Modified Kibria-Lukman (MKL) estimator for the Poisson Regression Model: application and simulation

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 was 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.

In this study, we propose the Modified Kibria-Lukman estimator to handle multicollinearity in PRM. The estimator is a single parameter estimator which makes it less computationally intensive as compared with the two-parameter estimators. Also, since the Kibria-Lukman estimator is found to outperform the Ridge and the Liu estimators, it is expected that the modification in this study will enhance the performance of the Kibria-Lukman estimator. 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 = e (x i β) , and In µ i = (x i β), x i is the i th row of matrix X which is a nÂ(p+1) data matrix with p explanatory 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:

REVISED Amendments from Version 1
The difference between this version and the first is that all corrections that were raised by the three reviewers were effected.
The new version included more equations to simplify methods earlier discussed as raised by the reviewers.
Any further responses from the reviewers can be found at the end of the article and the mean squared error is given as: where λ j is the j th eigen value of the X 0 b VX matrix.
The Ridge estimator was adopted by Månsson and Shukur (2011) to solve multicollinearity problem in count data. The estimator is defined as follows: The mean squared error is: β PRE is effective in practice but it is a complicated function of the biasing parameter k (Liu, 1993).
The MSE for the Liu estimator is defined as: where λ j is the j th eigenvalue of X 0 b LX 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 and k > 0:

The Poisson Modified KL estimator (PMKL)
The proposed estimator is obtained as follows: b β MLE in equation (2.11) is replaced with the ridge estimator. Thus, we have: The properties of the new estimator include: 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 (2.14) is extended to the PRM. It is referred to as the Poisson modified KL (PMKL) estimator and defined as: The mean squared error of the PMKL is defined as: :,λ p Þ: Where λ 1 ≥λ 2 ,:::,≥λ p , Λ is the matrix of eigen-values of X 0 b LX and Q is the matrix whose columns are the eigenvectors of X 0 b LX.
The mean squared error (MSEM) and 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 The difference of the variance is non-negative since 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 in equation (2.21) with respect to k as follows: By equating to 0 and dividing through by 2 we have the resulting equation as: Solving the equation above for k yields the biasing parameter k given below as: The shrinkage parameter estimated by Mansson and Shukur, (2011) and Kibria and Lukman (2020) was also adopted for this study as listed: (2.28) k 1 and k 2 is the biasing parameter for PMKL1 and PMKL2, while k 3 is the biasing parameters for 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 being the coefficient vector. The generation of the explanatory 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 squared 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 squared 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. The expressions for the biasing parameters are defined in equation (2.26)-(2.28).

Conclusion
The  On page 3, a comma is required before equation (2.2). ○ On page 5, line 1, "estimator" should be "estimators". ○ Section number is required for the sections, etc.

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The basic punctuation marks are missing throughout the paper.
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.

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

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It would be better if no abbreviations were used in the title. 15.

points.
Write one paragraph on count data models and their importance at the start of the Introduction and include some citations that demonstrate the importance of count data models, for example: Amin et al.,   There are some grammatical issues that should be corrected.
○ expertise to confirm that it is of an acceptable scientific standard, however I have