Modified ridge-type for the Poisson regression model: simulation and application

Abstract

The Poisson regression model (PRM) is employed in modelling the relationship between a count variable (y) and one or more explanatory variables. The parameters of PRM are popularly estimated using the Poisson maximum likelihood estimator (PMLE). There is a tendency that the explanatory variables grow together, which results in the problem of multicollinearity. The variance of the PMLE becomes inflated in the presence of multicollinearity. The Poisson ridge regression (PRRE) and Liu estimator (PLE) have been suggested as an alternative to the PMLE. However, in this study, we propose a new estimator to estimate the regression coefficients for the PRM when multicollinearity is a challenge. We perform a simulation study under different specifications to assess the performance of the new estimator and the existing ones. The performance was evaluated using the scalar mean square error criterion and the mean squared error prediction error. The aircraft damage data was adopted for the application study and the estimators’ performance judged by the SMSE and the mean squared prediction error. The theoretical comparison shows that the proposed estimator outperforms other estimators. This is further supported by the simulation study and the application result.

1. Introduction

The type of outcome variable helps to make informed choices on the regression models to adopt. The linear regression and the logistic regression model are used when the outcome (response) variable are continuous and binary, respectively. The gamma regression model is engaged when the response variable is continuous and positively skewed. The Poisson regression model (PRM) is a generalised linear model that is often employed when the outcome variable comes in the form of a count variable or non-negative integers. It is applicable in the following field: economics, social sciences, engineering medicine, physical sciences etc. Examples include the number of patents, takeover bids, bank failures, accident insurance and criminal careers, number of deaths, number of defects and others [29,6]. The parameters in a Poisson regression model are popularly estimated using the method of maximum likelihood [17,8]. In the fields as mentioned above, especially economics, most of the variables grow together; that is, they are usually correlated. This correlation results in the problem of multicollinearity. When multicollinearity is introduced into the model, the variance of the regression parameter obtained through the method of maximum likelihood becomes unstable and inflated [36,25,28]. Consequently, this makes statistical inference difficult due to inflated variance and unstable parameter estimates [27,28,30,9,35]. There are numerous methods of solving multicollinearity in the linear regression model. The long list of researchers that have developed estimators to circumvent the problem of multicollinearity in the linear regression model are not limited to Hoerl and Kennard [12], Swindel [40], Liu [20], Liu [21], Ozkale and Kaciranlar [33], Sakallioglu and Kaciranlar [39], Yang and Chang [42], Dorugade [10], Lukman et al. [25,26], Qasim et al. [34], Kibria and Lukman [19], Ahmad and Aslam [2], Lukman et al. [27] and others. These estimators can be classified into the following classes: single parameter estimators, single parameter estimators with prior information, two-parameter estimators and two-parameter estimator with prior information.

The ridge regression estimator is an ancient but effective method for dealing with multicollinearity, which was first introduced by Hoerl and Kennard [12]. The estimator was proposed by adding a small positive constant k (biasing parameter) to the diagonal elements of the $X^{\mathrm{\prime }}X$ matrix. Several efforts have been made on the estimation of the biasing parameter k through the following researchers: Kibria [16], Khalaf and Shukur [15], Alkhamisi et al. [4], Alkhamisi and Shukur [5], Muniz and Kibria [31], Lukman and Ayinde [22,23], Lukman and Arowolo [24], and others. Mansson and Shukur [29] introduced the ridge regression to the Poisson regression model and adopted some of the biasing parameters of the authors mentioned earlier in the study. The estimated parameters in the ridge regression estimator were non-linear functions of the biasing parameter, which makes it complicated and difficult to interpret [20]. Alternatively, he suggested the Liu estimator with shrinkage parameter d. [30] developed the Poisson Liu estimator and suggest its most appropriate biasing parameter. The ridge regression estimator and the Liu estimator are examples of a single parameter estimator. Recently, Kibria and Lukman [19] proposed another single parameter estimator that is called the K-L estimator. Also, Lukman et al. [25] developed the modification to the ridge regression estimator, which falls in the class of the two-parameter estimators. The estimator is referred to as the modified ridge-type estimator.

The focus of this study is to introduce the modified ridge-type estimator and its biasing parameter k and d to mitigate the problem of multicollinearity in the Poisson regression model. The properties of this estimator will be defined after which a theoretical comparison will be established with the Poisson maximum likelihood (PMLE), the Poisson ridge regression estimator (PRRE) and the Poisson Liu estimator (PLE) via the mean square error matrix (MSEM) and scalar mean square error (SMSE). The sample properties are investigated using a simulation experiment. Finally, the performance of the new estimator is investigated with a real-life example called the aircraft damage data.

