Poisson Inverse Gaussian Research Articles

Using contingent behaviour analysis to estimate benefits from coral reefs in Kume Island, Japan: A Poisson‐inverse Gaussian approach with on‐site correction

AbstractCoral reefs face a critical crisis worldwide because of rising ocean temperature, excessive use of resources and soil erosion. As reefs have great recreational and tourism value, the degradation of their quality may have a significant effect on tourism. This study employs a contingent behaviour approach to estimate the effect of reef extinction on the recreational demand for Kume Island, Okinawa, Japan. We propose a Poisson‐inverse Gaussian (PIG) model with correction for on‐site sampling issues to derive a more accurate estimate of consumer surplus (CS). The results show that the annual CS per‐person trip is 5898 yen (US$ 49.15 in 2015 currency) according to the random‐effects PIG model.

Analysis of Count Time Series: A Bayesian GARMA(p, q) Approach

Extensions of the Autoregressive Moving Average, ARMA(p, q), class for modeling non-Gaussian time series have been proposed in the literature in recent years, being applied in phenomena such as counts and rates. One of them is the Generalized Autoregressive Moving Average, GARMA(p, q), that is supported by the Generalized Linear Models theory and has been studied under the Bayesian perspective. This paper aimed to study models for time series of counts using the Poisson, Negative binomial and Poisson inverse Gaussian distributions, and adopting the Bayesian framework. To do so, we carried out a simulation study and, in addition, we showed a practical application and evaluation of these models by using a set of real data, corresponding to the number of vehicle thefts in Brazil.

ANALISIS REGRESI POISSON INVERSE GAUSSIAN UNTUK MENENTUKAN FAKTOR-FAKTOR YANG MEMPENGARUHI JUMLAH KASUS DEMAM BERDARAH DENGUE (DBD)

One of the census data is the number of dengue cases where to model the data can be done using poisson regression. Poisson regression has an assumptions that must be met, namely the average value dan variance must be the same or colled (equidispersion). But, in this study, the data experienced a violation in ehich the variance value was greater than the average value or colled (overdispersion). To overcome thie problem by using the mixed poisson method, namely poisson inverse gaussian regression for count data that is overdispersed and has a likelihood function that is cloce form, that is the parameters are know, so many researchers use this model. The number of DBD cases in Riau Province is counting data that has overdispersion. So therefore, to model the number of cases of DBD using poisson inverse gaussian regression. Based on the model used, it shows the factors that influence the number of dengue cases in Riau Province in 2019, namely the number of health facilities.

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.

Zero-inflated poisson quasi-Lindley regression for modeling number of doctor visit data

The Poisson regression is a popular approach in modeling count data. However, in many situations often the variance of data is greater than the mean (over-dispersed data) and the generalized Poisson or mixed Poisson models such as the Poisson gamma (negative binomial), Poisson inverse Gaussian, Poisson lognormal, and Poisson Lindley have been proposed as the alternatives to the Poisson for describing over-dispersed count data. In some situations, the source of over-dispersion is the large percentage of zeros in the dataset. In the other words, the dataset involves an excessive number of zeros than are expected in the common discrete distributions which are known as the zero-inflated events. In order to analyze these data, zero-inflated models such as the zero-inflated Poisson, zero-inflated generalized Poisson, and zero-inflated negative binomial have been applied. This work proposes the functional form and the regression model of the zero-inflated Poisson quasi-Lindley (ZIPQL) and then, beside the alternative models, it was fitted and compared to US National Medical Expenditure Survey data.

Modeling of Tuberculosis Cases in Sumatera Region using Poisson Inverse Gaussian Regression

In the Sumatra Region, tuberculosis (TB) is a disease that needs special attention because it tends to increase every year. Based on health theory, there are many factors that cause TB, but it is not easy to determine which factors have a significant effect. Therefore, in this study an analysis was carried out that could model, predict, and determine the factors causing TB disease in the Sumatra Region. The data used is data on TB cases in the Sumatra Region in 2018 taken from the Publication of the Central Statistics Agency. Poisson regression is an analysis that is suitable for modeling count data such as TB disease data. The assumption of Poisson regression is that the mean and variance of the response variables must be equal (equidispersion). However, the TB case data in the Sumatra Region in 2018 has an average value that is smaller than the variance (overdispersion) so it cannot be solved by Poisson regression. To overcome this problem, we need a method that can overcome overdispersion, namely Poisson Inverse Gaussian (PIG) ​​regression. From the results of the analysis using PIG regression, it can be concluded that the factors that have a significant effect on TB cases in the Sumatra Region are the percentage of the male population (X1), the percentage of the productive age population (X2), the percentage of households with a floor area of ≤ ​​19m2 (X3), and the percentage of households that have access to proper sanitation (X4), where the model formed is 
 
 Based on the model, the predicted results of TB cases in the Sumatra Region had an average of 596.04178 where the lowest cases occurred in Pringsewu of 154.8943 and the highest cases occurred in Bukittinggi of 2719.59400.

