Development of the generalized ridge estimator for the Poisson-Inverse Gaussian regression model with multicollinearity

Abstract

The Poisson-Inverse Gaussian regression model is a widely used method for analyzing count data, particularly in over-dispersion. However, the reliability of parameter estimates obtained through maximum likelihood estimation in this model can be compromised when multicollinearity exists among the explanatory variables. Multicollinearity means that high correlations between explanatory variables inflate the variance of the maximum likelihood estimates and increase the mean squared error. To handle this problem, the Poisson-Inverse Gaussian ridge regression estimator has been proposed as a viable alternative. This paper introduces a generalized ridge estimator to estimate regression coefficients in the Poisson-Inverse Gaussian regression model under multicollinearity. The performance of the proposed estimator is evaluated through a comprehensive simulation study, covering various scenarios and employing the mean squared error as the evaluation criterion. Furthermore, the practical applicability of the estimator is demonstrated using two real-life datasets, with its performance again assessed based on mean squared error. Theoretical analyses, supported by simulation and empirical findings, suggest that the proposed estimator outperforms existing methods, offering a more reliable solution in multicollinearity.

Introduction

Count data characterized by non-negative integer values frequently arises in various disciplines, necessitating the use of specialized regression models to explore relationships between the response variable and its predictors. These models facilitate the identification of significant explanatory variables and their impact on the response^1,2. Among the most commonly employed models is the Poisson regression model (PRM), which assumes that the mean and variance are equal (i.e., equi-dispersion). However, this assumption is often unrealistic in empirical datasets, where the variance either exceeds (over-dispersion) or is less than (under-dispersion) the mean. In such situations, applying the Poisson model may lead to biased estimates and unreliable conclusions³. To address the limitations of equi-dispersion, alternative count regression models, including the negative binomial (NBRM)⁴, Bell (BRM)⁵, quasi-Poisson (QPRM)⁶, generalized Poisson Lindley (GPLRM)⁷, Conway-Maxwell Poisson (CMPRM)⁸, and Poisson-Inverse Gaussian (PIGRM) models⁹, have been proposed. These models extend the Poisson framework to accommodate over-dispersed and under-dispersed data, thereby enhancing model accuracy and interpretability.

The PIGRM is highly effective for datasets with heavy tails, where a small number of extreme values differ significantly from most data near zero. This type of data is common in fields like actuarial science, biology, engineering, and medical research. The PIGRM is preferred over the NBRM for its ability to handle greater skewness and kurtosis, making it ideal for heavy-tailed count data⁹. Willmot³ highlighted the PIGRM as a better alternative to the NBRM for highly skewed and heterogeneous datasets. Putri et al.¹⁰ found PIGRM effective in managing overdispersion when compared to NBRM. Similarly, Saraiva et al.¹¹ used PIGRM for overdispersed dengue data, and Husna and Azizah¹² applied it to study dengue hemorrhagic fever in Central Java, showcasing its versatility for complex count data.

One of the fundamental assumptions of the generalized linear regression model is the absence of correlation among explanatory variables. However, in empirical settings, explanatory variables frequently exhibit strong or near-strong linear relationships, thereby violating this assumption and giving rise to the problem of multicollinearity¹³. This issue also manifests in the context of the PIGRM, where multicollinearity complicates parameter estimation, inflated variance, and increases mean squared error (MSE). The maximum likelihood estimator (MLE), commonly employed for estimating regression coefficients in the PIGRM, is particularly sensitive to multicollinearity, as it results in inflated variances and unreliable parameter estimates¹⁴. To tackle multicollinearity in regression models, ridge regression and the Liu estimator are two commonly studied techniques. Ridge regression, introduced by Hoerl and Kennard¹⁵, has been extensively explored, with notable studies focusing on optimal shrinkage parameters by Golub et al.¹⁶, and Alkhamisi et al.¹⁷. Segerstedt¹⁸ extended ridge regression to the generalized linear model (GLM), with further advancements by researchers like Månsson¹⁹, Saleh et al.²⁰, Amin et al.²¹, Sami et al.²², and Shahzad et al.²³. On anther hand Kejian²⁴ introduced the Liu estimator, which offers a linear shrinkage parameter and has been applied to different models. Kibria²⁵ and Alheety and Kibria²⁶ developed the Liu estimator for linear regression models (LRMs) and later Kurtoǧlu²⁷ defined the Liu estimator for GLMs. Then, inspiring further applications by Qasim et al.²⁸ and Bulut²⁹.

Several studies have focused on developing generalized versions of the ridge estimator to handle multicollinearity in different types of regression models. Rashad et al.³⁰ proposed a generalized ridge estimator for the NBRM, while Akram et al.³¹ extended this idea to both generalized ridge and generalized Liu estimators in the gamma regression. Fayose and Ayinde³² explored how to define an appropriate biasing parameter for the generalized ridge regression estimator, and Bhat and Raju³³ introduced a class of such estimators. Gómez et al.³⁴ discussed biased estimation using generalized ridge regression for multiple LRMs. Abdulazeez and Algamal³⁵ combined shrinkage estimation with particle swarm optimization to define a generalized ridge estimator. In addition, van Wieringen³⁶ applied the generalized ridge approach to estimate the inverse covariance matrix, and Mohammadi³⁷ suggested a test for detecting harmful multicollinearity based on this method. Bello and Ayinde³⁸ also proposed feasible generalized ridge estimators for models affected by both multicollinearity and autocorrelation. Recent works^30,35,39,40 have shown that the performance of biased estimators can be improved by allowing the biasing parameters to vary across observations (such as ) rather than using a fixed value (such as ). Based on this, the main goal of the present paper is to develop a more general form of the ridge estimator to better address multicollinearity in the PIGRM.

This paper discusses the advantages of the generalized ridge estimator in the PIGRM. We also present new methods for determining the optimal values of the shrinkage coefficients (kj), based on the methodologies of Rashad³⁰ and Dawood et al.³⁹ The careful selection of these coefficients is crucial to improving the performance of the biased generalized ridge estimation method. Monte Carlo simulations use the MSE to evaluate the performance of the proposed estimator compared to existing estimators. To validate the simulation results, we use two real-world data studies that confirm the effectiveness of the estimator, highlighting its improved accuracy and robustness in addressing problems related to multicollinearity.

The structure of this paper is as follows: Section 2 provides a summary of the PIGRM model and examines biased estimation approaches, including PIGRRE and the proposed estimator. It also discusses theoretical comparisons of the proposed estimator and methods for selecting biasing parameters. Section 3 describes the study and results of the Monte Carlo simulation used to evaluate the proposed estimator. Section 4 highlights the application of the estimator through two real-world datasets to confirm the simulation results. Finally, Section 5 summarizes the results and discusses their broader significance.

Methodology

The PIG distribution combines two probability distributions: the Poisson and the Inverse Gaussian. Let represent a random variable following a Poisson distribution with a mean of , where itself follows an Inverse Gaussian distribution with mean 1 and dispersion parameter . The probability mass function (PMF) of is expressed as^1,41:

1

where , and the probability density function of is given by⁴²:

2

The marginal PMF of , representing the PIG distribution, is derived by integrating over . This results in the following expression:

3

where , , and denotes the modified Bessel function of the third kind⁴³. The mean and variance of the distribution are and , respectively.

In the context of PIGRM, the linear predictor is used, which is equivalent to , where is the linear combination of the explanatory variables, defined as , where is the vector of regression coefficients, and is the vector of explanatory variables for the -th observation^1,43.

To estimate the parameters of the PIGRM, the MLE method is employed. The log-likelihood function for the model is as follows:

4

The log-likelihood function with respect to the regression coefficients is given by:

5

To obtain the MLE of the parameters, we differentiate the log-likelihood function concerning the parameters and set the derivatives equal to zero. The first derivative with respect to is¹⁰:

6

Similarly, thetove with respect to is:

7

To estimate the parameters and through MLE, numerical methods such as the Newton-Raphson algorithm or the iteratively reweighted least squares (IRLS) algorithm are utilized, given the non-linearity of the equations (Eqs. (6) and (7)). Upon completion of the iterative procedure, the MLE estimate for is derived from the following expression¹⁴:

8

where and , the adjusted response variable, is computed as . To evaluate the accuracy of the estimator, the mean squared error matrix (MSEM) and MSE of are determined through spectral decomposition of the matrix . where is the diagonal matrix of the eigenvalues of , and is the orthogonal matrix with columns corresponding to the eigenvectors of .

The MSEM for the estimator is given by:

9

The MSE is calculated as:

10

where is the estimated value of and computed as .