This paper structuring is as follows: The Poisson regression model, some existing estimators, the MSEM and SMSE properties of the estimators are discussed in section 2. A Monte Carlo simulation experiment has been conducted in Section 3. To illustrate the finding of the paper, aircraft damage data was analysed in Section 4. Some concluding remarks are presented in section 5.

2. Methodology

Let the response variable y_i be a count data, then the probability function is given as follows

$$f(y_i)=\frac{\exp (-{\mu }_i){\mu }_i^{y_i}}{y_i!},y_i=0,1,2,\dots ,{\mu }_i>0,$$ (1)

with mean and variance, $E(y_i)=V(y_i)={\mu }_i.$ We assume there exists a function, g, that relates the mean of the response to a linear predictor such that

$$g({\mu }_i)={\eta }_i={\beta }_0+{\beta }_1{x}_1+\dots +{\beta }_1{x}_1=x_i^{\mathrm{\prime }}\beta ,$$ (2)

where $g$ is the link function, $x_i$ is the i^th row of the design matrix X, and $\beta$ corresponds to a $(p+1)\times 1$ vector of the regression coefficients. The log link function is a popular type of this link function such that $g({\mu }_i)=\ln ({\mu }_i)=exp({x^{\mathrm{\prime }}}_i\beta ).$ The likelihood function is defined as:

$$l(\beta )=\prod _{i=1}^n\frac{exp(-{\mu }_i){{\mu }_i}^{yi}}{y_i!}.=\frac{\prod _{i=1}^n{{\mu }_i}^{yi}exp\left(-\sum _{i=1}^n{\mu }_i\right)}{\prod _{i=1}^n{y}_i!}$$ (3)

where ${\mu }_i=g^{-1}({x^{\mathrm{\prime }}}_i\beta ).$ The log-likelihood function is used to estimate the parameter vector  $\beta$

$$\ln l(\beta )=\sum _{i=1}^n{y}_i\ln ({\mu }_i)-\sum _{i=1}^n{\mu }_i-\sum _{i=1}^n\ln (y_i!)$$ (4)

We observed that equation (4) is nonlinear in $\beta$, then the solution is obtained using Fisher Scoring method and obtained the following equation

$${\beta }^{t+1}={\beta }^t+I^{-1}({\beta }^t)S({\beta }^t)$$ (5)

where $S(\beta )=\frac{\mathrm{\partial }l(\beta )}{\mathrm{\partial }\beta }$ and $I^{-1}(\beta )=(-E({\mathrm{\partial }}^{2}l(\beta )/\mathrm{\partial }\beta \mathrm{\partial }{\beta }^{\mathrm{\prime }}){)}^{-1}$. Consequently, the Poisson maximum likelihood estimator of $\beta$ is

$${\hat{\beta }}^{PMLE}=(X^{\mathrm{\prime }}\hat{W}X{)}^{-1}{X}^{\mathrm{\prime }}\hat{W}\hat{z}$$ (6)

where $\hat{W}=diag[{\hat{\mu }}_i]$ and $\hat{z}$ is a vector while the i^th element equals ${\hat{z}}_i=\log ({\hat{\mu }}_i)+\frac{y_i-{\hat{\mu }}_i}{{\hat{\mu }}_i}$.

The mean square error matrix and the scalar mean square error for the PMLE are defined respectively as

$$\begin{array}{rl}MSEM({\hat{\beta }}^{PMLE})& =(X^{\mathrm{\prime }}\hat{W}X{)}^{-1}\end{array}$$ (7)

$$\begin{array}{rl}SMSE({\hat{\beta }}^{PMLE})& =\sum _{j=1}^p\frac{1}{{\lambda }_j},\end{array}$$ (8)

where ${\lambda }_j$ is the j^th eigenvalue of the $X^{\mathrm{\prime }}\hat{W}X$ matrix.

When there is a correlation among the predictor variables, the $X^{\mathrm{\prime }}\hat{W}X$ is significantly affected which results to having a high variances and instability of the PMLE estimator. This led us to the review of some remedial measures of multicollinearity in PRM.

2.1. Poisson ridge regression estimator

The ridge regression estimator was first introduced by Hoerl and Kennard [12] to handle multicollinearity in the linear regression model (LRM). The estimator is defined as

$${\hat{\beta }}_k=(X^{\mathrm{\prime }}X+kI{)}^{-1}{X}^{\mathrm{\prime }}y$$ (9)

Månsson and Shukur [29] defined the Poisson ridge regression estimator as:

$${\hat{\beta }}^{PRRE}=(X^{\mathrm{\prime }}\hat{W}X+kI{)}^{-1}{X}^{\mathrm{\prime }}\hat{W}X{\hat{\beta }}^{PMLE}$$ (10)

