Poisson Inverse Gaussian Regression Model Research Articles

Performance of Alternative Estimators in the Poisson-Inverse Gaussian Regression Model: Simulation and Application

Abstract The study proposed and compared the biased estimators for the Poisson-Inverse Gaussian regression model to deal with correlated regressors. The limitations of each biased estimator are also discussed. Additionally, some biasing parameters for the Stein estimator are proposed. The performance of estimators is evaluated with the help of a simulation study and a real-life application based on the minimum mean squared error criterion. The simulation and application findings favor the ridge estimator with specific biasing parameters because it provides less variation than others.

On the performance of some new ridge parameter estimators in the Poisson-inverse Gaussian ridge regression

The Poisson Inverse Gaussian Regression model (PIGRM) is used for modeling the count datasets to deal with the issue of over-dispersion. Generally, the maximum likelihood estimator (MLE) is used to estimate the PIGRM estimates. In the PIGRM, when the explanatory variables are correlated, the MLE does not provide efficient results. To overcome this problem, we propose a ridge estimator for the PIGRM. The matrix mean square error (MSE) and the scalar MSE properties are derived and then compared with the MLE. In the ridge estimator, ridge parameter play a significant role, so, this study also proposes different ridge parameter estimators for the PIGRM. The performance of the proposed estimator is evaluated with the help of a simulation study and a real-life application using MSE as a performance evaluation criterion. The simulation study and the real-life application results show the superiority of the proposed parameter estimators as compared to the MLE.

The Geographically Weighted Multivariate Poisson Inverse Gaussian Regression Model and Its Applications

This study aims to develop a method for multivariate spatial overdispersion count data with mixed Poisson distribution, namely the Geographically Weighted Multivariate Poisson Inverse Gaussian Regression (GWMPIGR) model. The parameters of the GWMPIGR model are estimated locally using the maximum likelihood estimation (MLE) method by considering spatial effects. Therefore, the significance of the regression parameter differs for each location. In this study, four GWMPIGR models are evaluated based on the exposure variable and the spatial weighting function. We compare the performance of those four models in real-world application using data on the number of infant, under-5 and maternal deaths in East Java in 2019 using five predictor variables. In this study, the GWMPIGR model uses one exposure variable and three exposure variables. Compared to the fixed kernel Gaussian weighting function, the GWMPIGR model with the fixed kernel bisquare weighting function and one exposure variable has a better fit based on the AICc value. Furthermore, according to the best GWMPIGR model, there are several regional groups formed based on predictors that significantly affected each event in East Java in 2019.

A Comparison of Bivariate Zero-Inflated Poisson Inverse Gaussian Regression Models with and without Exposure Variables

In this paper, we focus on the comparison of the bivariate zero-inflated Poisson inverse Gaussian regression (BZIPIGR) type II model in two cases: with and without exposure variables. The BZIPIGR type II model is applied to analyze the occurrence of maternal and early neonatal mortality in South Sulawesi Province, Indonesia using 2019 data, which contain many zero values and have the issue of overdispersion in the response variable. Furthermore, to analyze the number of deaths in various areas, the exposure variable is considered. The maximum likelihood estimation (MLE) is used in parameter estimation, which involves numerical iteration and application of the Berndt–Hall–Hall–Hausman (BHHH) algorithm. Sum square error (SSE) serves as the criterion of model selection when exposure variables are included. The existence of exposure variables strongly affects the model’s accuracy, especially using the BZIPIGR type II model. According to the SSE and RMSE values, the BZIPIGR type II model with exposure variables performs better than the model without exposure variables in estimating parameter values. All predictors with exposure variables in this study had a significant influence on the number of maternal and early neonatal mortalities.

Bivariate Zero-Inflated Poisson Inverse Gaussian Regression Model and Its Application

Bivariate Zero-Inflated Poisson Inverse Gaussian Regression Model and Its Application

The multivariate mixed Negative Binomial regression model with an application to insurance a posteriori ratemaking

This paper is concerned with introducing a family of multivariate mixed Negative Binomial regression models in the context of a posteriori ratemaking. The multivariate mixed Negative Binomial regression model can be considered as a candidate model for capturing overdispersion and positive dependencies in multi-dimensional claim count data settings, which all recent studies suggest are the norm when the ratemaking consists of pricing different types of claim counts arising from the same policy. For expository purposes, we consider the bivariate Negative Binomial-Gamma and Negative Binomial-Inverse Gaussian regression models. An Expectation-Maximization type algorithm is developed for maximum likelihood estimation of the parameters of the models for which the definition of a joint probability mass function in closed form is not feasible when the marginal means are modelled in terms of covariates. In order to illustrate the versatility of the proposed estimation procedure a numerical illustration is performed on motor insurance data on the number of claims from third party liability bodily injury and property damage. Finally, the a posteriori, or Bonus-Malus, premium rates resulting from the bivariate Negative Binomial-Gamma and Negative Binomial-Inverse Gaussian regression model are compared to those determined by the bivariate Negative Binomial and Poisson-Inverse Gaussian regression models.