Poisson-Inverse Gaussian ridge regression estimator

Segerstedt¹⁸ proposed a ridge regression estimator to handle multicollinearity in GLM as an alternative to the MLE. Accordingly, Månsson¹⁹, Månsson and Shukur⁴⁴, Kibria et al.⁴⁵, and Ashraf⁴⁶ apply it to many GLMs. Following this research, Batool et al.¹⁴ proposed a PIGRRE to address multicollinearity in PIGRM. The PIGRRE is given by:

11

where represents a PIGRRE parameter, and is the identity matrix of . In particular, when , the PIGRRE reduces to MLE. The bias and covariance matrix associated with the PIGRRE are as follows:

12

13

where is a diagonal matrix consisting of the eigenvalues of the covariance matrix, adjusted by the ridge parameter.

The MSEM of the PIGRRE estimator combines both the covariance and the squared bias:

14

Expanding the above expression:

15

where . The MSE is calculated as the trace of the MSEM:

16

where denotes the -th component of the vector , representing the MLE-adjusted coefficient values.

Proposed estimator

Building on the work of Bhat and Raju³³ and extending the contributions of Rashad et al.³⁰ and Akram et al.³¹, we propose the generalized ridge estimator for the PIGRM. The Poisson-Inverse Gaussian generalized ridge estimator (PIGGRE) enhances standard ridge regression by assigning a unique shrinkage parameter to each regression coefficient, improving its ability to address multicollinearity.

In the PIGGRE method, the ridge parameter matrix is written as , where each controls how much shrinkage is applied to the -th regression coefficient. This setup gives more flexibility and improves prediction accuracy and addresses multicollinearity than the standard PIGRRE approach.

Key scenarios in the PIGGRE framework include:

1. for all : the estimator reduces to the MLE.

2. (constant) for all : the estimator corresponds to PIGRRE.

3. varying across coefficients: the estimator is referred to as PIGGRE

The PIGGRE is defined as:

17

The bias and covariance matrix associated with the PIGGRE are as follows:

18

19

The MSEM of the PIGGRE estimator is defined by:

20

where and . The MSE is calculated as the trace of the MSEM:

21

Superiority of PIGGRE

In this part, we compare the proposed PIGGRE method with other known estimators, such as MLE and PIGRRE. To support this comparison, we make use of the following lemma:

Lemma 1

Trenkler and Toutenburg⁴⁷. Consider two linear estimators of , denoted by for Let where is the covariance matrix of and assume Additionally, define the bias of as Then, the difference in MSEM between the two estimators can be expressed as:

where the MSEM of is given by:

This inequality is satisfied if and only if:

Theorem 2.1

The PIGGRE estimator, is superior to an alternative estimator if and only if:

Proof

The difference (D) between the covariance matrices of the two estimators is given by:

This can be expressed as:

The matrix is positive definite if and only if or . It is evident that for (), the term is positive, ensuring that is positive definite. Thus, by Lemma 3, the superiority condition of over is satisfied.

Theorem 2.2

The PIGGRE estimator, , is superior to an alternative estimator if and only if:

Proof

The difference (D) between the covariance matrices of the two estimators is given by:

This can be expressed as:

The matrix is positive definite if and only if or . It is evident that for (), the term is positive, ensuring that is positive definite. Thus, by Lemma 3, the superiority condition of over is satisfied.

The biasing parameters of the PIGGRE

To estimate the optimal values of for the proposed PIGRRE, the approach minimizes the MSE of the estimator, defined as:

To find the optimal values of , differentiate with respect to and set the derivative to zero Solving for , we obtain:

where and represent the estimated values of and , respectively.

Following Bhat and Raju³³ and Rashad et al.³⁰, we considered the following values of :

22

23

24

25

26

Monte Carlo simulation

Simulation design

The simulation procedure for evaluating the performance of the proposed estimator involves several steps, considering key factors such as sample size (), the number of explanatory variables (), levels of multicollinearity (), and dispersion parameters (), as detailed in Table 1. The simulation process is outlined as follows:

1. Choose the regression coefficients such that the sum of their squared values equals 1, i.e., . This normalization is a common assumption in regression modeling.

2. Use PIG() distribution to generate the response variable for the PIGRM, where . Here, represents the explanatory variables for observation , and are the regression coefficients.

3. Simulate the correlated explanatory variables using the formula:

   

   where are independent standard normal random variables. This ensures the correlation structure between the explanatory variables, with controlling the degree of correlation. This step is repeated for and .

4. Use the ‘gamlss‘ package in R to estimate the regression parameters based on the simulated data, selecting the PIG family for the model.

5. Repeat the generation of the entire data and estimation process for different combinations of ,, , and for 1000 replications to ensure the robustness of the results.

6. Evaluate the performance of the proposed estimator using the MSE criterion:

   27

   where represents the estimated parameters from the -th replication, is the true parameter vector, and is the total number of replications (1000 in this case). This criterion quantifies the deviation between the true parameters and the estimated ones, helping to assess the accuracy of the estimator.

   Following Segerstedt¹⁸, Månsson and Shukur⁴⁴, and Batool et al.¹⁴, we use the following values of for PIGRRE:

   

Table 1.

Different factors considered in the simulation study.

Factor	Notation	Values
Sample size	n	30, 75, 150, 200, 300, 400
Degree of correlation		0.8, 0.85, 0.9, 0.95, 0.99
Dispersion parameter		2, 4, 6
Number of explanatory variables	p	3, 6, 9

Simulation results

Tables 2, 3, 4, 5, 6, 7, 8, 9 and 10 introduced the simulated MSE values for the MLE, PIGRRE, and PIGGRE, based on the results of the simulation analysis:

1. An increase in the degree of multicollinearity (), the dispersion parameter (), or the number of explanatory variables () results in higher MSE values for all estimators, indicating a degradation in performance under these conditions.

2. The MSE values decrease as the sample size () increases, emphasizing the role of larger sample sizes in improving estimator reliability and precision.

3. The PIGRRE demonstrates superior performance compared to the MLE across all scenarios, consistently yielding lower MSE values irrespective of variations in , , , and .

4. The PIGGRE outperforms the PIGRRE, achieving the lowest MSE values among the evaluated estimators across all parameters irrespective of variations in , , , and .

5. The results show that the proposed PIGGRE estimator works efficiently, especially when the dispersion parameter () is high and the explanatory variables are highly correlated. This indicates that the estimator is reliable even in complex situations where traditional methods may struggle.

6. Figures 1a–d show the simulation results by presenting the mean squared errors (MSEs) for different estimators under various settings, such as different sample sizes (), levels of multicollinearity (), numbers of predictors (), and values of the dispersion parameter (). The results clearly show that PIGGRE performs better than both PIGRRE and MLE by consistently achieving lower MSE values. This improvement is especially clear when using the ridge parameters and , which help reduce estimation errors and improve the reliability of the method.

Table 2.

Estimated MSE for different estimators at p = 3 and .

