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Research Article
Revised

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

[version 2; peer review: 2 approved, 1 approved with reservations]
PUBLISHED 14 Dec 2021
Author details Author details
OPEN PEER REVIEW
REVIEWER STATUS

Abstract

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.

Keywords

Linear regression model, generalized regression model, Ridge estimator, Liu estimator, KL estimator.

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.

See the authors' detailed response to the review by Mohammad Arashi

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. Other count data models include: Bell regression model, Negative binomial regression model, zero inflated bell regression model, zero inflated regression model (Amin et al., 2020, 2021; Sami et al., 2021; Rashad and Algamal, 2019; Majid et al., 2021). The PRM is employed to model the relationship between a response variable and one or more explanatory 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 explanatory 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), Liu estimator by Liu (1993), Liu-type estimator by Liu (2003), two-parameter estimator by Özkale and Kaciranlar (2007), r-d class estimator Kaçiranlar and Sakallioǧlu (2007), k-d class estimator Sakallioglu and Kaciranlar (2008), a two-parameter estimator by Yang and Chang (2010), modified two-parameter estimator by Dorugade (2014), modified ridge-type estimator by Lukman et al. (2019), modified Liu estimator by Lukman et al. (2020), Kibria-Lukman (KL) estimator by Kibria and Lukman (2020), modified new two-parameter estimator by Ahmad and Aslam (2020), the modified Liu ridge type estimator by Aslam and Ahmad (2020) and the DK estimator by Dawoud and Kibria (2020) among others. Researchers have extended some of these existing estimators in LRM to the PRM. Mansson et al. (2012) introduced the Liu estimator into the PRM. The modified jackknifed ridge estimator for the PRM was introduced by Türkan and Özel (2016). The ridge estimator was introduced into the PRM by Månsson and Shukur (2011). A new two-parameter for PRM was developed by Asar and Genç (2017). Recently, Poisson KL estimator was developed by Lukman et al. (2021) for combating multicollinearity in the PRM.

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, yi is in the form of count data, then it is assumed to follow a Poisson distribution as Po (μi) where μi = e(xiβ), and In µi = (xiβ), xi is the ith row of matrix X which is a n×(p+1) data matrix with p explanatory variables and β is a (p+11 vector of coefficients. The log likelihood of the model is given as:

(2.1)
lμy=i=1nγilog(e(xiβ))(i=1ne(xiβ))i-logi=1nyi!

The most common method of maximizing the likelihood function is to use the iterated weighted least squares (IWLS) algorithm which results to:

(2.2)
β^MLE=XL^X1XL^z^

where L^=diagμ^i and z^ is a vector while the ith element equals z^i=log(μ^i)+yiμ^iμ^i.

The MLE is normally distributed with a covariance matrix that is equivalent to the inverse of the second derivative as:

(2.3)
Covβ^MLE=E2lβjβk1=XL^X1

and the mean squared error is given as:

(2.4)
Eβ^MLE=Eβ^MLEββ^MLEβ=trXL^X1=j=1P1λj

where λj is the jth eigen value of the XV̂X matrix.

The Ridge estimator was adopted by Månsson and Shukur (2011) to solve multicollinearity problem in count data. The estimator is defined as follows:

(2.5)
β^PRE=XL^X+kI1XL^Xβ^MLE

where k=1max(αi2) and (k>0).

The mean squared error is:

(2.6)
MSEβ̂PRE=j=1pλjλj+k2+k2j=1pα̂j2λj+k2

βPRE is effective in practice but it is a complicated function of the biasing parameter k (Liu, 1993).

Mansson et al. (2012) developed the Liu estimator to the Poisson regression model as:

(2.7)
β^PLE=XL^X+I1XL^X+dβ^β^ML

where

(2.8)
d^=max0α^j21α^j2+1λj,0d1.

The MSE for the Liu estimator is defined as:

(2.9)
MSEβ^PLE=j=1Pλj+d2λjλj+12+d12j1pαj2λj+12

where λj is the jth eigenvalue of XL^X and αj is the jth 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

(2.10)
β̂KL=XX+k1XXkβ̂MLE

By means of extension, the Poisson K-L estimator was proposed by Lukman et al. (2021) as follows:

(2.11)
β^PKL=XL^X+k1XL^Xkβ^MLE
(2.12)
MSEβ^PKL=j=1pλjk2λjλj+k2+4k2j=1pαj2λj+k2

where k=minαj22αj2+1λj and k>0.