The mean square error matrix and the scalar mean square error in canonical form for the PRRE are defined respectively as

$$MSEM({\hat{\alpha }}^{PRRE})=Q{\mathrm{\Lambda }}^k\mathrm{\Lambda }{\mathrm{\Lambda }}^k{Q}^{\mathrm{\prime }}+k^2{\mathrm{\Lambda }}^k\alpha {\alpha }^{\mathrm{\prime }}{\mathrm{\Lambda }}^k$$ (11)

where ${\mathrm{\Lambda }}^k=(\mathrm{\Lambda }+k{I}_p{)}^{-1}.$.

$$SMSE({\hat{\alpha }}^{PRRE})=\sum _{j=1}^p\left(\frac{{\lambda }_j}{{({\lambda }_j+k)}^2}\right)+k^2\sum _{j=1}^p\left(\frac{{\hat{\alpha }}_j^2}{{({\lambda }_j+k)}^2}\right)$$ (12)

where ${\hat{\alpha }}_j$ is the j^th component of $\hat{\alpha }=Q^{\mathrm{\prime }}\hat{\beta }$.

Hoerl and Kennard [13] suggested the following technique for the estimation of the ridge parameter k in LRM.

$${\hat{k}}_{HM}=\frac{p{\hat{\sigma }}^2}{\sum _{j=1}^p{\alpha }_j^2}$$ (13)

Following Schaefer, Kibria et al. [17], the biasing parameter k for the PRRE is given as

$${\hat{k}}_1=\frac{1}{max({\alpha }_j^2)}$$ (14)

2.2. Poisson Liu estimator

Also, to combat multicollinearity, the Liu estimator was introduced in multicollinear LRM by Liu [20] as

$${\hat{\beta }}_d=(X^{\mathrm{\prime }}X+I{)}^{-1}(X^{\mathrm{\prime }}y+d\hat{\beta })$$ (15)

where k > 0 and 0 < d < 1. Mansson et al. [30] defined the Poisson Liu estimator as

$${\hat{\beta }}^{PLE}=(X^{\mathrm{\prime }}\hat{W}X+I{)}^{-1}(X^{\mathrm{\prime }}\hat{W}X+d\hat{\beta }){\hat{\beta }}^{PMLE}$$ (16)

where

$$\hat{d}=max\left(0,\frac{{\hat{\alpha }}_j^2-1}{{\hat{\alpha }}_j^2+\frac{1}{{\lambda }_j}}\right)$$ (17)

where ${\hat{\alpha }}_j$ is the j^th component of $\hat{\alpha }=Q^{\mathrm{\prime }}\hat{\beta }$. The MMSE and the SMSE in canonical form are defined respectively as

$$MMSE({\hat{\alpha }}^{PLE})=Q{\mathrm{\Lambda }}_d{\mathrm{\Lambda }}^{-1}{\mathrm{\Lambda }}_d^{\mathrm{\prime }}{Q}^{\mathrm{\prime }}+({\mathrm{\Lambda }}_d-I)\alpha {\alpha }^{\mathrm{\prime }}({\mathrm{\Lambda }}_d-I{)}^{\mathrm{\prime }}$$ (18)

where ${\mathrm{\Lambda }}_d=(\mathrm{\Lambda }+I{)}^{-1}(\mathrm{\Lambda }+dI).$

$$SMSE({\hat{\alpha }}^{PLE})=\sum _{j=1}^p\frac{{({\lambda }_j+d)}^2}{{\lambda }_j{({\lambda }_j+1)}^2}+(d-1{)}^2\sum _{j=1}^p\frac{{\alpha }_j^2}{{({\lambda }_j+1)}^2}$$ (9)

where ${\lambda }_j$ is the j^th eigenvalue of $X^{\mathrm{\prime }}\hat{W}X$ and ${\alpha }_j$ is the j^th element of $\alpha .$

2.3. Poisson modified ridge-type estimator

Lukman et al. [25] developed modify the ridge regression estimator in the LRM and called it the modified ridge-type estimator (MRTE). This estimator is defined as

$${\hat{\beta }}^{MRTE}=(X^{\mathrm{\prime }}X+k(1+d)I{)}^{-1}{X}^{\mathrm{\prime }}X{\hat{\beta }}^{MLE}$$ (20)

where k > 0 and 0 < d < 1.

Lukman et al. [28] developed the logistic version of the modified ridge-type estimator. Thus, in this study, we propose the Poisson modified ridge-type estimator as follows:

$${\hat{\beta }}^{PMRTE}=(X^{\mathrm{\prime }}\hat{W}X+k(1+d)I{)}^{-1}{X}^{\mathrm{\prime }}\hat{W}X{\hat{\beta }}^{PMLE}$$ (21)

where k > 0 and 0 < d < 1.