	n	PIGMLE	PIGRRE					PIGGRE
		MLE
0.80	30	1.50903	1.24908	1.05844	1.19373	0.93372	0.78300	0.64993	0.51268	0.38956	0.80052	0.59035
	75	0.43763	0.35645	0.30387	0.33531	0.24895	0.20746	0.16700	0.12728	0.10926	0.16970	0.14486
	150	0.17445	0.16225	0.15243	0.15606	0.12085	0.10453	0.08896	0.07361	0.06486	0.08581	0.08132
	200	0.13210	0.12172	0.11362	0.11633	0.08769	0.07615	0.06554	0.05902	0.05903	0.06267	0.06149
	300	0.09989	0.09490	0.09065	0.09181	0.07149	0.06248	0.05440	0.04846	0.04628	0.05124	0.05142
	400	0.06679	0.06459	0.06261	0.06302	0.05011	0.04448	0.03945	0.03627	0.03584	0.03737	0.03779
0.85	30	1.33903	1.03215	0.81491	0.93519	0.67985	0.54769	0.43190	0.31404	0.23184	0.52534	0.37503
	75	0.65692	0.57372	0.51132	0.55719	0.43693	0.36691	0.30437	0.23738	0.17464	0.34472	0.27680
	150	0.22981	0.20421	0.18578	0.19403	0.14468	0.12191	0.10076	0.08150	0.07053	0.09801	0.09105
	200	0.19751	0.18076	0.16739	0.17354	0.13115	0.11071	0.09180	0.07249	0.05783	0.08903	0.08302
	300	0.10427	0.09946	0.09528	0.09643	0.07626	0.06648	0.05727	0.04806	0.04097	0.05369	0.05319
	400	0.09503	0.09069	0.08688	0.08787	0.06831	0.05898	0.05041	0.04229	0.03676	0.04573	0.04674
0.90	30	3.58361	2.85458	2.30876	2.54826	1.56711	1.20000	0.95493	0.74698	0.54494	1.36040	0.87907
	75	0.88513	0.74347	0.64152	0.72270	0.53783	0.43328	0.34501	0.25344	0.17309	0.41951	0.30692
	150	0.35437	0.29885	0.26082	0.28156	0.19809	0.15990	0.12506	0.08985	0.06812	0.12571	0.10756
	200	0.22758	0.20208	0.18371	0.19342	0.14633	0.12096	0.09833	0.07481	0.05595	0.09713	0.08806
	300	0.14540	0.13580	0.12779	0.13047	0.09861	0.08297	0.06836	0.05235	0.03993	0.06313	0.06118
	400	0.14155	0.13121	0.12290	0.12548	0.09150	0.07649	0.06253	0.04802	0.03815	0.05602	0.05570
0.95	30	3.97676	3.08556	2.44899	2.73670	1.69227	1.27262	0.96068	0.68571	0.46335	1.43603	0.84215
	75	1.20052	0.87025	0.65469	0.80304	0.53416	0.40297	0.29942	0.20375	0.13700	0.38052	0.25500
	150	0.70068	0.56333	0.46752	0.53661	0.36893	0.28392	0.21534	0.14823	0.09576	0.24890	0.18634
	200	0.45861	0.38447	0.33131	0.36771	0.25950	0.20289	0.15709	0.11229	0.07600	0.17257	0.13814
	300	0.30176	0.26922	0.24446	0.25939	0.19246	0.15486	0.12314	0.09034	0.05957	0.12884	0.11027
	400	0.21079	0.18927	0.17206	0.18030	0.12914	0.10319	0.08094	0.05781	0.03919	0.07693	0.07099
0.99	30	31.38604	22.74318	16.44907	17.68168	4.77521	2.92910	1.88350	1.14688	0.61308	5.07014	1.57104
	75	6.78964	4.53029	2.91014	3.48779	1.20196	0.74147	0.45384	0.23894	0.13677	0.88678	0.34049
	150	3.38447	2.79009	2.31554	2.60513	1.43551	0.98270	0.68667	0.44689	0.23208	1.15075	0.60028
	200	2.08629	1.47794	1.05982	1.31217	0.73104	0.49374	0.33699	0.20953	0.11263	0.50359	0.28436
	300	1.58204	1.05168	0.70881	0.90450	0.41763	0.26520	0.16730	0.09091	0.05055	0.21931	0.12933
	400	1.01811	0.70303	0.50180	0.63887	0.35199	0.23691	0.15914	0.09560	0.05173	0.19491	0.13119

Bolded values indicate the lowest MSE.

Table 3.

Estimated MSE for different estimators at p = 3 and .

	n	PIGMLE	PIGRRE					PIGGRE
		MLE
0.80	30	2.15564	1.62867	1.26034	1.48080	1.13092	0.91920	0.73896	0.56563	0.42561	0.95081	0.65980
	75	0.62399	0.44114	0.33711	0.40462	0.30984	0.25322	0.19872	0.14992	0.13231	0.20734	0.16961
	150	0.28150	0.24072	0.21508	0.22843	0.17555	0.14915	0.12432	0.10156	0.09059	0.12330	0.11225
	200	0.21743	0.18767	0.16806	0.17591	0.13249	0.11321	0.09564	0.08547	0.08455	0.09314	0.08920
	300	0.14747	0.13257	0.12124	0.12518	0.09427	0.08075	0.06923	0.06381	0.06458	0.06681	0.06582
	400	0.10194	0.09507	0.08956	0.09106	0.07080	0.06225	0.05492	0.05249	0.05386	0.05418	0.05326
0.85	30	2.12999	1.59841	1.21674	1.42659	1.03492	0.82703	0.64929	0.47434	0.35167	0.82755	0.56511
	75	0.85339	0.66763	0.54621	0.62403	0.47125	0.38334	0.30635	0.23052	0.17563	0.35370	0.27082
	150	0.32325	0.25889	0.21966	0.24128	0.17382	0.14389	0.11674	0.09707	0.09097	0.11582	0.10499
	200	0.28602	0.24337	0.21510	0.23000	0.17163	0.14315	0.11746	0.09388	0.07967	0.11609	0.10566
	300	0.15523	0.14109	0.12985	0.13407	0.09973	0.08410	0.06993	0.05734	0.05135	0.06550	0.06350
	400	0.12983	0.11635	0.10633	0.10981	0.08100	0.06861	0.05759	0.04984	0.04786	0.05331	0.05320
0.90	30	5.95678	4.45542	3.31046	3.62954	1.85292	1.38258	1.05263	0.76455	0.52849	1.65059	0.92996
	75	1.49481	1.13027	0.88044	1.06255	0.77398	0.60630	0.47192	0.34292	0.23640	0.59671	0.41647
	150	0.46122	0.32829	0.25511	0.30053	0.21332	0.17059	0.13194	0.09783	0.08358	0.13285	0.11247
	200	0.35818	0.29545	0.25558	0.27933	0.20138	0.16281	0.12847	0.09377	0.07187	0.12901	0.11151
	300	0.21791	0.19108	0.17142	0.18057	0.13106	0.10847	0.08775	0.06745	0.05537	0.08265	0.07756
	400	0.17353	0.14721	0.12981	0.13643	0.09369	0.07638	0.06094	0.04827	0.04352	0.05559	0.05384
0.95	30	5.85553	4.23101	3.06481	3.50715	2.05555	1.53282	1.14824	0.82090	0.57591	1.78159	0.99895
	75	1.89321	1.20766	0.77528	1.00843	0.60700	0.43625	0.30484	0.19204	0.13538	0.40659	0.24463
	150	0.88355	0.61185	0.44801	0.55063	0.36559	0.27905	0.20833	0.14320	0.10411	0.24542	0.17560
	200	0.84905	0.62570	0.48226	0.58130	0.38819	0.29221	0.21706	0.14866	0.10070	0.24856	0.18581
	300	0.51988	0.40558	0.32990	0.38636	0.26687	0.20256	0.15180	0.10410	0.06906	0.16151	0.13076
	400	0.28943	0.23742	0.20348	0.22272	0.15762	0.12396	0.09570	0.06783	0.04804	0.09500	0.08289
0.99	30	45.55769	29.48378	18.00619	19.99563	4.23131	2.63187	1.70194	1.02752	0.62923	4.79892	1.37238
	75	10.41812	6.79721	4.30754	4.97318	1.47997	0.91597	0.55986	0.29412	0.17674	1.19741	0.41395
	150	4.93233	3.78519	2.89201	3.38328	1.67366	1.10657	0.75795	0.48877	0.26770	1.35780	0.65364
	200	3.45022	2.33896	1.54919	1.90179	0.83609	0.51968	0.32810	0.18562	0.10020	0.53587	0.26210
	300	2.48682	1.48487	0.83898	1.09458	0.44296	0.27583	0.16921	0.08911	0.05474	0.23260	0.12536
	400	1.68915	1.09843	0.70556	0.92756	0.44143	0.28157	0.17855	0.09780	0.05495	0.23097	0.13824

Bolded values indicate the lowest MSE.

Table 4.