The Poisson Modified KL estimator (PMKL)

The proposed estimator is obtained as follows: β̂MLE in equation (2.11) is replaced with the ridge estimator. Thus, we have:

(2.13)
β^MKL=XX+k1XXkXX+k1Xy

The properties of the new estimator include:

(2.14)
Eβ̂MKL=XX+kI1XXkIXX+kI1X
(2.15)
Biasβ̂MKL=XX+kI1XXkIXX+kI1Xβ=XX+kI2k3XXkIβ

The bias can be written in scalar form as:

(2.16)
Biasβ̂MKL=kj=1p3λjkβλj+k2
(2.17)
Vβ̂MKL=σ2XX+kI1XXkIXX+kI1XXXX+kI1XXkIXX+kI1

Vβ̂MKL can be represented in scalar form as follows:

(2.18)
Vβ̂MKL=j=1pλjλjk2λj+k4

Thus, the MSE is obtained as:

(2.19)
MSEβ̂MKL=σ2j=1pλjλjk2λj+k4+k2j=1p3λj+k2β2λj+k4

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:

(2.20)
β^PMKL=XL^X+k1XL^XkXL^X+k1XL^Xβ^MLE

The mean squared error of the PMKL is defined as:

(2.21)
MSEβ^PMKL=j=1pλjλjk2λj+k4+k2j=1p3λj+k2αj2λj+k4
where k=min(3λiα2+σ2)24α2σ2λi-3λia2+σ22β2-(3λiα2+σ2) and k > 0.

Suppose α=Qβ and QXL^XQ=Λ=diag(λ1,λ2,..,λp). Where λ1λ2,...,λp, Λ is the matrix of eigen-values of XL^X and Q is the matrix whose columns are the eigenvectors of XL^X.

The mean squared error (MSEM) and 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 Aaa0 if and only if (iff) aA1a1 (Farebrother, 1976).

Lemma 2.2 MSEMβ^1MSEMβ^2=δ2D+b1b1b2b2>0, if and only if b2[σ2D+b1b1]1b2<1 where MSE(β^j)=V(β^j)+bjbj,b1=bias(β^1) and b2=bias(β^2) (Trenker and Toutenburg, 1990).

Theorem 2.1: α^PMKL is preferred to α^PMLE iff, MSEMα^PMLEMSEMα^PMKL>0 provided k > 0.

Proof

Vα̂PMLEVα̂PMKL=Qdiag1λjλjλjk2λj+k4j=1PQ

It is observed that λj+k4λj2λjk2>0 such that the expression above is non-negative for k > 0

Theorem 2.2: α^PMKL is preferred to α^PRE iff, MSEMα^PREMSEMα^PMKL>0 provided k > 0.

Proof

Vα^PREVα^PMKL=Qdiagλjλj+k2λjλjk2λj+k4j=1pQ

We can observe that the difference of the variance of the estimator is non-negative since λj+k2λj2k2>0 for k > 0.

Theorem 2.3: α^PMKL is preferred to α^PLE iff, MSEMα^PLEMSEMα^PMKL>0 provided k > 0 and 0 < d < 1.

Proof

Vα^PLECovα^PMKL=Qdiagλj+d2λjλj+12λjλjk2λj+k4j=1pQ

The difference of the variance is non-negative since

λj+kλj+dλjλj+1λjk>0 for 0 < d < 1 and k > 0.

Theorem 2.4: α^PMKL is preferred to α^PKL iff, MSEMα^PKLMSEMα^PMKL>0 provided k > 0.

Proof

Vα^PKLCovα^PMKL=Qdiagλjk2λjλj+k2λjλjk2λj+k4j=1pQ

The difference of the variance is non-negative since λj+kλjkλjλjk>0 for k > 0.

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:

(2.22)
MSE(β^MKL)=2σ2j=1pλj(λjk)(λj+k)4+4σ2j=1pλj(λjk)2(λj+k)5+2kβ2j=1p(3λj+k)[(3λj+k)+K](λj+k)44k2j=1p(3λj+k)2β2(λj+k)5
By equating to 0 and dividing through by 2 we have the resulting equation as:
(2.23)
j=1pσ2(λj(λjk)(λj+k)4+2σ2λj(λjk)2(λj+k)5+j=1pkβ2(3λj+k)(3λj+2k)(λj+k)42k2(3λj+k)2β2(λj+k)5=0
(2.24)
σ2λj(λjk)[λj+k+2λj2k)]=k(3λj+k)β2[6kλj+2k23λj25kλj2k2]σ2λj(λjk)[3λjk)]=k(3λj+k)β2[kλj3λj2]
Solving the equation above for k yields the biasing parameter k given below as:
(2.25)
kMKL=min3λiα2+σ22+4α2σ2λi3λiα2+σ22β23λiα2+σ2

