\mu }}_{WLSE}>$ Bias of ${\check{\mu }}_{LSE}$ and ${\check{\xi }}_{MLE}<$ Bias of ${\check{\xi }}_{LSE}<$ Bias of ${\check{\xi }}_{WLSE}$.
• •Generally, the smallest MSEs of $\check{\mu }$ and $\check{\xi }$ are noticed under MLE, although the least MSE of $\check{\lambda }$ is recorded under LSE(check right panel of Fig. 7).
• •As shown in right panel of Fig. 7, the maximum likelihood technique of estimation outperforms alternative approaches in terms of MSE.
• •The LSE is usually the next best estimator, followed by MLE in other situations. According to Fig. 7, when the $n$ grows, all bias and MSE plots for all parameters eventually reach zero. That highlights the accuracy of various estimation techniques.

Real data practices

In this portion, the MKNPF model’s usefulness for two real data sets is presented. The MKITL (modified Kies inverted Topp-Leone) model [25], MKEx (modified Kies exponential) model [26], and NPF model are all considered viable alternatives to the MKNPF model. The analytical measures, including, -Log-likelihood (-LL), the AIC (Akaike information criterion), AICC (Akaike Information Criterion Corrected), HQIC (Hannan–Quinn information) and the BIC (Bayesian information criterion) have all been used to compare these models. we also analyze the Kolmogorov–Smirnov (K–S) statistic and its P-value (PV). The model with the lowest analytical measures scores for the real data set may be the best fit. The results of these examinations are shown in Table 6, Table 7. The first COVID-19 data comes from the United Kingdom and covers the period from May 1 to July 16, 2021, a total of 82 days, as seen at [https://covid19.who.int/]. Daily new deaths (ND), daily cumulative cases (CC), and daily cumulative deaths (CD) were used to create this data. The detail of this dataset can be seen in Abu El Azm et al. [45]. The second real data collection presented COVID-19 data from Italy, which spans 111 days from April 1 to July 20, 2020 and is available at https://covid19.who.int/. This data is calculated by dividing daily new deaths by new cases. Hassan et al. [46] investigated it. See [47], [48] for other examples of COVID-19 data applications. To assess the pertinent parameters of models, the MLE method has been utilized. Table 4, Table 5 provide ML estimates and their standard errors (SEs) in parenthesis, for two real data sets. For the two real data sets analyzed, the results in these tables prove that proposed distribution gives better fits than competing models. Also, Table 6, Table 7 show that MKNPF model has the highest P-value and the smallest Kolmogorov–Smirnov (K–S) distance. Fig. 8, Fig. 9, Fig. 10, Fig. 11 demonstrate the MKNPF distribution’s fitted PDF, CDF and P-P layouts for two real data sets, respectively. Finally, MKNPF model emerges as the most appropriate model for both datasets, demonstrating its usefulness in a real context. We make the plots for the log-likelihood function as we can see in Fig. 12, Fig. 13 and by studying the plots of the log-likelihood function and the data, we can see that Fig. 12, Fig. 13 confirm that the estimates conducted from the MLEs for the proposed model parameters are global maximum, not local maximum for all model’s parameters.

Table 6.

The values of the considered goodness-of-fit indicators for Data I.

Distribution	-LL	AIC	CAIC	BIC	HQIC	$K-S$	$PV$
MKNPF $\left(\mu ,\lambda ,\xi \right)$	−195.9468	−385.8936	−385.5859	−378.6734	−382.9948	0.04680	0.9939
MKITL $\left(\mu ,\beta \right)$	−178.85	−353.7001	−353.5482	−348.8866	−351.7675	0.23956	0.0002
MKIEx $\left(\xi ,\beta \right)$	−186.7006	−369.4012	−369.2493	−364.5877	−367.4687	0.13983	0.0810
NPF $\left(\vartheta ,\kappa \right)$	−191.6261	−379.2522	−379.1003	−374.4387	−377.3197	0.11996	0.1887

Table 7.

The values of the considered goodness-of-fit indicators for Data II.

Distribution	-LL	AIC	CAIC	BIC	HQIC	$K-S$	$PV$
MKNPF $\left(\mu ,\lambda ,\xi \right)$	−127.363	−248.7258	−248.5015	−240.5972	−245.4282	0.07692	0.5273
MKITL $\left(\mu ,\beta \right)$	−122.942	−241.8847	−241.7735	−236.4656	−239.6863	0.12186	0.0740
MKIEx $\left(\xi ,\beta \right)$	−124.913	−245.826	−245.7148	−240.4069	−243.6276	0.09839	0.2328
NPF $\left(\vartheta ,\kappa \right)$	−101.930	−199.8599	−199.7488	−194.4409	−197.6616	0.24203	0.0000

Table 4.

MLEs and SEs of the parameters of considered models for Data Set I.