Estimated MSE for different estimators at p = 3 and .

	n	PIGMLE	PIGRRE					PIGGRE
		MLE
0.80	30	2.90019	2.20724	1.72061	1.99130	1.45367	1.15795	0.91159	0.67491	0.48868	1.24105	0.80376
	75	0.84714	0.54942	0.39919	0.48663	0.37825	0.30893	0.24367	0.19449	0.17728	0.25995	0.21300
	150	0.36724	0.30063	0.26087	0.28459	0.21971	0.18564	0.15421	0.12760	0.11351	0.15621	0.14014
	200	0.29467	0.23654	0.20356	0.22104	0.16577	0.14063	0.11804	0.10805	0.10841	0.11667	0.11084
	300	0.16791	0.14091	0.12478	0.13085	0.09982	0.08647	0.07549	0.07559	0.08026	0.07609	0.07423
	400	0.13481	0.12420	0.11571	0.11844	0.09039	0.07874	0.06856	0.06386	0.06462	0.06633	0.06554
0.85	30	2.70445	1.97343	1.45363	1.69922	1.19842	0.94863	0.73445	0.52226	0.38589	0.97028	0.62825
	75	0.96421	0.73353	0.59520	0.68536	0.53059	0.43440	0.35153	0.27059	0.21318	0.41135	0.31338
	150	0.41371	0.30945	0.25267	0.28514	0.21272	0.17604	0.14245	0.11879	0.11133	0.14395	0.12802
	200	0.36780	0.29487	0.25045	0.27571	0.20295	0.16708	0.13430	0.10385	0.08829	0.13348	0.11807
	300	0.21338	0.18539	0.16653	0.17541	0.13271	0.11217	0.09332	0.07629	0.06716	0.08944	0.08459
	400	0.15396	0.13281	0.11836	0.12368	0.09092	0.07688	0.06437	0.05658	0.05553	0.06044	0.05957
0.90	30	6.95244	5.23459	3.92789	4.32501	2.21047	1.66753	1.29739	0.97659	0.69283	2.05466	1.16542
	75	1.88609	1.41759	1.10233	1.30716	0.90559	0.69904	0.53414	0.37684	0.26102	0.70709	0.46271
	150	0.74052	0.50768	0.37802	0.46105	0.33130	0.26086	0.20057	0.14946	0.12194	0.21673	0.17339
	200	0.43575	0.33393	0.27271	0.31363	0.22773	0.18191	0.14234	0.10541	0.08278	0.14591	0.12396
	300	0.25732	0.20929	0.17918	0.19336	0.13832	0.11236	0.08858	0.06625	0.05458	0.08446	0.07674
	400	0.25021	0.19843	0.16748	0.18120	0.12670	0.10205	0.08004	0.06167	0.05381	0.07400	0.06984
0.95	30	9.73192	7.17397	5.33250	5.87416	2.35866	1.63297	1.13038	0.71996	0.49419	2.13076	0.92197
	75	2.80585	1.87235	1.25238	1.51528	0.90806	0.66265	0.47272	0.30603	0.21738	0.65863	0.38505
	150	1.43859	1.00031	0.70947	0.88018	0.54209	0.39793	0.28712	0.18951	0.13070	0.35492	0.23885
	200	0.84493	0.58278	0.42076	0.53114	0.34931	0.26018	0.19142	0.13139	0.09567	0.22258	0.16201
	300	0.60130	0.44141	0.34106	0.41020	0.27474	0.20693	0.15341	0.10421	0.07325	0.16366	0.12972
	400	0.38389	0.29919	0.24516	0.27964	0.18930	0.14532	0.10909	0.07426	0.05266	0.10918	0.09213
0.99	30	52.69233	35.16393	22.90962	24.88603	5.10796	3.02614	1.87835	1.08187	0.63873	6.05239	1.49412
	75	12.94604	7.31371	3.86940	4.55806	1.54210	0.96409	0.61110	0.35747	0.21923	1.36305	0.48390
	150	5.50229	4.02173	2.89635	3.37138	1.41976	0.90134	0.58921	0.35467	0.20141	1.12812	0.48469
	200	4.22484	2.78537	1.78618	2.18736	0.92028	0.57639	0.36450	0.20462	0.12036	0.62550	0.28512
	300	3.14565	1.82022	0.97316	1.27665	0.48990	0.30300	0.18485	0.09801	0.06276	0.26413	0.13621
	400	2.27878	1.46754	0.92196	1.18953	0.53024	0.33265	0.20825	0.11376	0.06519	0.28260	0.16051

Bolded values indicate the lowest MSE.

Table 5.

Estimated MSE for different estimators at p = 6 and .

	n	PIGMLE	PIGRRE					PIGGRE
		MLE
0.80	30	3.35069	2.98528	2.67866	2.94989	2.03342	1.70706	1.40527	0.75811	0.53697	1.72209	0.83496
	75	1.17938	0.93639	0.76699	0.92204	0.56131	0.45736	0.34374	0.14161	0.11721	0.39978	0.15548
	150	0.47088	0.41992	0.38250	0.40788	0.26458	0.22035	0.17135	0.07385	0.05830	0.18506	0.08185
	200	0.35761	0.34573	0.33476	0.34131	0.25489	0.22057	0.18424	0.09452	0.06781	0.19773	0.10498
	300	0.22132	0.20958	0.19933	0.20365	0.13242	0.11207	0.08846	0.04409	0.04063	0.08890	0.04669
	400	0.16608	0.16349	0.16098	0.16182	0.12188	0.10607	0.08916	0.04710	0.03606	0.09065	0.05181
0.85	30	5.18336	4.54392	3.98710	4.37839	2.57827	2.07467	1.65996	0.83043	0.54828	2.16599	0.92901
	75	1.37263	1.12642	0.94529	1.11162	0.69646	0.56443	0.42732	0.16973	0.12349	0.51496	0.19164
	150	0.54982	0.51432	0.48418	0.50807	0.35667	0.30207	0.24401	0.11025	0.07663	0.26897	0.12459
	200	0.42809	0.40186	0.37962	0.39423	0.27310	0.23014	0.18448	0.08142	0.05688	0.19967	0.09211
	300	0.27777	0.26318	0.25060	0.25653	0.17187	0.14493	0.11523	0.05345	0.04279	0.12079	0.05874
	400	0.19416	0.19021	0.18644	0.18785	0.13703	0.11808	0.09766	0.04777	0.03502	0.10085	0.05326
0.90	30	7.89813	6.63853	5.58868	6.31368	3.75147	3.00520	2.37028	1.15168	0.74315	3.19024	1.29298
	75	1.56800	1.42166	1.30084	1.42954	1.04987	0.89206	0.73807	0.38219	0.26099	0.87122	0.42526
	150	0.82835	0.73337	0.66308	0.72703	0.49254	0.40756	0.31910	0.13024	0.08668	0.36649	0.14925
	200	0.65576	0.61679	0.58272	0.61079	0.43592	0.36768	0.29864	0.13789	0.09102	0.33662	0.15637
	300	0.45124	0.42645	0.40497	0.41924	0.29151	0.24505	0.19684	0.08756	0.05895	0.21521	0.09939
	400	0.29690	0.28727	0.27816	0.28300	0.20004	0.16840	0.13583	0.05950	0.03966	0.14494	0.06792
0.95	30	22.17859	18.49087	15.42387	17.43801	8.96908	6.93258	5.29576	2.43417	1.58691	7.99886	2.74114
	75	3.30286	2.69005	2.21612	2.63372	1.61660	1.28847	0.96945	0.36613	0.22725	1.26511	0.42442
	150	1.71742	1.43479	1.21540	1.44446	0.90702	0.72071	0.53986	0.18799	0.11302	0.67734	0.22179
	200	1.22567	0.97596	0.79944	0.97891	0.58060	0.45658	0.33075	0.10130	0.06392	0.40404	0.12049
	300	0.88743	0.83390	0.78553	0.83189	0.58524	0.48308	0.38516	0.16445	0.09923	0.44785	0.18991
	400	0.59477	0.54001	0.49498	0.52972	0.33346	0.27050	0.20464	0.06987	0.04366	0.22968	0.08245
0.99	30	56.96305	50.66928	45.23884	48.75429	27.22787	21.51509	17.03938	8.65323	5.58702	26.05924	9.66559
	75	22.99812	19.26258	16.07941	18.43814	8.57154	6.26691	4.44097	1.48240	0.80892	7.56976	1.76183
	150	7.72570	6.54383	5.54156	6.48948	3.81450	2.90583	2.14274	0.76654	0.41155	3.18183	0.90842
	200	6.51381	5.72780	5.03562	5.70110	3.27868	2.49684	1.83423	0.64327	0.35857	2.69226	0.76141
	300	4.40918	3.69318	3.09365	3.67075	1.98315	1.48364	1.05112	0.31062	0.16682	1.51364	0.37694
	400	3.12077	2.68126	2.31130	2.72016	1.57603	1.19396	0.86355	0.27008	0.14135	1.18622	0.32688

Bolded values indicate the lowest MSE.

Table 6.