Following Lukman et al. [26] and Lukman et al. [28], the MMSE and the SMSE in the canonical form are defined respectively as

$$MSEM({\hat{\alpha }}^{PMRTE})=Q{\tilde{\mathrm{\Lambda }}}_k{\mathrm{\Lambda }}^{-1}{\tilde{\mathrm{\Lambda }}}_k^{\mathrm{\prime }}{Q}^{\mathrm{\prime }}+({\tilde{\mathrm{\Lambda }}}_k-I)\alpha {\alpha }^{\mathrm{\prime }}({\tilde{\mathrm{\Lambda }}}_k-I{)}^{\mathrm{\prime }}$$ (22)

where ${\tilde{\mathrm{\Lambda }}}_k=\mathrm{\Lambda }(\mathrm{\Lambda }+k(1+d)I{)}^{-1}.$

$$SMSE({\hat{\alpha }}^{PMRTE})=\sum _{j=1}^p\frac{{\lambda }_j}{{\lambda }_j{({\lambda }_j+k)}^2}+{(k(1+d))}^2\sum _{j=1}^p\frac{{\hat{\alpha }}_j^2}{{\lambda }_j{({\lambda }_j+k)}^2}$$ (23)

The following lemmas are adopted for theoretical comparisons among the estimators.

Lemma 2.1:

Let A be a positive definite (pd) matrix, that is A > 0, and $a$ be some vector, then $A-a{a}^{\mathrm{\prime }}\ge 0$ if and only if (iff) $a^{\mathrm{\prime }}{A}^{-1}a\le 1$ [11].

Lemma 2.2:

$MSEM({\hat{\beta }}_1)-MSEM({\hat{\beta }}_2)={\sigma }^{2}D+b_1{b}_2^{\mathrm{\prime }}-b_2{b}_2^{\mathrm{\prime }}>0$ if and only if $b_2^{\mathrm{\prime }}[{\sigma }^{2}D+b_1{b^{\mathrm{\prime }}}_1{]}^{-1}{b}_2<1$ where $MSE({\hat{\beta }}_j)=Cov({\hat{\beta }}_j)+b_j^{\mathrm{\prime }}{b}_j$ [41].

Theorem 2.1:

${\hat{\alpha }}^{PMRTE}$ is preferred to ${\hat{\alpha }}^{PMLE}$ if $MSEM({\hat{\alpha }}^{PMLE})-MSEM({\hat{\alpha }}^{PMRTE})>0$ provided k > 0 and 0 < d < 1.

Proof:

$$Cov({\hat{\alpha }}^{PMLE})-Cov({\hat{\alpha }}^{PMRTE})=Q[{\mathrm{\Lambda }}^{-1}-{\tilde{\mathrm{\Lambda }}}_k{\mathrm{\Lambda }}^{-1}{{\tilde{\mathrm{\Lambda }}}^{\mathrm{\prime }}}_k]Q^{\mathrm{\prime }}$$

$$=Qdiag{\left\{\frac{1}{{\lambda }_j}-\frac{{\lambda }_j}{{({\lambda }_j+k(1+d))}^2}\right\}}_{j=1}^p$$

The matrix ${\mathrm{\Lambda }}^{-1}-{\tilde{\mathrm{\Lambda }}}_k{\mathrm{\Lambda }}^{-1}{\tilde{\mathrm{\Lambda }}}_k^T$ is non-negative since $({\lambda }_j+k(1+d){)}^2-{\lambda }_j^2>0$ for 0 < d < 1 and k > 0.

Theorem 2.2:

${\hat{\alpha }}^{PMRTE}$ is preferred to ${\hat{\alpha }}^{PRRE}$ if $MSEM({\hat{\alpha }}^{PRRE})-MSEM({\hat{\alpha }}^{PMRTE})>0$ provided k > 0 and 0 < d < 1.

Proof:

$$\begin{array}{rl}Cov({\hat{\alpha }}^{PRRE})-Cov({\hat{\alpha }}^{PMRTE})& =Q[{\mathrm{\Lambda }}_k\mathrm{\Lambda }{{\mathrm{\Lambda }}^{\mathrm{\prime }}}_k-{\tilde{\mathrm{\Lambda }}}_k{\mathrm{\Lambda }}^{-1}{{\tilde{\mathrm{\Lambda }}}^{\mathrm{\prime }}}_k]Q^T\\ & =Qdiag{\left\{\frac{{\lambda }_j}{{({\lambda }_j+k)}^2}-\frac{{\lambda }_j}{{({\lambda }_j+k(1+d))}^2}\right\}}_{j=1}^p{Q}^T\end{array}$$

The matrix ${\mathrm{\Lambda }}_k\mathrm{\Lambda }{\mathrm{\Lambda }}_k^{\mathrm{\prime }}-{\tilde{\mathrm{\Lambda }}}_k{\mathrm{\Lambda }}^{-1}{\tilde{\mathrm{\Lambda }}}_k^{\mathrm{\prime }}$ is non-negative since $({\lambda }_j+k(1+d){)}^2-({\lambda }_j+k{)}^2>0$ for 0 < d < 1 and k > 0.