Using Saturated Count Models for User-Friendly Synthesis of Large Confidential Administrative Databases

AbstractOver the past three decades, synthetic data methods for statistical disclosure control have continually evolved, but mainly within the domain of survey data sets. There are certain characteristics of administrative databases, such as their size, which present challenges from a synthesis perspective and require special attention. This paper, through the fitting of saturated count models, presents a synthesis method that is suitable for administrative databases. It is tuned by two parameters, σ and α. The method allows large categorical data sets to be synthesized quickly and allows risk and utility metrics to be satisfied a priori, that is, prior to synthetic data generation. The paper explores how the flexibility afforded by two-parameter count models (the negative binomial and Poisson-inverse Gaussian) can be utilised to protect respondents'—especially uniques'—privacy in synthetic data. Finally, an empirical example is carried out through the synthesis of a database which can be viewed as a good substitute to the English School Census.

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.

COUNT DATA MODELING USING GAMLSS APPROACH AND ITS APPLICATION IN DENGUE HEMORRHAGIC FEVER CASES IN EAST KALIMANTAN PROVINCE, INDONESIA

Generalized Additive Models for Location, Scale, and Shape (GAMLSS) is a robust approach used to model various types and characteristics of data. Therefore, this research aims to model the count data using the GAMLSS approach through Poisson Regression (PR), Poisson Inverse Gaussian Regression (PIGR), and Negative Binomial Regression (NBR). PIGR and NBR are the best models compared to PR based on their application to modeling the number of dengue hemorrhagic fever (DHF) cases in East Kalimantan Province, Indonesia, in 2019. Furthermore, both models produced varying results on the factors with a significant effect on DHF. Only one factor of the PIGR model, namely altitude, significantly affected these cases. Meanwhile, the NBR model produced three factors that affected the number of dengue cases: altitude, population density, and health workers.

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

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

Bivariate Mixed Poisson Regression Models with Varying Dispersion

The main purpose of this article is to present a new class of bivariate mixed Poisson regression models with varying dispersion that offers sufficient flexibility for accommodating overdispersion and accounting for the positive correlation between the number of claims from third-party liability bodily injury and property damage. Maximum likelihood estimation for this family of models is achieved through an expectation-maximization algorithm that is shown to have a satisfactory performance when three members of this family, namely, the bivariate negative binomial, bivariate Poisson–inverse Gaussian, and bivariate Poisson–Lognormal distributions with regression specifications on every parameter are fitted on two-dimensional motor insurance data from a European motor insurer. The a posteriori, or bonus-malus, premium rates that are determined by these models are calculated via the expected value and variance principles and are compared to those based only on the a posteriori criteria. Finally, we present an extension of the proposed approach with varying dispersion by developing a bivariate Normal copula-based mixed Poisson regression model with varying dispersion and dependence parameters. This approach allows us to consider the influence of individual and coverage-specific risk factors on the mean, dispersion, and copula parameters when modeling different types of claims from different types of coverage. For expository purposes, the Normal copula paired with negative binomial distributions for marginals and regressors on the mean, dispersion, and copula parameters is fitted on a simulated dataset via maximum likelihood.

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 Poisson Inverse Gaussian Regression Model in the Analysis of Clustered Counts Data

We explore the possibility of modeling clustered count data using the Poisson Inverse Gaussian distribution. We develop a regression model, which relates the number of mastitis cases in a sample of dairy farms in Ontario, Canada, to various farm level covariates, to illustrate the method ology. Residual plots are constructed to explore the quality of the fit. We compare the results with a negative binomial regression model using max imum likelihood estimation, and to the generalized linear mixed regression model fitted in SAS.

Discrimination between Some Over Dispersed Count Distributions

The Poisson inverse Gaussian and generalized Poisson distributions are widely used in modelling overdispersed count data which are commonly found in healthcare, insurance, engineering, econometric and ecology. The inverse trinomial distribution is a relatively new count distribution arising from a one-dimensional random walk model (Shimizu & Yanagimoto, 1991). The Poisson inverse Gaussian distribution is a popular count model that has been proposed as an alternative to the negative binomial distribution. The inverse trinomial and generalized Poisson models possess a common characteristic of having a cubic variance function, while the Poisson inverse Gaussian has a quadratic variance function. The nature of the variance function seems to be an important property in modelling overdispersed count data. Hence it is of interest to be able to select among the three models in practical applications. This paper considers discrimination of three models based on the likelihood ratio statistic and computes via Monte Carlo simulation the probability of correct selection.