Estimated MSE for different estimators at p = 6 and .

	n	PIGMLE	PIGRRE					PIGGRE
		MLE
0.80	30	5.70916	4.93744	4.28906	4.81213	3.08376	2.50145	1.98185	0.95094	0.63230	2.58249	1.06594
	75	1.69674	1.20216	0.85119	1.14056	0.69408	0.56191	0.40785	0.15666	0.13575	0.49567	0.17038
	150	0.76054	0.61209	0.51674	0.59309	0.38762	0.32124	0.24519	0.10356	0.08370	0.27317	0.11403
	200	0.57336	0.53693	0.50523	0.52987	0.37279	0.31549	0.25467	0.11644	0.08291	0.28135	0.13093
	300	0.33520	0.30553	0.28201	0.29455	0.18805	0.15845	0.12323	0.05820	0.05305	0.12744	0.06198
	400	0.21962	0.21365	0.20799	0.21025	0.14715	0.12606	0.10289	0.05148	0.04220	0.10634	0.05618
0.85	30	8.46109	7.14163	5.98629	6.66511	3.36577	2.65362	2.05185	0.92821	0.60414	2.81312	1.04919
	75	2.25170	1.70572	1.30270	1.65262	1.02561	0.82426	0.60991	0.22394	0.16378	0.76412	0.25451
	150	0.76991	0.69548	0.63584	0.68730	0.46873	0.39184	0.30999	0.13259	0.09227	0.35378	0.15029
	200	0.65829	0.59368	0.54180	0.58487	0.38573	0.31983	0.24960	0.10185	0.07220	0.27808	0.11582
	300	0.45626	0.41454	0.38100	0.40409	0.26327	0.21911	0.17033	0.07256	0.05704	0.18436	0.08067
	400	0.31913	0.30758	0.29685	0.30246	0.21126	0.17916	0.14462	0.06504	0.04722	0.15296	0.07319
0.90	30	11.46938	9.41220	7.72538	8.74723	4.68385	3.59193	2.66762	1.03873	0.63890	3.93899	1.20112
	75	2.42968	2.07989	1.79738	2.08476	1.39272	1.14437	0.90189	0.39664	0.26433	1.12924	0.44997
	150	1.25033	0.99579	0.81721	0.98780	0.63142	0.51123	0.38311	0.13839	0.09637	0.45726	0.15935
	200	0.95100	0.85094	0.76887	0.84870	0.56330	0.46049	0.35698	0.13837	0.09038	0.41769	0.16019
	300	0.64643	0.58445	0.53368	0.57591	0.37684	0.30876	0.23825	0.09136	0.06143	0.27023	0.10544
	400	0.50014	0.47268	0.44763	0.46576	0.31575	0.26067	0.20477	0.08205	0.05476	0.22563	0.09457
0.95	30	38.58210	32.28457	27.12338	29.96805	11.79339	8.76006	6.44526	2.66813	1.61814	10.74345	3.05467
	75	4.87018	3.73969	2.86747	3.54226	2.00294	1.54897	1.10800	0.35146	0.22295	1.56029	0.41418
	150	2.65203	2.12344	1.71606	2.11806	1.27725	1.00682	0.74042	0.24623	0.15068	0.97164	0.29095
	200	1.98940	1.44749	1.06547	1.43288	0.81690	0.63415	0.44520	0.12608	0.08182	0.57153	0.15021
	300	1.16438	1.04569	0.94557	1.04668	0.67812	0.54491	0.41578	0.15191	0.09252	0.50337	0.17826
	400	0.91651	0.77584	0.67044	0.76989	0.46213	0.36596	0.26767	0.08364	0.05228	0.31288	0.09943
0.99	30	86.84169	74.29102	63.57600	70.46644	33.87303	25.75647	19.69364	9.34368	5.99847	32.73497	10.50118
	75	35.09694	28.43725	22.85759	26.33208	11.09987	7.91067	5.44613	1.70266	0.92544	10.06332	2.03519
	150	12.67409	10.31242	8.31080	10.09648	4.95917	3.57352	2.44390	0.66887	0.55196	4.07855	0.81918
	200	9.01714	7.59321	6.37332	7.53075	4.04242	3.00462	2.14228	0.67853	0.36244	3.34492	0.81598
	300	5.98490	4.77790	3.77690	4.71455	2.36668	1.73272	1.18620	0.31455	0.17060	1.81922	0.38567
	400	4.12574	3.33075	2.67274	3.33279	1.76216	1.30033	0.90189	0.25244	0.13618	1.31595	0.30738

Bolded values indicate the lowest MSE.

Table 7.

Estimated MSE for different estimators at p = 6 and .

	n	PIGMLE	PIGRRE					PIGGRE
		MLE
0.80	30	7.18003	5.96725	4.95220	5.65691	3.43017	2.72024	2.06997	0.84094	0.53800	2.87566	0.96569
	75	2.16899	1.49165	0.99617	1.36881	0.84479	0.68046	0.49076	0.18804	0.15851	0.61702	0.20495
	150	0.86676	0.65707	0.52660	0.62939	0.39995	0.32912	0.24483	0.09814	0.08226	0.27872	0.10737
	200	0.66804	0.61011	0.56204	0.60211	0.41545	0.34953	0.27908	0.12621	0.09206	0.31281	0.14130
	300	0.40789	0.36347	0.33030	0.35021	0.22821	0.19257	0.15029	0.07255	0.06477	0.15840	0.07725
	400	0.31116	0.29780	0.28569	0.29244	0.20175	0.17121	0.13784	0.06497	0.05267	0.14408	0.07151
0.85	30	10.14016	8.48593	7.04951	7.88894	4.06232	3.18657	2.45156	1.11959	0.77145	3.43226	1.25556
	75	2.68506	1.90842	1.35977	1.77110	1.07970	0.86152	0.62508	0.22662	0.17170	0.80587	0.25533
	150	0.98303	0.85630	0.75711	0.84855	0.55362	0.45568	0.35068	0.13520	0.09582	0.40985	0.15471
	200	0.85518	0.73529	0.64939	0.72495	0.47859	0.39667	0.30818	0.12716	0.08925	0.35018	0.14409
	300	0.60915	0.53869	0.48518	0.52807	0.34403	0.28468	0.22019	0.09294	0.07253	0.24267	0.10348
	400	0.37681	0.35578	0.33698	0.34848	0.23258	0.19406	0.15270	0.06486	0.04946	0.16341	0.07279
0.90	30	13.14303	10.60679	8.52664	9.67515	5.15026	4.02647	3.06061	1.38919	0.97241	4.38099	1.55263
	75	2.97547	2.46797	2.05120	2.50301	1.63325	1.32316	1.02410	0.42405	0.28170	1.31799	0.48440
	150	1.64494	1.28766	1.03472	1.26495	0.79908	0.64378	0.47922	0.17359	0.12091	0.58764	0.19932
	200	1.10051	0.95294	0.83500	0.95261	0.61667	0.50022	0.38336	0.14826	0.09859	0.45771	0.17084
	300	0.82955	0.71539	0.62936	0.70889	0.45245	0.36832	0.28020	0.10575	0.07320	0.32251	0.12143
	400	0.55415	0.51468	0.47972	0.50684	0.33374	0.27466	0.21368	0.08508	0.05906	0.23877	0.09744
0.95	30	47.89147	38.17942	30.49429	34.15357	14.82216	11.25518	8.38152	3.73163	2.37455	13.86783	4.20847
	75	5.93897	4.43688	3.26921	4.14677	2.38095	1.85000	1.33513	0.45088	0.28393	1.89978	0.52735
	150	3.77671	2.94719	2.29815	2.88179	1.62622	1.25264	0.89004	0.26519	0.16417	1.21939	0.31635
	200	2.50137	1.79143	1.28696	1.73304	1.00482	0.78128	0.55048	0.16227	0.10631	0.72471	0.19204
	300	1.39646	1.20607	1.05069	1.21629	0.77962	0.62152	0.46969	0.16892	0.10304	0.58134	0.19835
	400	1.10468	0.91389	0.77171	0.91029	0.54009	0.42692	0.31007	0.09435	0.05997	0.37108	0.11227
0.99	30	104.99316	88.77856	75.12642	82.60415	33.77235	24.75321	18.05264	7.25678	4.23941	33.20595	8.37287
	75	41.75106	32.20102	24.40079	28.96857	10.65210	7.39667	4.84938	1.27197	0.70639	9.76427	1.54692
	150	15.15633	12.07267	9.49770	11.59891	5.29019	3.80320	2.55909	0.68119	0.37146	4.41116	0.83063
	200	12.85437	10.69363	8.83174	10.47499	5.11777	3.71635	2.57294	0.76477	0.41932	4.22717	0.92073
	300	8.18779	6.54940	5.18827	6.43319	3.18094	2.31601	1.58019	0.42582	0.22992	2.49836	0.52003
	400	5.38889	4.35928	3.50626	4.33666	2.24028	1.63563	1.11685	0.28836	0.15509	1.69409	0.35542

Bolded values indicate the lowest MSE.

Table 8.