Theorem 2.3:

${\hat{\alpha }}^{PMRTE}$ is preferred to ${\hat{\alpha }}^{PLE}$ if $MSEM({\hat{\alpha }}^{PLE})-MSEM({\hat{\alpha }}^{PMRTE})>0$ provided k > 0 and 0 < d < 1.

Proof:

$$\begin{array}{rl}Cov({\hat{\alpha }}^{PLE})-Cov({\hat{\alpha }}^{PMRTE})& =Q[{\mathrm{\Lambda }}_d{\mathrm{\Lambda }}^{-1}{{\mathrm{\Lambda }}^{\mathrm{\prime }}}_d-{\tilde{\mathrm{\Lambda }}}_k{\mathrm{\Lambda }}^{-1}{{\tilde{\mathrm{\Lambda }}}^{\mathrm{\prime }}}_k]Q^{\mathrm{\prime }}\\ & =Qdiag{\left\{\frac{{({\lambda }_j+d)}^2}{{\lambda }_j{({\lambda }_j+1)}^2}-\frac{{\lambda }_j}{{({\lambda }_j+k(1+d))}^2}\right\}}_{j=1}^p{Q}^T\end{array}$$

The matrix ${\mathrm{\Lambda }}_d{\mathrm{\Lambda }}^{-1}{\mathrm{\Lambda }}_d^{\mathrm{\prime }}-{\tilde{\mathrm{\Lambda }}}_k{\mathrm{\Lambda }}^{-1}{\tilde{\mathrm{\Lambda }}}_k^{\mathrm{\prime }}$ is non-negative since $({\lambda }_j+k(1+d){)}^2({\lambda }_j+d{)}^2-{\lambda }_j^2({\lambda }_j+1{)}^2>0$ for 0 < d < 1 and k > 0.

2.4. Selection of biasing parameter

In this study, the biasing parameter k for the proposed estimator is determined using the Poisson ridge regression biasing estimator defined in equation (14). Following Asar and Genc [8], the biasing parameter d is estimated using the following equation.

$$\hat{d}=min\left(\frac{{\lambda }_j{\alpha }_j^2}{1+{\lambda }_j{\hat{\alpha }}_j^2}\right)$$ (24)

3. Simulation experiment

The Monte-Carlo simulation study is further carried out in this section. The response variable is generated to follow the Poisson distribution $P_0({\mu }_i)$ where ${\mu }_i=exp(x_i\beta )i=1,2,\dots ,n,\beta =({\beta }_0,{\beta }_1,{\beta }_2,\dots ,{\beta }_p{)}^{\mathrm{\prime }}$ such that $x_i$ is the i^th row of the design matrix X. The predictor variables were derived following Kibria [16]; Kibria and Banik [18]; Akdeniz and Roozbeh [3] and Roozbeh et al. [38] as follows:

$$x_{ij}=(1-{\rho }^2{)}^{1/2}{w}_{ij}+\rho w_{ip+1},i=1,2,\dots ,n;j=1,2,\dots p,p+1$$ (25)

where $w_{ij}$ are generated from standard normal, ${\rho }^2$ is the correlation between the explanatory variables. The values of $\rho$ are chosen to be 0.8, 0.9, 0.95, 0.99 and 0.999. We obtained the mean function for p = 4 and 7 predictor variables, respectively. According to Kibria et al. [17], the intercept values are chosen to be −1, 0 and 1 to change the average intensity of the Poisson process. The slope coefficients chosen so that $\sum _{j=1}^p{\beta }_j^2=1$ and ${\beta }_1={\beta }_2=\dots ={\beta }_p$ for sample sizes 50, 75, 100 and 200 [35,1]. Simulation experiment conducted through R programming language. For each replicate, we compute the mean square error (MSE) of the estimators by using the following equation

$$MSE({\alpha }^{\ast })=\frac{1}{5000}\sum _{i=1}^{5000}({\alpha }^{\ast }-\alpha {)}^{\mathrm{\prime }}({\alpha }^{\ast }-\alpha )$$ (26)