The shrinkage parameter estimated by Mansson and Shukur, (2011) and Kibria and Lukman (2020) was also adopted for this study as listed:

(2.26)
k1=1maxαj2
(2.27)
k2=p2αj2+1λje
(2.28)
k3=kMKL=k3=min(3λjα2+σ2)2+4α2σ2λj-3λjα2+α22β2-(3λjα2+σ2)

k1 and k2 is the biasing parameter for PMKL1 and PMKL2, while k3 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 Po (μi) where μi=exiβi=1,2,,n and Xi is the ith row of the design matrix with β=β0β1βp being the coefficient vector. The generation of the explanatory variables with different levels of correlation is obtained using

(3.1)
xij=1ρ212zij+ρzip+1;i=1,2,...,nandj=1,2,...,p.

where ρ is the level of multicollinearity between the explanatory variables (Kibria et al. 2015; Kibria and Banik, 2016; Lukman et al., 2019b, Lukman et al. 2020b). zij 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 j=1pβj2=1 and β1=β2==βp. Also, the effect of the intercept value is also being investigated as values are taken to be 1, 0 and -1 (Kibria et al. 2014). The different levels of correlation taken are 0.8, 0.9, 0.95, 0.99 and 0.999. The other factors varied in the simulation study are the sample size n and the number of explanatory variable p. We assume n = 50, 100 and 200 observations and p = 4 and 8 explanatory variables.

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.

Table 1. Simulation result for mean squared error (MSE) when P = 4 and intercept = 1.

β0NρMLEPREPLEPKLPMKL1PMKL2PMKL3
1500.80.03890.03760.03840.03830.03660.03660.0383
0.90.05340.04940.05200.05150.04400.04460.0515
0.950.08520.07290.08080.07910.05530.05740.0806
0.990.35480.20130.28000.24350.06960.07500.2835
0.9993.42440.87571.81910.23020.11870.10411.4952
1000.80.01070.01080.01070.01070.01130.01110.0107
0.90.01250.01240.01250.01250.01250.01250.0125
0.950.01820.01770.01810.01810.01710.01720.0181
0.990.06910.05950.06700.06650.04510.04800.0682
0.9990.64650.29950.50610.44040.08520.09790.6098
2000.80.00570.00560.00560.00560.00600.00590.0056
0.90.00680.00680.00680.00680.00670.00680.0067
0.950.01050.01040.01050.01050.01030.01040.0105
0.990.04220.03940.04160.04150.03470.03570.0419
0.9990.52340.18970.43550.23220.04190.03240.5211

Table 2. Simulation result for mean squared error (MSE) when P = 4 and intercept = 0.

β0nρMLEPREPLEPKLPMKL1PMKL2PMKL3
0500.80.10910.08210.10030.10000.04740.05320.1036
0.90.14790.09780.13030.12950.05050.05560.1386
0.950.23560.12870.19100.18660.05360.05880.2158
0.990.93930.28470.54100.34250.05650.05930.6883
0.9999.47571.93494.80302.21840.25620.18172.9303
1000.80.02950.02660.02910.02910.02300.02380.0294
0.90.03400.03010.03350.03350.02430.02580.0339
0.950.05000.04160.04880.04880.02970.03250.0497
0.990.18960.10880.17120.17060.04200.05030.1867
0.9993.56241.18971.54320.71680.09450.09911.5706
2000.80.01540.01530.01530.01530.01380.01390.0154
0.90.01780.01870.01870.01870.01610.01660.0188
0.950.02620.02840.02840.02840.02230.02340.0286
0.990.82920.10830.10830.10820.04540.05290.1126
0.9991.51850.32220.85480.11830.05430.07430.9527

Table 3. Simulation result for mean squared error (MSE) when P = 4 and intercept = -1.