Estimated MSE for different estimators at p = 9 and .

	n	PIGMLE	PIGRRE					PIGGRE
		MLE
0.80	30	7.06866	6.52061	6.01562	6.44942	4.25378	3.5949	3.02618	1.42379	0.97861	3.66112	1.52125
	75	2.04102	1.77469	1.56616	1.78387	1.1806	0.98454	0.78045	0.24191	0.15483	0.94489	0.26642
	150	0.79185	0.74977	0.71275	0.74773	0.51018	0.42981	0.34364	0.10325	0.06801	0.39445	0.11383
	200	0.47735	0.46198	0.44757	0.45771	0.3103	0.26314	0.21152	0.06693	0.04846	0.23778	0.07294
	300	0.36382	0.3563	0.34909	0.35324	0.24516	0.20947	0.16983	0.0543	0.03974	0.18639	0.05916
	400	0.23405	0.22976	0.2256	0.22736	0.1501	0.12686	0.10041	0.03031	0.0261	0.10826	0.03244
0.85	30	12.2065	11.96343	9.22308	10.20689	5.29031	4.58022	3.59615	1.396	0.82003	3.43257	1.27171
	75	2.15174	1.93216	1.76066	1.97197	1.36983	1.15279	0.93559	0.32023	0.20626	1.11523	0.35115
	150	0.99095	0.93454	0.88553	0.93354	0.64419	0.54648	0.44385	0.15374	0.10228	0.51256	0.16767
	200	0.66282	0.63483	0.60928	0.63056	0.42024	0.35439	0.28219	0.0811	0.05387	0.32245	0.08974
	300	0.46491	0.45237	0.4405	0.44864	0.30967	0.26296	0.21237	0.06424	0.04226	0.23762	0.07101
	400	0.3686	0.35947	0.35071	0.35571	0.23483	0.19676	0.15513	0.04161	0.02914	0.17084	0.04604
0.90	30	16.1216	14.12896	12.38458	13.55324	6.62773	5.25457	4.16014	1.57106	0.9913	5.48195	1.70696
	75	3.15944	2.71129	2.34182	2.74773	1.69349	1.3901	1.07534	0.30093	0.18758	1.34573	0.33379
	150	1.36055	1.26435	1.18044	1.27819	0.87593	0.73646	0.59065	0.18277	0.11283	0.69982	0.20213
	200	1.22283	1.10382	1.00116	1.1175	0.70701	0.57373	0.43502	0.09034	0.05384	0.52193	0.10317
	300	0.70297	0.66275	0.62604	0.65764	0.39805	0.32487	0.24441	0.05014	0.03332	0.28604	0.05662
	400	0.49001	0.47002	0.45166	0.46402	0.29606	0.24658	0.19181	0.04826	0.03177	0.2169	0.05384
0.95	30	57.5364	50.65413	45.10653	47.68283	16.88082	12.80335	9.99353	3.85353	2.51441	14.40752	4.16851
	75	6.07069	5.34264	4.71659	5.37574	3.32054	2.72268	2.14339	0.71872	0.46314	2.75744	0.78464
	150	3.11469	2.79998	2.5222	2.86519	1.76555	1.42049	1.08569	0.26084	0.14953	1.39067	0.295
	200	2.2451	1.94156	1.69362	1.98443	1.17039	0.94121	0.70439	0.15085	0.08831	0.89358	0.17148
	300	1.14368	1.03288	0.94162	1.03812	0.63089	0.51417	0.38779	0.08171	0.04835	0.4712	0.09305
	400	0.95289	0.90168	0.85529	0.90192	0.598	0.49409	0.38736	0.1001	0.05768	0.45761	0.11276
0.99	30	108.40914	101.72837	95.56783	100.73019	56.64332	45.47928	36.82599	15.56412	10.22591	51.99598	16.75204
	75	34.77464	30.75336	27.24856	30.51649	16.1249	12.52277	9.30831	2.29935	1.28055	13.95642	2.58621
	150	19.55027	16.98758	14.74125	16.98293	9.065	7.00506	5.14027	1.14291	0.59838	7.65428	1.30046
	200	10.72963	8.95639	7.4008	9.18044	4.64182	3.49441	2.42921	0.36963	0.20193	3.6507	0.43094
	300	6.71319	5.63807	4.69959	5.78008	3.15813	2.44236	1.75644	0.3161	0.16521	2.47979	0.36583
	400	4.92142	4.18446	3.53909	4.36078	2.45692	1.88906	1.3606	0.24214	0.12627	1.88982	0.28059

Bolded values indicate the lowest MSE.

Table 9.

Estimated MSE for different estimators at p = 9 and .

	n	PIGMLE	PIGRRE					PIGGRE
		MLE
0.80	30	10.73294	9.45433	8.34504	9.2466	4.77015	3.77006	2.95387	1.04324	0.66517	3.89265	1.13721
	75	3.39965	2.8339	2.37349	2.8566	1.67625	1.36618	1.04177	0.28177	0.18911	1.31229	0.31112
	150	1.00687	0.91899	0.84481	0.91956	0.57895	0.48226	0.37387	0.10312	0.07488	0.4418	0.11288
	200	0.76086	0.71416	0.67315	0.70936	0.45036	0.37757	0.29493	0.08426	0.06381	0.34071	0.09168
	300	0.51897	0.49806	0.47875	0.49283	0.32741	0.27601	0.21887	0.06557	0.04934	0.24586	0.0713
	400	0.37566	0.3643	0.35349	0.35965	0.23012	0.19321	0.15092	0.04331	0.03603	0.16682	0.0466
0.85	30	19.27733	16.4411	13.97169	15.41655	7.34859	6.03932	4.86817	1.95523	1.39088	6.31112	2.10618
	75	3.01251	2.54954	2.17559	2.60826	1.57734	1.28672	0.98978	0.2674	0.16871	1.24286	0.29726
	150	1.39509	1.27017	1.16427	1.28516	0.83681	0.6946	0.54567	0.15332	0.0978	0.65557	0.1702
	200	0.9971	0.92815	0.86744	0.92901	0.60136	0.49869	0.38857	0.10122	0.06688	0.45827	0.11262
	300	0.72715	0.68888	0.65405	0.68541	0.44354	0.36892	0.28808	0.07651	0.0516	0.33138	0.08479
	400	0.46915	0.44992	0.43196	0.4442	0.2797	0.23352	0.18064	0.04551	0.03394	0.20371	0.05009
0.90	30	23.48545	20.23299	17.35121	19.28553	8.81518	6.97209	5.49392	2.20602	1.47766	7.3625	2.37001
	75	4.88456	4.03043	3.32542	3.97971	2.10156	1.66492	1.2086	0.25931	0.16393	1.62338	0.29102
	150	2.0514	1.82449	1.63481	1.87283	1.20509	0.9851	0.7599	0.18865	0.11348	0.94259	0.21218
	200	1.5734	1.36041	1.18742	1.40096	0.84245	0.67646	0.50151	0.09732	0.06092	0.62088	0.11088
	300	1.0206	0.9376	0.86443	0.93869	0.5753	0.46924	0.35383	0.07563	0.04978	0.4235	0.08522
	400	0.72157	0.67685	0.63723	0.67197	0.41639	0.34302	0.26226	0.06088	0.0402	0.30564	0.06817
0.95	30	91.65183	76.54036	63.70994	68.79082	22.89659	18.98429	15.77151	7.78501	5.67074	20.83585	8.24485
	75	9.25171	8.01005	6.91761	8.019	4.59294	3.63626	2.73381	0.68391	0.39583	3.75901	0.76761
	150	4.25302	3.65305	3.1465	3.78068	2.24924	1.79866	1.35828	0.33327	0.19578	1.79209	0.37437
	200	3.16039	2.62687	2.19173	2.71861	1.59826	1.27863	0.95523	0.21441	0.12605	1.2391	0.24246
	300	1.8119	1.53173	1.30864	1.57621	0.89114	0.71774	0.52614	0.10264	0.06424	0.66215	0.11651
	400	1.34627	1.22863	1.12705	1.23887	0.75511	0.61133	0.45988	0.09511	0.05655	0.5654	0.1084
0.99	30	152.83988	139.29117	127.14329	137.69195	73.0238	58.05439	46.53535	19.79738	13.08846	67.35226	21.25746
	75	47.1158	38.7285	31.72315	38.05415	18.62169	14.46648	10.68585	2.9046	1.6622	16.39355	3.22694
	150	25.1333	21.27287	17.90344	21.41746	10.35064	7.86377	5.58007	1.06329	0.57424	8.79321	1.21762
	200	15.09767	12.36091	9.96464	12.48608	5.6498	4.17964	2.80434	0.37499	0.21025	4.4955	0.43872
	300	8.83032	6.92371	5.3008	7.16387	3.54861	2.68545	1.83668	0.25979	0.14606	2.76853	0.30308
	400	7.23955	6.10533	5.10174	6.29475	3.30984	2.52143	1.77695	0.29117	0.15611	2.57024	0.33795

Bolded values indicate the lowest MSE.