Parameter estimation and hypothesis testing of geographically and temporally weighted bivariate Poisson inverse Gaussian regression model

One of the appropriate methods used to model count data response and its corresponding predictors is Poisson regression. Poisson regression strictly assumes that the mean and variance of response variables should be equal (equidispersion). Nonetheless, some cases of the count data unsatisfied this assumption because variance can be larger than mean (over-dispersion). If overdispersion is violated, causing the underestimate standard error. Furthermore, this will lead to incorrect conclusions in the statistical test. Thus, a suitable method for modelling this kind of data needs to develop. One alternative model to outcome the overdispersion issue in bivariate response variable is the Bivariate Poisson Inverse Gaussian Regression (BPIGR) model. The BPIGR model can produce a global model for all locations. On the other hand, each location and time have different geographic conditions, social, cultural, and economical so that Geographically and Temporally Bivariate Poisson Inverse Gaussian Regression (GTWBPIGR)) is needed. The weighting function spatial-temporal in GTWBPIGR generates a different local model for each period. GTWBPIGR model solves the overdispersion case and generates global models for each period and location. The parameter estimation of the GTWBPIGR model uses the Maximum Likelihood Estimation (MLE) method, followed by Newton Raphson iteration. Meanwhile, the test statistics on the hypothesis testing is simultaneously testing of the GTWBPIGR model is obtained with the Maximum Likelihood Ratio Test (MLRT) approach, using n large samples of the statistical test is chi-square distribution. Moreover, the test statistics for partially testing used the Z-test statistic.

The climate index-length of stay nexus

This study sought to analyze the effect of the climate of tourists’ region of origin on their length of stay in a specific inland destination as climate of origin has been ignored in previous analyses. The present study collected data from 674 valid surveys of visitors in the selected destination and applied a zero-truncated negative binomial regression model and a Poisson-inverse Gaussian regression model. The results for this destination suggest that climate of origin affects tourists’ length of stay. This finding was obtained via the Poisson-inverse Gaussian regression model because of its greater tolerance to long tail distributions. Similarities and differences were found regarding results for other destinations found in the literature. The present findings further include the non-significant effect of reasons for traveling and tourists’ satisfaction and the significant influence of tourists’ mode of transport, income, and age on length of stay. Cheaper lodging categories also have an important impact on visitors who prefer extended stays.

Flexible regression models for counts with high-inflation of zeros

In this paper, we introduce a flexible class of regression models for counts with high-inflation of zeros that cannot be predicted by the Poisson, the zero-inflated Poisson, the negative binomial and the Poisson-inverse Gaussian regression models. Our proposed flexible regression models are based on a class of zero-inflated mixed Poisson distributions and contain the zero-inflated negative binomial (ZINB) and the zero-inflated Poisson-inverse Gaussian (ZIPIG) distributions, as particular cases, among others. We consider regression structures for the mean, the dispersion, and the zero-inflation parameters. Consequently, we generalize existing models, such as the ZINB regression (with non-varying dispersion), and also open the possibility of introducing new models, such as the ZIPIG and the zero-inflated generalized hyperbolic secant regressions. We propose an Expectation-Maximization (in short EM) algorithm for estimating the parameters and the associated information matrix. Simulation results are presented to compare the finite-sample performance of our proposed EM-algorithm with a direct maximization of the log-likelihood function based on the GAMLSS approach. These simulated results show some advantages of our EM-algorithm concerning the GAMLSS proposal. We also discuss a measure of influence based on the EM approach and propose simulated envelopes for checking the adequacy of our zero-inflated regression models. An empirical application, about the number of roots produced by 270 micropropagated shoots of the columnar apple cultivar Trajan, illustrates the usefulness of the proposed class of regression models for dealing with count data presenting high-inflation of zeros and shows that one cannot use the GAMLSS approach in some practical situations due to numerical problems.