where ${\alpha }^{\ast }$ would be any of the estimators (PMLE, PRRE, PLE and PMRTE). Estimator with the least MSE is considered best. The experiment is replicated 5000 times. The simulated SMSE values of the estimators for p = 4 and intercepts = −1, 0 and 1 are presented in Tables 1–3 respectively and p = 7 and intercepts = −1, 0 and 1 are presented in Tables 4–6 respectively. The estimated SMSE of the PRM are presented in Tables 1–6. We investigated the effect of the following factors on the performance of each of the estimators: the degree of correlation, sample sizes and the number of explanatory variables. From Tables 1–6, we observed that the proposed estimator (PMRTE) is generally preferred in this study because it possesses the least SMSE among all other biased estimators. As expected PMLE performance is the least among all the considered estimators and not to be recommended for parameter estimation in the PRM with multicollinearity. The simulation results show that the simulated SMSE values for each of the estimators increase as the level of multicollinearity (ρ) increase keeping other factors constant. Also, an increase in the sample size (n) resulted in a decrease in the SMSE values keeping all other factors fixed. Increasing the number of explanatory variables (p) from 4 to 7 inflated the SMSE for each of the estimators. The SMSE for all the estimators decreases when we change the intercept value from −1 to +1. The results show that the proposed estimator will be appropriate for parameter estimation in the Poisson regression model with multicollinearity. The proposed estimator performance in this study is consistent, as observed from the simulation result. The simulation results agree with the theoretical result in Section 2. The MSE of all the estimators are not far from each other when the level of multicollinearity is low say ρ = 0.8 especially when there is large sample. Though, the proposed still has the least MSE with a negligible difference. The proposed maintain its performance even at severe multicollinearity level of ρ = 0.99 and 0.999.

Table 1.

Simulated SMSE when p = 4 and intercept = −1.

n	ρ	PMRT	PLE	PRRE	PMLE
50	0.8	0.2371	0.2569	0.2592	0.3089
	0.9	0.2779	0.3327	0.3301	0.4295
	0.95	0.3743	0.4891	0.4800	0.6925
	0.99	1.0248	1.5114	1.4882	2.7803
	0.999	9.4429	14.1059	13.8943	26.9726
75	0.8	0.1533	0.1771	0.1742	0.2087
	0.9	0.2049	0.2454	0.2418	0.2966
	0.95	0.2820	0.3550	0.3501	0.4557
	0.99	0.7060	1.0551	1.0286	1.7619
	0.999	5.6564	8.4838	8.4305	16.0243
100	0.8	0.1621	0.1653	0.1672	0.1800
	0.9	0.2105	0.2270	0.2271	0.2575
	0.95	0.2896	0.3392	0.3371	04106
	0.99	0.7561	1.0984	1.0568	1.6915
	0.999	5.5141	8.3621	8.1781	15.5668
200	0.8	0.0434	0.0425	0.0431	0.0436
	0.9	0.0532	0.0547	0.0546	0.0569
	0.95	0.0757	0.0810	0.0801	0.0860
	0.99	0.2348	0.2753	0.2740	0.3261
	0.999	1.1208	1.6990	1.6528	2.9956

Table 2.

Simulated SMSE when p = 4 and intercept = 0.

n	ρ	PMRT	PLE	PRRE	PMLE
50	0.8	0.0910	0.1003	0.0992	0.1092
	0.9	0.1146	0.1303	0.1291	0.1479
	0.95	0.1576	0.1910	0.1893	0.2356
	0.99	0.3730	0.5410	0.5380	0.9393
	0.999	3.2565	4.80300	4.8225	9.4758
75	0.8	0.0669	0.0704	0.0703	0.0740
	0.9	0.0903	0.0967	0.0965	0.1036
	0.95	0.1316	0.1465	0.1460	0.1635
	0.99	0.3180	0.4331	0.4304	0.6302
	0.999	2.1135	3.1112	3.1256	5.8753
100	0.8	0.0582	0.0617	0.0611	0.0647
	0.9	0.0815	0.0880	0.0870	0.0935
	0.95	0.1181	0.1328	0.1308	0.1463
	0.99	0.3048	0.4235	0.4115	0.6069
	0.999	2.0403	2.9952	3.0079	5.6855
200	0.8	0.0156	0.0158	0.0157	0.0159
	0.9	0.0202	0.0205	0.0205	0.0208
	0.95	0.0306	0.0313	0.0312	0.0319
	0.99	0.0999	0.1090	0.1086	0.1186
	0.999	0.4617	0.6723	0.6674	1.1149

Table 3.