Table 10.

Estimated MSE for different estimators at p = 9 and .

	n	PIGMLE	PIGRRE					PIGGRE
		MLE
0.80	30	14.13639	12.45102	11.08498	11.85672	7.21457	4.32150	1/3/54	1.02540	0.95413	2.36540	1.02457
	75	4.31753	3.51981	2.84032	3.49948	2.01266	1.61894	1.21913	0.32555	0.21513	1.57290	0.35911
	150	1.51394	1.34503	1.20689	1.36470	0.84308	0.69803	0.53451	0.13305	0.09395	0.64457	0.14722
	200	0.98297	0.90530	0.83749	0.90699	0.57686	0.47721	0.36956	0.10018	0.07406	0.43427	0.10973
	300	0.66995	0.63527	0.60389	0.63025	0.41583	0.34913	0.27526	0.08073	0.06002	0.31379	0.08795
	400	0.44410	0.42688	0.41074	0.42114	0.26788	0.22438	0.17478	0.05099	0.04279	0.19549	0.05469
0.85	30	22.93924	18.86938	15.41333	17.65680	8.79002	7.19663	5.81838	2.52928	1.85809	7.63006	2.69973
	75	4.22081	3.58501	3.05923	3.64595	2.19375	1.77302	1.35584	0.35521	0.22341	1.73835	0.39590
	150	1.83722	1.61645	1.43452	1.65789	1.02933	0.84240	0.64326	0.15406	0.09941	0.79429	0.17253
	200	1.25434	1.14301	1.04836	1.15582	0.73589	0.60676	0.46868	0.11666	0.07612	0.56166	0.13029
	300	0.81613	0.76278	0.71520	0.75910	0.47592	0.39411	0.30408	0.07770	0.05437	0.35475	0.08597
	400	0.60656	0.57509	0.54626	0.56914	0.35331	0.29323	0.22523	0.05505	0.03975	0.25791	0.06088
0.90	30	34.13813	28.81708	24.15645	26.80033	9.16679	6.93045	5.13035	1.55475	1.03765	7.44783	1.69806
	75	6.67024	5.41400	4.38328	5.38376	3.10448	2.51187	1.89912	0.53536	0.34887	2.51546	0.58852
	150	2.75325	2.42688	2.14861	2.49620	1.59435	1.30587	1.00847	0.26155	0.16086	1.26186	0.29226
	200	1.89986	1.61865	1.39050	1.67339	0.97862	0.78705	0.58094	0.11742	0.07774	0.72838	0.13247
	300	1.16488	1.04188	0.93762	1.05129	0.63383	0.51421	0.38406	0.07967	0.05311	0.46751	0.08977
	400	0.87179	0.79856	0.73537	0.79607	0.48303	0.39772	0.30213	0.07142	0.04851	0.35609	0.07954
0.95	30	94.90217	77.26117	61.86054	68.25513	22.65027	18.12375	14.35692	5.72101	3.72485	20.47085	6.16699
	75	11.79514	9.92881	8.30918	9.88617	5.32410	4.17139	3.09222	0.74950	0.43346	4.37701	0.84170
	150	5.77443	4.93719	4.20032	5.05762	2.76499	2.16220	1.56668	0.29284	0.17236	2.16803	0.33532
	200	4.04025	3.38149	2.82350	3.47303	1.90766	1.50210	1.08991	0.20740	0.12447	1.46820	0.23670
	300	2.29645	1.89752	1.57940	1.96681	1.07548	0.85600	0.61524	0.10914	0.06944	0.79606	0.12435
	400	1.77928	1.56894	1.38976	1.61018	0.96332	0.77022	0.57165	0.11425	0.06927	0.71677	0.13011
0.99	30	185.21400	162.97272	153.94068	165.69072	98.36454	65.25440	21.64878	15.21401	10.36545	65.64515	25.01201
	75	62.66777	50.24417	39.81006	48.35449	20.48084	15.33929	10.64582	2.00869	1.09794	17.95028	2.29288
	150	32.62897	27.41424	22.82419	27.34288	12.28139	9.09425	6.21548	0.98725	0.54912	10.40860	1.13958
	200	20.90990	17.41000	14.29089	17.56324	7.81490	5.73117	3.82612	0.49668	0.27212	6.28415	0.58335
	300	12.17176	10.01420	8.13842	10.22618	5.04651	3.81327	2.63285	0.40241	0.22126	4.01746	0.46742
	400	8.85903	7.36665	6.05705	7.60977	3.91514	2.97142	2.06779	0.30997	0.16999	3.06943	0.36153

Bolded values indicate the lowest MSE.

Fig. 1.

MSE of different estimators under various factors in the simulation study.

Applications

In this section, we compare our proposed estimator with existing estimators such as MLE and PIGRRE, using two real-world datasets to demonstrate the advantages and superiority of the proposed estimator and highlight its effectiveness in improving estimation accuracy, especially in the presence of multicollinearity.

Number of equations and citations

The performance of the proposed estimator is assessed using a real dataset from Fawcett and Higginson⁴⁸, which is available in R through the AER package under the name EquationCitations. This data investigates the relationship between the number of citations received by evolutionary biology publications and the number of equations per page in the cited papers. This dataset comprises 649 observations, with the response variable representing the total number of equations (y). The explanatory variables include the total number of citations received (), the number of equations present in appendix (), the number of equations present in the main text (), the number of citations made by the authors of the paper (), the number of citations from theoretical papers (), and the number of citations from non-theoretical papers ().

The response variable in this study is count data, which requires specialized models. We compared the PRM, NBRM, COMPRM, and PIGRM to identify the best fit for the relationship between the response variable and explanatory variables in the number of equations and citation data. The selection criteria were the log-likelihood (LL), the Akaike Information Criterion (AIC), and the Bayesian Information Criterion (BIC), where a lower AIC, BIC, and higher LL indicate a better model. Results in Table 11 show that the PIGRM outperforms the other models with the lowest AIC, BIC, and highest LL.

Table 11.

Comparison of model performance for the number of equations and citation data.

Model	PRM	NBRM	CMPRM	PIGRM
LL	− 1853.099	− 1852.959	− 1578.121	− 1095.091
AIC	3720.197	3721.917	3172.241	2206.182
BIC	3751.525	3757.721	3208.045	2241.985

Bolded values indicate the fitted model for the data.

With six explanatory variables in the dataset, the eigenvalues of the matrix are: 1754069.240, 76832.270, 59927.316, 17295.812, 4131.207, 129.039, and 63.036. Multicollinearity was assessed using the condition number (CN) and variance inflation factors (VIFs). The CN, defined as where and are the largest and smallest eigenvalues of , respectively, was calculated to be 166.81, indicating a high level of multicollinearity. In addition, the VIF for each explanatory variable, computed as: where is the coefficient of determination from regressing the -th variable on all other predictors, yielded values of 1201.15, 1.68, 1.57, 23.77, 255.37, and 463.52. These values confirm the presence of severe multicollinearity among several variables. This is further supported by the correlation matrix shown in Fig. 2, emphasizing the need for caution in interpreting parameter estimates and potentially adopting bias-reducing methods to improve model stability.

Fig. 2.

Correlation matrix for explanatory variables in the number of equations and citations data.

The coefficients of the PIGRM are estimated using three methods: MLE, PIGRRE, and PIGGRE, as specified in Eqs. 8, 11, and 17, respectively. The corresponding MSEs for each estimator are computed using Eqs. 10, 16, and 21. The results indicate that MLE performs less favorably than PIGRRE across all ridge parameter estimators, as reflected by its higher MSE. Conversely, PIGGRE demonstrates improved performance compared to both MLE and PIGRRE, yielding lower MSE values. A comparison of the ridge parameter estimators used in PIGGRE, presented in Table 12, highlights that through outperform other existing estimators. These findings are consistent with the results observed in the simulation studies, supporting the efficacy of the proposed estimators.

Table 12.

Coefficients and MSEs of estimators in the number of equations and citation data.