Models	MLEs	Standard errors
MKNPF $\left(\mu ,\lambda ,\xi \right)$	102.9867375, 0.4283564, 1.7290155	85.1797881, 0.1702234, 0.4221061
MKITL $\left(\mu ,\beta \right)$	0.4436225, 223.1059521	0.04081851, 26.78938001
MKIEx $\left(\xi ,\beta \right)$	0.8623125, 15.0042180	0.07684104, 1.28409033
NPF $\left(\vartheta ,\kappa \right)$	−0.2584536, 36.1414106	1.163282, 55.314676

Table 5.

MLEs and SEs of the parameters of considered models for Data Set II.

Models	MLEs	Standard errors
MKNPF $\left(\mu ,\lambda ,\xi \right)$	27.4046947, 0.3418181, 2.9171457	6.39169, 0.03375, 0.28293
MKITL $\left(\mu ,\beta \right)$	0.9011534, 24.3806218	0.0698631, 1.7450298
MKIEx $\left(\xi ,\beta \right)$	1.561634, 3.542441	0.1146064, 0.1586595
NPF $\left(\vartheta ,\kappa \right)$	−0.9046125, 49.7649243	0.0809519, 41.671355

Fig. 8.

Fitted densities plotted over the sample histogram of dataset I(left panel) and fitted CDFs on empirical CDF of dataset I(right panel).

Fig. 9.

Fitted densities plotted over the sample histogram of dataset II (left panel) and fitted CDFs on empirical CDF of dataset II(right panel).

Fig. 10.

P-P layouts of the MKNPF model for datasets I.

Fig. 11.

P-P layouts of the MKNPF model for datasets II.

Fig. 12.

Profile of log-likelihood function of $\mu ,\lambda$ and $\xi$ for dataset I.

Fig. 13.

Profile of log-likelihood function of $\mu ,\lambda$ and $\xi$ for dataset II.

Closing remarks on both applications

1. MKNPF has the highest $P$-value and the lowest K–S distance, according to both datasets.

2. As shown in Fig. 8, Fig. 9 MKNPF is the most effective model for fitting datasets I and II

3. MKNPF is the best model for modeling datasets I and II, as noticed in Fig. 10, Fig. 11.

4. The MKITL and NPF distributions demonstrate poor fit for the first dataset I and II respectively, as shown in Table 6, Table 7.

Concluding remarks

The three-parameter modified Kies new power function (MKNPF) model is proposed in this content as a novel 3-parameter model. The MKNPF model is more adaptable than other known models when it comes to studying lifespan data. The linear version of PDF, SF, hrf, chrf, QF and moments of the MKNPF model are obtained. The estimation methodologies like MLE, LSE, and WLSE are employed to evaluate the parameters of MKNPF model and compared. A emulation study is used to assess the model’s performance under various estimating approaches. We represent two accomplishment based on the COVID 19 mortality rate, proving that the MKNPF distribution is the finest model for fitting this type of data among its counterparts. Based on two real-life examples, the model gives a good fit than the MKITL, MKIEx, and NPF distributions.

Nomenclature

Symbols
$f\left(\left(t\right|\Psi \right)$	PDF	$F\left(\left(t\right|\Psi \right)$	CDF
$S\left(\left(t\right|\Psi \right)$	SF	$h\left(\left(t\right|\Psi \right)$	HRF/FRF
$H\left(\left(t\right|\Psi \right)$	CHRF	$Q(q;\Psi )$	QF
$Q^{\prime }(q;\Psi )$	Quantile Density Function	$M\left(\left(t\right|\Psi \right)$	MGF
${\rho }_{r,s}$	Probability weighted moments
Abbreviations
MLE	Maximum likelihood Estimation	PF	Power Function
MK	Modified Kies	MKNPF	Modified Kies New Power Function
PDF	Probability Density Function	FGF	Factorial Generating Function
MPSM	Maximum product spacing method	CHRF	Cumulative Hazard Rate Function
SF	Survival Function	FRF	Failure Rate Function
WLSEs	Weighted Least Square Estimates	LSEs	Least Square Estimates
hrf	hazard rate function	MMEs	Method of Moments Estimates
L-I	Lehmann Type I	MSE	Mean square error
CF	Characteristic Function	CDF	Cumulative Distribution Function
MGF	Moment Generating Function	QF	Quantile Function

CRediT authorship contribution statement

Anum Shafiq: Conceptualization, Methodology, Writing – review & editing, Data curation, Formal analysis, Methodology, Supervision. Tabassum Naz Sindhu: Methodology, Writing – review & editing, Data curation, Formal analysis, Methodology, Validation, Conceptualization. Naif Alotaibi: Writing – review & editing, Data curation, Formal analysis, Methodology, Validation.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Data availability

All data generated or analyzed during this study are included in this article.