Simulated SMSE when p = 4 and intercept = 1.

n	ρ	PMRT	PLE	PRRE	PMLE
50	0.8	0.0380	0.0384	0.0385	0.0389
	0.9	0.0507	0.0520	0.0520	0.0534
	0.95	0.0768	0.0808	0.0808	0.0852
	0.99	0.2251	0.2800	0.2768	0.3548
	0.999	1.2127	1.8191	1.7855	3.4245
75	0.8	0.0263	0.0265	0.0265	0.0267
	0.9	0.0372	0.0378	0.0378	0.0383
	0.95	0.0581	0.0597	0.0597	0.0613
	0.99	0.1882	0.2117	0.2111	0.2391
	0.999	0.8339	1.2553	1.2272	2.1638
100	0.8	0.0230	0.0231	0.0232	0.0233
	0.9	0.0335	0.0337	0.0337	0.0340
	0.95	0.0516	0.0525	0.0525	0.0535
	0.99	0.1845	0.2027	0.2020	0.2226
	0.999	0.8481	1.2950	1.2353	2.1185
200	0.8	0.005747	0.005749	0.005749	0.005754
	0.9	0.007650	0.007666	0.007666	0.007683
	0.95	0.011630	0.011681	0.011681	0.011734
	0.99	0.042467	0.043408	0.043403	0.044377
	0.999	0.269914	0.326152	0.323417	0.396677

Table 4.

Simulated SMSE when p = 7 and intercept = −1.

n	ρ	PMRT	PLE	PRRE	PMLE
50	0.8	0.3294	0.3804	0.3769	0.4637
	0.9	0.4089	0.5015	0.4940	0.6461
	0.95	0.5827	0.7687	0.7555	1.0860
	0.99	1.9069	2.7232	2.7199	4.6936
	0.999	18.3393	26.1740	26.1958	46.0358
75	0.8	0.1831	0.2080	0.2039	0.2360
	0.9	0.2379	0.2777	0.2733	0.3244
	0.95	0.3475	0.4231	0.4178	0.5243
	0.99	0.9523	1.3495	1.3345	2.1762
	0.999	8.2666	11.7522	11.7935	20.7648
100	0.8	0.0788	0.0789	0.0793	0.0807
	0.9	0.0933	0.0940	0.0946	0.0981
	0.95	0.1253	0.1353	0.1330	0.1449
	0.99	0.3494	0.4383	0.4287	0.5536
	0.999	2.0836	2.9960	2.9699	5.1593
200	0.8	0.0530	0.0535	0.0536	0.0548
	0.9	0.0643	0.0659	0.0658	0.0681
	0.95	0.0904	0.0985	0.0963	0.1040
	0.99	0.2983	0.3456	0.3442	0.4065
	0.999	1.5766	2.2399	2.2197	3.7609

Table 5.

Simulated SMSE when p = 7 and intercept = 0.

n	ρ	PMRT	PLE	PRRE	PMLE
50	0.8	0.1299	0.1405	0.1401	0.1521
	0.9	0.1792	0.1988	0.1984	0.2221
	0.95	0.2729	0.3194	0.3186	0.3813
	0.99	0.7817	1.0797	1.0797	1.7025
	0.999	6.6939	9.3978	9.4980	16.5277
75	0.8	0.0808	0.0842	0.0840	0.0876
	0.9	0.1060	0.1121	0.1120	0.1186
	0.95	0.1604	0.1753	0.1749	0.1921
	0.99	0.4491	0.5846	0.5822	0.8096
	0.999	2.9545	4.1938	4.2278	7.4702
100	0.8	0.0288	0.0293	0.0292	0.0297
	0.9	0.0342	0.0347	0.0347	0.0353
	0.95	0.0506	0.0518	0.0517	0.0530
	0.99	0.1721	0.1890	0.1885	0.2079
	0.999	0.8516	1.1919	1.1898	1.9389
200	0.8	0.0199	0.0202	0.0202	0.0204
	0.9	0.0243	0.0246	0.0246	0.0249
	0.95	0.0363	0.0370	0.0369	0.0377
	0.99	0.1267	0.1365	0.1362	0.1472
	0.999	0.6563	0.8957	0.8932	1.3862

Table 6.

Simulated SMSE when p = 7 and intercept = 1.

n	ρ	PMRT	PLE	PRRE	PMLE
50	0.8	0.0524	0.0529	0.0529	0.0536
	0.9	0.0769	0.0787	0.0787	0.0807
	0.95	0.1272	0.1330	0.1329	0.1395
	0.99	0.4119	0.4963	0.4935	0.6117
	0.999	2.3644	3.3598	3.3722	5.8804
75	0.8	0.0311	0.0313	0.0313	0.0315
	0.9	0.0415	0.0420	0.0420	0.0426
	0.95	0.0670	0.0687	0.0687	0.0704
	0.99	0.2380	0.2641	0.2635	0.2947
	0.999	1.1712	1.6782	1.6561	2.7591
100	0.8	0.010908	0.010911	0.010911	0.010922
	0.9	0.013103	0.013129	0.013129	0.013160
	0.95	0.019258	0.019343	0.019343	0.019435
	0.99	0.072319	0.074162	0.074152	0.076084
	0.999	0.464532	0.571076	0.565579	0.715018
200	0.8	0.007382	0.007386	0.007386	0.007393
	0.9	0.009120	0.009136	0.009137	0.009154
	0.95	0.013730	0.013785	0.013785	0.013842
	0.99	0.053231	0.054346	0.054341	0.055501
	0.999	0.358218	0.423967	0.421192	0.510032