Estimates	Parameters								MSE
PIGMLE	–	0.0683	0.027	0.1014	0.1054	− 0.0265	− 5e−04	− 0.0422	0.0237
PIGRRE		0.0424	0.0215	0.101	0.1055	− 0.0198	0.0051	− 0.0365	0.0114
		0.0303	0.0179	0.1003	0.1055	− 0.0155	0.0089	− 0.0327	0.0078
		0.0451	0.0222	0.1011	0.1055	− 0.0206	0.0044	− 0.0372	0.0124
		0.0173	0.0128	0.0981	0.1056	− 0.0096	0.0144	− 0.0273	0.0058
		0.006	0.006	0.0171	0.0646	− 0.0036	0.0209	− 0.0102	0.0158
PIGGRE		0.0012	0.0038	0.1155	0.0932	− 0.0012	0.0238	− 0.0167	0.0056
		0.0012	0.0037	0.1156	0.0931	− 0.0011	0.0239	− 0.0167	0.0056
		0.0012	0.0037	0.1157	0.093	− 0.001	0.024	− 0.0166	0.0055
		0.0005	0.002	0.1306	0.0566	− 0.0039	0.0294	− 0.0114	0.0068
		0.0069	0.0168	0.1014	0.1048	− 0.0135	0.01	− 0.0313	0.0066

Bolded values indicate the best biasing parameter.

Australian institute of sports

To empirically evaluate the proposed estimator’s performance, we utilize the Australian Institute of Sport dataset, as presented by Telford and Cunningham⁴⁹ and is available in the R software through the GLMsData package under the name AIS. This dataset encompasses physical and blood measurements of high-performance athletes from various sports, including basketball (BBall), field events, gymnastics (Gym), netball, rowing, swimming (Swim), track events over 400 meters (T400m), tennis, track sprints (TPSprnt), and water polo (WPolo). The dataset contains information for 202 athletes, comprising 102 males and 100 females. The response variable in this dataset is the plasma ferritin (in ng per decilitre) (y). In addition, the dataset includes 10 explanatory variables lean body mass (in kg) (), height (in cm) (), weight (in kg) (), body mass index (in ) (), the sum of skin folds (), percentage body fat (), red blood cell count (in per liter) (), white blood cell count (in per liter) (), hematocrit (in percent) (), and hemoglobin concentration (in grams per deciliter) ()

The dependent variable in this analysis is count data, which necessitates appropriate models. To determine the best model for the relationship between the dependent variable and the explanatory variables in the context of the number of equations and citation data, we compared four models: the PRM, NBRM, COMPRM, and PIGRM. We evaluated the models based on three criteria: LL, AIC, and BIC. A lower AIC, BIC, and higher LL indicate a more suitable model. As shown in Table 13, the PIGRM provides the best fit, with the lowest AIC and BIC values and the highest LL.

Table 13.

Comparison of model performance for the Australian Institute of Sports data.

Model	PRM	NBRM	CMPRM	PIGRM
LL	− 2750.089	− 1007.920	− 2040.085	− 1007.269
AIC	5522.179	2039.840	4104.171	2038.538
BIC	5558.570	2079.539	4143.870	2078.238

Bolded values indicate the fitted model for the data.

The dataset includes ten explanatory variables, the eigenvalues of the matrix are: 30689803.431, 702460.991, 114401.412, 8572.263, 1996.989, 1704.116, 1019.529, 123.657, 94.052, 19.553, and 0.037. Multicollinearity was assessed using the CN and VIFs. The observed CN value of 28699.25, alongside VIF values of 442.07, 56.21, 516.80, 79.96, 22.75, 7.27, 1.09, 15.95, 12.23, and 62.66, indicates substantial multicollinearity among the variables. These findings are further corroborated by the correlation matrix shown in Fig. 3. Such evidence highlights the necessity for careful interpretation and potential adjustments in subsequent analyses to mitigate the impact of multicollinearity.

Fig. 3.

Correlation matrix for explanatory variables in the Australian Institute of Sports data.

The coefficients for the PIGRM were estimated using three methods: MLE, PIGRRE, and PIGGRE, as outlined in equations 8, 11, and 17. The corresponding MSEs were calculated based on equations 10, 16, and 21. The results indicate that MLE performs less effectively than PIGRRE, as reflected by its higher MSE values. In contrast, PIGGRE shows superior performance, with consistently lower MSEs compared to both MLE and PIGRRE. Additionally, a comparison of the ridge parameter estimators in PIGGRE, shown in Table 14, highlights that estimators and outperform , , and in PIGGRE. This can be attributed to the nature of the data, including the degree of multicollinearity among the predictors, the level of dispersion, and the sample size. These factors collectively influence the performance of the estimators and the sensitivity of each to the choice of the biasing parameter K. Overall, the PIGGRE estimator demonstrates superior performance, which is consistent with both the simulation results and the findings of the first application.

Table 14.

Coefficients and MSEs of Estimators for the Australian Institute of Sports Data.

Estimates	Parameters												MSE
PIGMLE	–	8.4170	0.0652	− 0.0352	− 0.0224	− 0.0534	0.0115	0.0224	0.0164	− 0.0392	0.1340	− 0.0269	103.5890
PIGRRE		5.9212	0.0651	−  0.0213	− 0.0395	0.0022	0.0113	0.0239	0.0163	− 0.0394	0.1337	− 0.0266	57.6193
		3.4255	0.0650	− 0.0074	− 0.0566	0.0579	0.0111	0.0254	0.0161	− 0.0395	0.1334	− 0.0263	42.2953
		3.4269	0.0650	− 0.0074	− 0.0566	0.0578	0.0111	0.0254	0.0161	− 0.0395	0.1334	− 0.0263	42.2953
		0.0040	0.0309	0.0120	− 0.0462	0.1160	0.0086	0.0033	0.0130	0.0014	0.0434	− 0.0367	70.8322
		9.53e−05	0.0055	0.0157	0.0058	0.0021	0.0032	0.0005	0.0007	0.0042	0.0015	0.0006	70.8747
PIGGRE		0.0006	0.0007	0.0053	0.0029	7.6e−06	0.0055	0.0083	0.0031	0.0640	0.0053	− 0.0174	42.0372
		0.0020	0.0020	0.0079	− 0.0005	2.97e−05	0.0077	0.0031	0.0075	0.0529	0.0177	− 0.0296	59.7067
		0.0068	0.0066	0.0091	− 0.0057	0.0001	0.0086	− 0.0078	0.0125	0.0365	0.0563	− 0.0314	86.2329
		0.0008	0.0008	0.0060	0.0021	1.02e−05	0.0061	0.0075	0.0039	0.0618	0.0068	− 0.0208	42.9759
		0.0078	0.0200	0.0090	− 0.0183	0.0065	0.0090	0.0008	0.0130	0.0329	0.0529	− 0.0257	67.0565

Bolded values indicate the best biasing parameter.

Conclusion

The PIGRM is one of the most widely used models for analyzing overdispersed count data. In the PIGRM, MLE is used to estimate regression coefficients. However, when the explanatory variables are highly correlated, this leads to multicollinearity, a problem that reduces the reliability of the regression coefficients and inflates the variance. To address this, we introduced in this paper a new biased estimation method called the PIGGRE, which uses shrinkage parameter techniques to reduce the effect of multicollinearity. We evaluated the performance of the proposed PIGGRE through simulation studies with different scenarios, such as sample sizes, levels of dispersion, multicollinearity, and number of predictors. The results showed that PIGGRE outperformed both MLE and PIGRRE. In particular, using shrinkage parameters and resulted in the lowest MSE, demonstrating higher accuracy. To confirm the simulation study results, we also applied the proposed PIGGRE to two real datasets. The results of these applications provided significant support and confirmation of the simulation results, demonstrating the advantages of using PIGGRE. Therefore, we recommend using PIGGRE with values of and to address multicollinearity in PIGRM. This approach could be extended in future work to other models, such as the zero-inflated negative binomial model, the zero-inflated Poisson model, and the Conway-Maxwell-Poisson model. Although PIGGRE outperforms in the context of PIGRM under multicollinearity, there is one important limitation worth mentioning. The estimator’s performance depends mainly on the K values, which balance bias and variance, and suboptimal K values can affect the estimator’s performance, making the estimator sensitive to the choice of parameters. Further improvements could be achieved by incorporating accurate estimation techniques, such as those proposed by Omara⁵⁰ and Lukman et al.⁵¹.

Author contributions

All authors have worked equally to write and review the manuscript.

Data availability

The data that support the findings of this study are available on request from the corresponding author. The data are not publicly available due to privacy or ethical restrictions.

Declarations

Competing interests

The authors declare no competing interests.