β0nρMLEPREPLEPKLPMKL1PMKL2PMKL3
-1500.80.30890.22300.25690.22110.20930.21460.2478
0.90.42950.26040.33270.27020.22990.23660.3205
0.950.69240.33720.48900.33640.25620.27550.4573
0.992.78020.77231.51140.33020.30200.37341.0038
0.99926.97265.492814.10599.20160.74861.93804.1156
1000.80.08090.07720.07750.07640.05330.07390.0778
0.90.09350.08340.08910.08860.10430.10040.0812
0.950.13890.11160.12900.12870.11750.10990.1296
0.990.51610.26660.40240.38780.12810.12310.4790
0.9994.68051.12572.67750.75170.19940.20433.5989
2000.80.04210.04260.04170.04090.04040.04230.0409
0.90.05110.04980.05010.04980.04460.05350.0499
0.950.07670.07040.07410.07410.06720.06330.0741
0.990.31070.21150.27660.27310.12260.13090.3016
0.9994.12470.84502.02750.71020.13690.15282.6541

Table 4. Simulation result for mean squared error (MSE) when P = 8 and intercept = 1.

β0nρMLEPREPLEPKLPMKL1PMKL2PMKL3
1500.80.09800.08830.09600.09510.07960.08040.0969
0.90.14040.11630.13550.13290.09060.09380.1369
0.950.22560.16310.21130.20360.10040.10720.2201
0.990.92550.39560.71900.58370.11330.12560.8598
0.9998.47131.94144.88161.04530.30700.27885.8787
1000.80.02320.02290.02310.02310.02270.02270.0231
0.90.03400.03290.03370.03350.03140.03160.0336
0.950.05340.04990.05250.05180.04400.04490.0526
0.990.22260.16340.20260.18910.08530.07920.2081
0.9992.11850.72841.29500.44990.08820.09511.2901
2000.80.00570.00570.00570.00570.00560.00570.0057
0.90.00760.00760.00760.00760.00750.00740.0076
0.950.01170.01150.01160.01160.011300.011340.0116
0.990.04430.04120.04340.04300.03560.03650.0436
0.9991.87220.20710.54910.26710.05660.05620.8730

Table 5. Simulation result for mean squared error (MSE) when P = 8 and intercept = 0.

β0nρMLEPREPLEPKLPMKL1PMKL2PMKL3
0500.80.27380.14730.23770.23520.08080.08790.2682
0.90.39270.18340.32580.32010.08290.08990.3825
0.950.61140.23820.46770.44480.09420.09520.5888
0.992.48580.62181.48820.78650.12050.14282.1778
0.99923.35484.686013.25736.52910.70550.463612.7807
1000.80.06460.05540.06170.06170.04910.04910.0635
0.90.09340.07500.08790.08780.05260.05780.0907
0.950.14620.10410.13270.13240.05530.05930.1395
0.990.60680.25890.42340.25770.05870.06430.5154
0.9995.68541.41782.99510.38780.14330.10702.2332
2000.80.01590.01510.01570.01570.01410.01430.0158
0.90.02070.01960.02050.02050.01760.01810.0206
0.950.03190.02900.03120.03120.02420.02540.0315
0.990.11850.08570.10890.10870.04410.05060.1141
0.9991.75960.46121.08890.18100.07940.09211.0001

Table 6. Simulation result for mean squared error (MSE) when P = 8 and intercept = -1.

β0nρMLEPREPLEPKLPMKL1PMKL2PMKL3
-1500.80.82480.2590.64690.45120.21590.67000.7314
0.91.13550.49450.82530.56990.46930.55260.9927
0.951.77010.62641.17450.69210.58480.56141.4256
0.997.19641.68654.20931.00610.47350.44934.8258
0.99965.876012.825138.072637.73052.04090.901724.8129
1000.80.18000.15420.16530.14930.14220.16160.1547
0.90.25750.20010.22690.20730.20200.18580.2131
0.950.41050.27290.33920.30190.23540.21150.3419
0.991.69140.65361.09830.50390.24200.21801.1012
0.99915.56673.71888.36204.05650.37100.22213.8870
2000.80.04360.04220.04250.04190.03030.04750.0419
0.90.05680.05350.05460.05440.05120.05450.0542
0.950.08600.07660.08100.08080.06930.06910.0813
0.990.32600.21680.27530.25620.10220.10870.2890
0.9998.75941.56393.32662.00650.24330.16211.7855