Comparing Poisson-Inverse Gaussian Model and Negative Binomial Model on case study: horseshoe crabs data

Poisson Regression analysis is commonly used for dependent variable that has non-negative value, called count data. Poisson Regression has an assumption that mean of dependent variable equal to its variance. On over dispersion case where the variance is greater than mean, poisson regression is inconvenient to used because it may underestimate the standard error of regression parameters and consequently giving misleading inference. Poisson-Inverse Gaussian and Negative Binomial regression model can be used on over dispersion data. This paper will discuss about Poisson-Inverse Gaussian regression model, Negative Binomial regression model and comparing them in terms of Goodness-of-fit (GOF) statistics on case study of horseshoe crabs data. According to the result, pseudo R-squared value of P-IG regression model is greater than the Negative Binomial regression model. It shows that P-IG regression model is better than Negative Binomial regression model.

Insurance ratemaking using the Exponential-Lognormal regression model

AbstractThis paper is concerned with presenting the Exponential-Lognormal (ELN) regression model as a competitive alternative to the Pareto, or Exponential-Inverse Gamma, regression model that has been used in a wide range of areas, including insurance ratemaking. This is the first time that the ELN regression model is used in a statistical or actuarial context. The main contribution of the study is that we illustrate how maximum likelihood estimation of the ELN regression model, which does not have a density in closed form, can be accomplished relatively easily via an Expectation-Maximisation type algorithm. A real data application based on motor insurance data is examined in order to emphasise the versatility of the proposed algorithm. Finally, assuming that the number of claims is distributed according to the classic Negative Binomial and Poisson-Inverse Gaussian regression models, both the a priori and a posteriori, or Bonus–Malus, premium rates resulting from the ELN regression model are calculated via the net premium principle and compared to those determined by the Pareto regression model that has been traditionally used for modelling claim sizes.

The negative binomial-inverse Gaussian regression model with an application to insurance ratemaking

This paper presents the Negative Binomial-Inverse Gaussian regression model for approximating the number of claims as an alternative to mixed Poisson regression models that have been widely used in various disciplines including actuarial applications. The Negative Binomial-Inverse Gaussian regression model can be considered as a plausible model for highly dispersed claim count data and this is the first time that it is used in a statistical or actuarial context. The main achievement is that we propose a quite simple Expectation-Maximization type algorithm for maximum likelihood estimation of the model. Finally, a real data application using motor insurance data is examined and both the a priori and a posteriori, or Bonus-Malus, premium rates resulting from the Negative Binomial-Inverse Gaussian model are calculated via the net premium principle and compared to those determined by the Negative Binomial Type I and the Poisson-Inverse Gaussian regression models that have been traditionally used for a priori and a posteriori ratemaking.

The Poisson–Inverse-Gaussian regression model with cure rate: a Bayesian approach and its case influence diagnostics

This paper proposes a new survival model, called Poisson Inverse-Gaussian regression cure rate model (PIGcr), which enables different underlying activation mechanisms that lead to the event of interest. The number of competing causes of the event of interest follows a Poisson distribution and the time for the event follows an Inverse-Gaussian distribution. The model takes into account the presence of censored data and covariates. For inferential purposes, a Bayesian approach via Markov Chain Monte Carlo was considered. Discussions on the model selection criteria, as well as a case deletion influence diagnostics are addressed for a joint posterior distribution based on the \(\psi \)-divergence, which has several divergence measures as particular cases, such as Kullback–Leibler (K–L), \(J\)-distance, \(L_1\) norm and \(\chi ^2\)-square divergence measures. The procedures are illustrated in artificial and real data.

Influence analysis for Poisson inverse Gaussian regression models based on the EM algorithm

For Poisson inverse Gaussian regression models, it is very complicated to obtain the influence measures based on the traditional method, because the associated likelihood function involves intractable expressions, such as the modified Bessel function. In this paper, the EM algorithm is employed as a basis to derive diagnostic measures for the models by treating them as a mixed Poisson regression with the weights from the inverse Gaussian distributions. Several diagnostic measures are obtained in both case-deletion model and local influence analysis, based on the conditional expectation of the complete-data log-likelihood function in the EM algorithm. Two numerical examples are given to illustrate the results.

A general EM approach for maximum likelihood estimation in mixed Poisson regression models

An EM type algorithm for maximum likelihood estimation is proposed for the case of mixed Poisson regression models. The algorithm makes use of the mixture representation of such models. Two members of this family are examined in depth, the negative binomial regression model and the Poisson-inverse Gaussian regression model. Closed form expressions are derived for both models leading to easily programmable algorithms. Especially for the case of the Poisson-inverse Gaussian model no special numerical techniques are needed. The algorithms are applied to a real data set concerning crime data from Greece.