4. Real-life application

There is a need to further examine the efficiency of the proposed estimator by considering a real-life application. Different researchers have fit the Poisson regression model to the aircraft damage dataset [32,8,6], among others. The response variable, y which denote the number of locations with damage on the aircraft follows a Poisson distribution [32,8,6]. Amin et al. [6] examine if the model follows a Poisson regression model by adopting the Pearson chi-square goodness of fit test. The test confirms that the response variable is well fitted to the Poisson distribution with test statistic (p-value) is given as 6.89812 (0.07521). The dataset provides the information about two types of aircraft, the McDonnell Douglas A-4 Skyhawk and the Grumman A-6 Intruder. This data describe 30 strike missions of these two aircraft. The explanatory variables are as follows: x₁ is a binary variable representing the aircraft type (A-4 coded as 0 and A-6 coded as 1), x₂ and x₃ denote bomb load in tons and total months of aircrew experience, respectively. According to Myers et al. [32], there is evident of multicollinearity problem in the data. The eigenvalues of the data matrix X are 4.3333, 374.8961 and 2085.2251. Thus, the condition number is 219.3654, indicates there is multicollinearity problem [8,6]. The estimators’ performance is judged through the scalar mean square error. The regression coefficients and the SMSE for all the estimators are available in Table 7. The MSE of the estimators is computed using equations (8), (12), (19) and (23), respectively. We also adopt the leave-one-out cross-validation to validate the performance of the estimators (see [14]). The performance of the estimator is assessed through the mean square prediction error (MSPE). From Table 7, we observed that all the coefficients have a similar sign. PMLE has the highest mean squared error, while the proposed estimator (PMRTE) has the least MSE and MSPE which established its superiority. The estimated k and d values are 2.5444 and 0.0967, respectively. The result also agrees with the simulation and the theoretical result.

Table 7.

Regression coefficients and SMSE.

coef.	${\hat{\alpha }}^{PMLE}$	${\hat{\alpha }}^{PRRE}$	${\hat{\alpha }}^{PLE}$	${\hat{\alpha }}^{PMRTE}$
${\hat{\alpha }}_0$	−0.4060	−0.1675	−0.2555	−0.1641
${\hat{\alpha }}_1$	0.5688	0.3799	0.4789	0.3751
${\hat{\alpha }}_2$	0.1654	0.1705	0.1665	0.1708
${\hat{\alpha }}_3$	−0.0135	−0.0153	−0.0147	−0.0153
SMSE	1.0290	0.2727	0.4320	0.1422
MSPE	6.5468	5.7491	5.9448	5.6133

5. Concluding remarks

Lukman et al. [25] developed the modified ridge-type estimator (MRTE) to deal effectively with parameter estimation in the linear regression model (LRM) with multicollinearity. This estimator is in the class of the two-parameter estimator. MRTE was found to perform better than the ordinary least squares estimator, the ridge regression and the Liu estimator in LRM. This result is the motivation behind introducing MRTE to handle the problem of multicollinearity in the Poisson regression model (PRM). Thus, the estimator is called Poisson modified ridge-type estimator (PMRTE). The statistical properties of this new estimator were established in this study. We compared the performance of PMRTE with the Poisson maximum likelihood estimator (PMLE), Poisson ridge regression estimator (PRRE) and the Poisson Liu estimator (PLE) theoretically. The proposed estimator PMRTE performs better in terms of matrix mean squared error (MMSE) and scalar mean square error (SMSE). We further assess the performance of PMRTE with the help of a simulation study using the SMSE. The developed estimator possesses the least SMSE. We investigated the effects of some factors on the performance of all the estimators. We observed that increasing the sample size led to a decrease in the MSE values of all the estimators. Increasing the level of multicollinearity and explanatory variables increases the simulated SMSE for each of the estimators. Also, the real application supported the fact that PMRTE is generally preferred with least MSE and MSPE. The performance of the PMLE is the worst due to the presence of multicollinearity.

In conclusion, the theoretical result, the simulation result, and the real-life application result agree. PMRTE is generally preferred to all the estimators considered in this study. We, therefore, recommend this technique as an alternative to PMLE, PRRE and PLE for parameter estimation in PRM. In future study, we will explore the generalised cross-validation (GCV) method of selecting the biasing parameter [7,37].

Disclosure statement

No potential conflict of interest was reported by the author(s).