Real Life Application

Having carried out a simulation study, the efficacy of the proposed estimator needs to be further investigated by considering a real-life application. The Poisson regression model has been applied to the aircraft damage dataset initially by Myers et al. (2012) and subsequently by other researchers such as Asar and Genc (2017) and Amin et al. (2020) among others. By following the Pearson chi-square goodness of fit test, Amin et al. (2020) was able to ascertain that the data fits a Poisson regression model. The test confirms the suitability of the response variable to Poisson distribution with P-value of 6.898122 (0.07521). The dataset provides some detail on two separate aircrafts: The McDonnell Douglas A-4 Skyhawk and the A-6 Grumman Itruder. The dependent variable denotes the number of locations with damage on the aircraft and this follows a Poisson distribution (Asar and Genc, 2017; Amin et al., 2020). The data set has three explanatory variables, X1 shows the type of aircraft which makes the outcome binary (A-4 is coded as 0 and A-6 is coded as 1). X2 is the bomb load in tons and X3 is the number of months of aircrew experience. Meyers et al. (2012) was able to ascertain that the data set is greatly affected by multicollinearity. The eigenvalues of the matrix X were obtained as 4.3333, 374.8961 and 2085.2251. The condition number of 219.3654 was also obtained which is an indication of the problem of multicollinearity since it is greater than 30 (Asar and Genc, 2017). The performance of the estimators is judged based on the mean squared error of each of the estimators.

From 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).

Table 7. Regression coefficients and MSE.

coef.MLEPREPLEPKLPMKL1
(k1)
PMKL2
(k2)
PMKL3
(k3)
α^0-0.406-0.167-0.255-0.107-0.019-0.002-0.077
α^10.5690.3800.4790.3910.1200.1790.322
α^20.1650.1710.1670.1680.1830.1790.172
α^3-0.014-0.015-0.015-0.016-0.017-0.017-0.016
MSE1.0290.2730.4320.2250.0830.0950.092
k2.54440.11201.94092.54441.94090.9905

Conclusion

The parameters in the PRM are commonly estimated using the Maximum Likelihood Estimator. However, literature had shown that the estimator suffers a setback when the explanatory variables are correlated. This problem led to the implementation of alternative estimators with single shrinkage parameters such as the Poisson Ridge Regression Estimator (PRE), Poisson Liu Estimator (PLE) and the Poisson KL Estimator (PKLE). The KL estimator was generally preferred to the ridge regression and Liu estimator in the linear regression model. According to Lukman et al. (2021), the Poisson KL estimator outperforms PRE and PLE. This study modified the KL estimator to propose a new estimator called the Poisson Modified KL estimator (PMKL). The new estimator falls in the same class with the ridge, Liu and KL estimators since they possessed a single shrinkage parameter. We investigated the performance of the estimators with a simulation study and a real-life application. From the results, we observed that the new estimator consistently performed well in the presence of multicollinearity with the lowest MSE. Finally, the new estimator is more suitable to combat multicollinearity in the PRM.

Data availability

All data underlying the results are available as part of the article and no additional source data are required.

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Aladeitan BB, Adebimpe O, Lukman AF et al. Modified Kibria-Lukman (MKL) estimator for the Poisson Regression Model: application and simulation [version 2; peer review: 2 approved, 1 approved with reservations]. F1000Research 2021, 10:548 (https://doi.org/10.12688/f1000research.53987.2)
NOTE: If applicable, it is important to ensure the information in square brackets after the title is included in all citations of this article.
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Open Peer Review

Current Reviewer Status: ?
Key to Reviewer Statuses VIEW
ApprovedThe paper is scientifically sound in its current form and only minor, if any, improvements are suggested
Approved with reservations A number of small changes, sometimes more significant revisions are required to address specific details and improve the papers academic merit.
Not approvedFundamental flaws in the paper seriously undermine the findings and conclusions
Version 2
VERSION 2
PUBLISHED 14 Dec 2021
Revised
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7
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Reviewer Report 21 Dec 2021
Muhammad Amin, Department of Statistics, University of Sargodha, Sargodha, Pakistan 
Approved
VIEWS 7
This paper can be accepted for indexing after incorporation of the following minor points:
  1. Check equations 2.1 and 3.1.
     