Parameter Estimation and Hypothesis Testing of Geographically Weighted Bivariate Zero Inflated Poisson Inverse Gaussian Regression Models

Count data is data that can be counted within a certain range of time. The count data itself can be accommodated in the Poisson distribution which is always positive value. Poisson regression is often used to model count data, but Poisson regression requires equality in means and variance (equidispersion). The count data often contains many zero values and is unable to meet the equidipersion assumption. Zero Inflated Poisson Inverse Gaussian Regression (ZIPIGR) is a mix poisson regression that can accommodate under/overdispersion and also the number of zero values in the data. ZIPIGR is global regression model, so the interpretation of the model obtained is global in all observational data. The ZIPIGR method that accommodate the location factor is the Geographically Weighted ZIPIGR (GWZIPIGR). Bivariate GWZIPIGR (GWBZIPIGR) can be used when there are two response variables that correlate each other. This paper will discuss parameter estimation of GWBZIPIGR using Maximum Likelihood Estimation (MLE) and numerical iteration of Berndt Hall Hall Hausman (BHHH) is used when the result of estimation equation does not close form, Meanwhile Maximum Likelihood Ratio Test (MLRT) is used to test the parameters where simultaneous hypothesis testing used the G2 test statistic at large sample with X2 distribution.

Bivariate Poisson Inverse Gaussian Regression Model with Exposure Variable: Infant and Maternal Death Case Study

One of appropriate methods that could be used to model positive integers response and its corresponding predictors is Poisson regression. This method has a strict assumption to be achieved, that is the mean and variance of response variable must be equal (equidispersion). In many practical issues, sometimes, the variance is larger than the mean (overdispersion). Bivariate Poisson Inverse Gaussian Regression (BPIGR) is proposed to overcome the overdispersion issue. In this study, the BPIGR model is developed by adding the exposure variable as the weight of each sample unit. This study applies the proposed method to model the infant and maternal death data (as the responses) and the factors that influence them. The data of East Java 2017 are used for the application. Maximum Likelihood Estimation (MLE) method is used to estimate the parameters, while the hypothesis testing of the BPIGR model is derived using Maximum Likelihood Ratio Test (MLRT) approach. The results showed that the BPIGR model is a proper model for modeling the number of infant and maternal deaths. Furthermore, all predictor variables significantly influence those two responses.

Inference Procedures on the Generalized Poisson Distribution from Multiple Samples: Comparisons with Nonparametric Models for Analysis of Covariance (ANCOVA) of Count Data

Count data that exhibit over dispersion (variance of counts is larger than its mean) are commonly analyzed using discrete distributions such as negative binomial, Poisson inverse Gaussian and other models. The Poisson is characterized by the equality of mean and variance whereas the Negative Binomial and the Poisson inverse Gaussian have variance larger than the mean and therefore are more appropriate to model over-dispersed count data. As an alternative to these two models, we shall use the generalized Poisson distribution for group comparisons in the presence of multiple covariates. This problem is known as the ANCOVA and is solved for continuous data. Our objectives were to develop ANCOVA using the generalized Poisson distribution, and compare its goodness of fit to that of the nonparametric Generalized Additive Models. We used real life data to show that the model performs quite satisfactorily when compared to the nonparametric Generalized Additive Models.

Parameter Estimation and Hypothesis Testing of Multivariate Poisson Inverse Gaussian Regression

Multivariate Poisson regression is used in order to model two or more count response variables. The Poisson regression has a strict assumption, that is the mean and the variance of response variables are equal (equidispersion). Practically, the variance can be larger than the mean (overdispersion). Thus, a suitable method for modelling these kind of data needs to be developed. One alternative model to overcome the overdispersion issue in the multi-count response variables is the Multivariate Poisson Inverse Gaussian Regression (MPIGR) model, which is extended with an exposure variable. Additionally, a modification of Bessel function that contain factorial functions is proposed in this work to make it computable. The objective of this study is to develop the parameter estimation and hypothesis testing of the MPIGR model. The parameter estimation uses the Maximum Likelihood Estimation (MLE) method, followed by the Newton–Raphson iteration. The hypothesis testing is constructed using the Maximum Likelihood Ratio Test (MLRT) method. The MPIGR model that has been developed is then applied to regress three response variables, i.e., the number of infant mortality, the number of under-five children mortality, and the number of maternal mortality on eight predictors. The unit observation is the cities and municipalities in Java Island, Indonesia. The empirical results show that three response variables that are previously mentioned are significantly affected by all predictors.

EM Estimation for the Poisson-Inverse Gamma Regression Model with Varying Dispersion: An Application to Insurance Ratemaking

This article presents the Poisson-Inverse Gamma regression model with varying dispersion for approximating heavy-tailed and overdispersed claim counts. Our main contribution is that we develop an Expectation-Maximization (EM) type algorithm for maximum likelihood (ML) estimation of the Poisson-Inverse Gamma regression model with varying dispersion. The empirical analysis examines a portfolio of motor insurance data in order to investigate the efficiency of the proposed algorithm. Finally, both the a priori and a posteriori, or Bonus-Malus, premium rates that are determined by the Poisson-Inverse Gamma model are compared to those that result from the classic Negative Binomial Type I and the Poisson-Inverse Gaussian distributions with regression structures for their mean and dispersion parameters.