Modified Kibria-Lukman (MKL) estimator for the Poisson Regression Model: application and simulation [version 1; peer review: 3 approved with reservations]

Background: Multicollinearity greatly affects the Maximum Likelihood Estimator (MLE) efficiency in both the linear regression model and the generalized linear model. Alternative estimators to the MLE include the ridge estimator, the Liu estimator and the Kibria-Lukman (KL) estimator, though literature shows that the KL estimator is preferred. Therefore, this study sought to modify the KL estimator to mitigate the Poisson Regression Model with multicollinearity. Methods: A simulation study and a real-life study were carried out and the performance of the new estimator was compared with some of the existing estimators. Results: The simulation result showed the new estimator performed more efficiently than the MLE, Poisson Ridge Regression Estimator (PRE), Poisson Liu Estimator (PLE) and the Poisson KL (PKL) estimators. The real-life application also agreed with the simulation result. Conclusions: In general, the new estimator performed more efficiently than the MLE, PRE, PLE and the PKL when multicollinearity was present.


Introduction
A special case of the Generalized Linear Models (GLM) is the Poisson Regression Model (PRM) which is generally applied for count or frequency data modelling. It is employed to model the relationship between a response variable and one or more independent variable where the response variable denotes a rare event or count data. The response variable also takes the form of a non-negative variable, and it is applicable in the following fields: economics, health, social and physical sciences. The Maximum Likelihood Estimation (MLE) method is popularly used to estimate the regression coefficient in a PRM. In both a Linear Regression Model (LRM) and Generalized Linear Model (GLM), MLE suffers a setback when the independent variables are correlated, which implies multicollinearity. Multicollinearity effects include large variance and regression coefficient covariances, negligible t-ratio and a high coefficient of determination (R-square) values. Alternative estimators to the MLE in the linear regression model include the ridge regression estimator by Hoerl and Kennard (1970) In this study, we modified the KL estimator to handle multicollinearity in PRM. Furthermore, we compared the performance of the estimator with the Poisson Maximum Likelihood Estimator (PMLE), Poisson Ridge Regression Estimator (PRE), Poisson Liu Estimator (PLE) and the Poisson KL estimator (PKLE).

Methods
Given that the response variable, y i is in the form of count data, then it is assumed to follow a Poisson distribution as P o (μ i ) where μ i = exp (x i β), such that x i is the i th row of X which is a nÂ(p+1) data matrix with p independent variables and β is a (p+1)Â1 vector of coefficients. The log likelihood of the model is given as: The most common method of maximizing the likelihood function is to use the iterated weighted least squares (IWLS) algorithm which results to: The MLE is normally distributed with a covariance matrix that is equivalent to the inverse of the second derivative as: and the mean square error is given as: where λ j is the j th eigen value of the X 0 b VX matrix.
The Poisson Ridge Estimator (PRE) was introduced by Månsson and Shukur (2011) as a solution to multicollinearity in PRM. The estimator is defined as follows: The mean square error (MSE) is: Mansson et al. (2012) developed the Liu estimator to the Poisson regression model as: The means square error for the Liu estimator is defined as: where λ j is the j th eigenvalue of X 0 b VX and α j is the j th element of α.
The KL estimator was proposed by Kibria and Lukman (2020) as a means of mitigating the effect of multicollinearity on parameter estimation. The estimator is defined as By means of extension, the Poisson K-L estimator was proposed by Lukman et al. (2021) as follows:

The Poisson Modified KL estimator (PMKL)
The proposed estimator is obtained as follows: b β MLE in equation (2.10) is replaced with the ridge estimator. Thus, we have: The properties of the new estimator include: 14) The bias can be written in scalar form as: can be represented in scalar form as follows: Thus, the MSE is obtained as: The proposed estimator in (2.13) is extended to the PRM. It is referred to as the Poisson modified KL (PMKL) estimator and defined as: The mean square error of the PMKL is defined as: The following lemmas are adopted for theoretical comparisons among the estimators.
We can observe that the difference of the variance of the estimator is non-negative since λ j þ k À Á 2 λ j À λ j λ j À k À Á 2 > 0 for The difference of the variance is non-negative since

Selection of biasing parameter
The biasing parameter k for the estimator is obtained by differentiating the MSE with respect to k and obtained as: The shrinkage parameter estimated by Mansson and Shukur, (2011) and Kibria and Lukman (2020) was also adopted for this study as listed: k 1 is the biasing parameter for PMKL1, while k 2 and k 3 are the biasing parameters for PMKL2 and PMKL3.

Simulation Design and Real-Life Application
Simulation study and result In this section, a simulation study is carried out to compare the performance of the different estimators. The generation of the dependent variables are done using pseudo-random numbers from P o (μ i ) where μ i ¼ e βxi i ¼ 1, 2,…, n and X i is the i th row of the design matrix with β ¼ β 0 , β 1 , …,β p À Á being the coefficient vector. The generation of the independent variables with different levels of correlation is obtained using . z ij are pseudo-random numbers generated using the standard normal distribution such that i ranges from 1 to n and j from 1 to p. As a common restriction used in simulation studies, it is assumed that P p j¼1 β 2 j ¼ 1 and The simulation results in Tables 1 to 6 that for each of the estimators, the simulated MSE values increase as the multicollinearity level increases, keeping other factors constant. There is also an increase in the mean square error as the sample size increases for all estimators compared while other factors were kept constant. As the intercept values varied from -1 to +1, the values of the mean square error reduced for all estimators. Result shows that the PMKL1 performed best with minimum MSE at varying sample sizes. It was closely followed by PMKL2. They are both considered more suitable for estimation of parameters in the Poisson regression model than the MLE as it performed worst when multicollinearity is a challenge. In general, the PMKL1 estimator consistently performed more efficiently than the MLE, PRE, PLE and the PKL estimators.

Real Life Application
Having carried out a simulation study, the efficacy of the proposed estimator needs to be further investigated by  Table 7, it is evident that all of the regression coefficients had identical signs. The estimator with the highest mean squared error is the MLE due to the presence of multicollinearity. The suggested estimator (PMKL1, PMKL2, PMKL3) has the lowest MSE that has established its dominance. We also observed that the performance of the estimator is highly dependent on the biasing parameter k.

Data availability
All data underlying the results are available as part of the article and no additional source data are required.  In the whole article, there are some inappropriate uses of the abbreviations. The authors should rearrange the use of abbreviations. In some places, previously made abbreviations are repeated.

3.
The use of "hat" is missing while presenting some estimators. 4.
In the Introduction section, the manuscript Defining a two-parameter estimator: a mathematical programming evidence by Üstündağ Şiray et al. (2021) 1 may be mentioned since this is a more recent article in which a new biased estimator is proposed to mitigate multicollinearity.

5.
On page 4, the authors should explain what lambdas are. 6.
On page 5, the authors should explain what lambdas are. Do the authors use "V" to Show variance? If so, some explanations should be added about it.

10.
On page 5, there is the incorrect use of "MSEM". This abbreviation does not exist, although it is used while representing the lemmas and theorems.

11.
I think the authors employ the canonical form in the proof of the theorems. Unfortunately, I did not find some information about the canonical model.

12.
The selection of the biasing parameter section is insufficient. A detailed derivation and more information should be given.

13.
In the simulation section, on page 7, why does the mean square error increase as the sample size increases?

14.
It would be better if no abbreviations were used in the title. 15.
the Introduction section.
Change independent variables to explanatory variables in the whole study.