  2. Change "PMKL" to "PMKLE" in the whole manuscript.
     
... Continue reading
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HOW TO CITE THIS REPORT
Amin M. Reviewer Report For: Modified Kibria-Lukman (MKL) estimator for the Poisson Regression Model: application and simulation [version 2; peer review: 2 approved, 1 approved with reservations]. F1000Research 2021, 10:548 (https://doi.org/10.5256/f1000research.78664.r113239)
NOTE: it is important to ensure the information in square brackets after the title is included in all citations of this article.
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2
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Reviewer Report 15 Dec 2021
Mohammad Arashi, Department of Statistics, Faculty of Mathematical Sciences, Ferdowsi University of Mashhad, Mashhad, Iran 
Approved
VIEWS 2
The authors covered all the raised comments ... Continue reading
CITE
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HOW TO CITE THIS REPORT
Arashi M. Reviewer Report For: Modified Kibria-Lukman (MKL) estimator for the Poisson Regression Model: application and simulation [version 2; peer review: 2 approved, 1 approved with reservations]. F1000Research 2021, 10:548 (https://doi.org/10.5256/f1000research.78664.r113241)
NOTE: it is important to ensure the information in square brackets after the title is included in all citations of this article.
Version 1
VERSION 1
PUBLISHED 08 Jul 2021
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14
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Reviewer Report 09 Aug 2021
Nimet Özbay, Department of Statistics, Faculty of Science and Letters, Çukurova University, Adana, Turkey 
Approved with Reservations
VIEWS 14
This article focuses on proposing a modified KL estimator to mitigate the Poisson Regression Model with multicollinearity. Some theoretical properties of the new estimator are examined. A numerical example is conducted to show the performance of the new estimator. I ... Continue reading
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Özbay N. Reviewer Report For: Modified Kibria-Lukman (MKL) estimator for the Poisson Regression Model: application and simulation [version 2; peer review: 2 approved, 1 approved with reservations]. F1000Research 2021, 10:548 (https://doi.org/10.5256/f1000research.57427.r89263)
NOTE: it is important to ensure the information in square brackets after the title is included in all citations of this article.
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17
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Reviewer Report 03 Aug 2021
Muhammad Amin, Department of Statistics, University of Sargodha, Sargodha, Pakistan 
Approved with Reservations
VIEWS 17
In this paper, the authors introduced a new estimator by modified KL estimator for the Poisson regression model to overcome the effect of multicollinearity. The paper is original and deals with a topic of interest. This paper could be accepted ... Continue reading
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HOW TO CITE THIS REPORT
Amin M. Reviewer Report For: Modified Kibria-Lukman (MKL) estimator for the Poisson Regression Model: application and simulation [version 2; peer review: 2 approved, 1 approved with reservations]. F1000Research 2021, 10:548 (https://doi.org/10.5256/f1000research.57427.r89258)
NOTE: it is important to ensure the information in square brackets after the title is included in all citations of this article.
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15
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Reviewer Report 21 Jul 2021
Mohammad Arashi, Department of Statistics, Faculty of Mathematical Sciences, Ferdowsi University of Mashhad, Mashhad, Iran 
Approved with Reservations
VIEWS 15
The paper extends the Liu estimator in generalized linear modeling. Specifically, the authors propose a new biased estimator for the estimation of regression coefficients in the discrete Poisson regression.

The results are interesting and the topic is ... Continue reading
CITE
CITE
HOW TO CITE THIS REPORT
Arashi M. Reviewer Report For: Modified Kibria-Lukman (MKL) estimator for the Poisson Regression Model: application and simulation [version 2; peer review: 2 approved, 1 approved with reservations]. F1000Research 2021, 10:548 (https://doi.org/10.5256/f1000research.57427.r89260)
NOTE: it is important to ensure the information in square brackets after the title is included in all citations of this article.
  • Author Response 27 Jul 2021
    BENEDICTA Aladeitan, Department of Physical Sciences, Landmark University, Omu-Aran, +234, Nigeria
    27 Jul 2021
    Author Response
    Thanks for your observations and corrections. All will be duely implemented.
    Competing Interests: No competing interests were disclosed.
COMMENTS ON THIS REPORT
  • Author Response 27 Jul 2021
    BENEDICTA Aladeitan, Department of Physical Sciences, Landmark University, Omu-Aran, +234, Nigeria
    27 Jul 2021
    Author Response
    Thanks for your observations and corrections. All will be duely implemented.
    Competing Interests: No competing interests were disclosed.

Comments on this article Comments (0)

Version 2
VERSION 2 PUBLISHED 08 Jul 2021
Comment
Alongside their report, reviewers assign a status to the article:
Approved - the paper is scientifically sound in its current form and only minor, if any, improvements are suggested
Approved with reservations - A number of small changes, sometimes more significant revisions are required to address specific details and improve the papers academic merit.
Not approved - fundamental flaws in the paper seriously undermine the findings and conclusions
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