Finite Poisson Mixtures Research Articles

Spatial identification of component-based relative risks

This article aims at identifying the high risk provinces in Iraq using a finite Poisson mixture. Through this methodology, the levels of relative risk is determined through identifying the number of components. In this article we do not investigate spatial correlation among regions and assume that the levels of risk observed in different regions are independent each other. The estimation of the model parameters and the model selection are performed using the Bayesian approach which allow to allocate each province to an identified risk level. We consider the data of the Coronavirus disease (COVID-19) infections in 18 provinces in Iraq and determining the levels of relative risks of this pandemic. The results are spatially shown in map which illustrates that the best Bayesian model fitted the data is 3 components model (high, medium and low risk).

Optimal Bonus-Malus Systems Using Generalized Additive Models for Location, Scale and Shape

This paper presents the design of optimal Bonus-Malus Systems (BMS) using generalized additive models for location, scale and shape (GAMLSS), extending the work of Tzougas, Frangos and Vrontos (2014). Specifically, for the frequency component we employ a Negative Binomial Type I, a Poisson-Inverse Gaussian, a Sichel and a finite Poisson mixture GAMLSS model, while for the severity component we employ a Pareto and a finite Exponential mixture GAMLSS models. In the path towards actuarial relevance the Bayesian view is taken and the premiums are calculated by updating the posterior mean and posterior probability of the policyholders' classes of risk. Our analysis shows that the employment of more advanced models can provide a measure of uncertainty regarding the credibility updates of claim frequency/severity of each specific risk class and the difference in the premium that they imply can act as a cushion against adverse experience. Finally, these tailor-made premiums are compared to those which correspond to the 'univariate',without regression components, models.

PM-Seq: Using Finite Poisson Mixture Models for RNA-Seq Data Analysis and Transcript Expression Level Quantification

RNA-Seq has emerged as a powerful technique for transcriptome study. As much as the improved sensitivity and coverage, RNA-Seq also brings about challenges for data analysis. The massive amount of sequence reads data, excessive variability, uncertainties, and bias and noises stemming from multiple sources all make the analysis of RNA-Seq data difficult. Despite much progress, RNA-Seq data analysis still has much room for improvement, especially on the quantification of transcript/gene expression levels. In this article, using finite Poisson mixture models, we propose a two-step approach, called PM-Seq, to characterizing base pair level RNA-Seq data and quantifying transcript/gene expression levels. Finite Poisson mixture models combine the strength of fully parametric models with the flexibility of fully nonparametric models, and are extremely suitable for modeling heterogeneous count data such as RNA-Seq data. In particular, we consider three types of Poisson mixture model and propose to use a BIC-based model selection procedure to adapt the models to individual transcripts. A unified quantification method based on the Poisson mixture models is developed to measure transcript/gene expression levels. The Poisson mixture models and the proposed quantification method were applied to analyze two RNA-Seq data sets and demonstrated excellent performances in comparison with other existing methods. Our approach resulted in better characterization of the data and more accurate measurements of transcript expression levels. We believe that finite Poisson mixture models provide a flexible framework to model RNA-Seq data, and methods developed based on this framework have the potential to become powerful tools for RNA-Seq data analysis.

Minimum Hellinger Distance Estimation for k‐Component Poisson Mixture with Random Effects

The k-component Poisson regression mixture with random effects is an effective model in describing the heterogeneity for clustered count data arising from several latent subpopulations. However, the residual maximum likelihood estimation (REML) of regression coefficients and variance component parameters tend to be unstable and may result in misleading inferences in the presence of outliers or extreme contamination. In the literature, the minimum Hellinger distance (MHD) estimation has been investigated to obtain robust estimation for finite Poisson mixtures. This article aims to develop a robust MHD estimation approach for k-component Poisson mixtures with normally distributed random effects. By applying the Gaussian quadrature technique to approximate the integrals involved in the marginal distribution, the marginal probability function of the k-component Poisson mixture with random effects can be approximated by the summation of a set of finite Poisson mixtures. Simulation study shows that the MHD estimates perform satisfactorily for data without outlying observation(s), and outperform the REML estimates when data are contaminated. Application to a data set of recurrent urinary tract infections (UTI) with random institution effects demonstrates the practical use of the robust MHD estimation method.

Analysis of Defaulters' Behaviour Using Poisson-Mixture Approachs

Credit scoring is of major interest to financial institutions as a mean of controlling their capital allocation and increasing their profitability. Classical credit-scoring approaches are based on a ‘good’ or ‘bad’ basis, i.e. by classifying new applicants into one of the two possible categories based on a series of their characteristics. This strongly depends on the definition of good and bad customers. We adopt a different approach. Instead of trying to predict the class of a new applicant, we try to predict the number of defaults in the near future. Previous attempts toward this idea have used negative binomial regression models to fit the data. We fit finite Poisson mixtures treating the number of components unknown which have to be estimated from the data. Using covariates in all components, we reveal the impact of several demographic variates in creating the different groups of customers but we are able to predict for each customer the group to which he belongs, as well his expected number of defaults. This is of particular interest for financial purposes as it allows the estimation of the expected loss from each customer and thus the ability to create rules for providing the loan on a personal basis. This classification allows the means to determine the optimal interest rate to maximize bank profit under asymmetrical information. This paper uses real data and compares various models focusing on the finite Poisson mixture regression models.

Analysis of defaulters' behaviour using the Poisson-mixture approach

Credit scoring is of major interest to financial institutions as a mean of controlling their capital allocation and increasing their profitability. Classical credit-scoring approaches are based on a ‘good’ or ‘bad’ basis, i.e. by classifying new applicants into one of the two possible categories based on a series of their characteristics. This strongly depends on the definition of good and bad customers. We adopt a different approach. Instead of trying to predict the class of a new applicant, we try to predict the number of defaults in the near future. Previous attempts toward this idea have used negative binomial regression models to fit the data. We fit finite Poisson mixtures treating the number of components unknown which have to be estimated from the data. Using covariates in all components, we reveal the impact of several demographic variates in creating the different groups of customers but we are able to predict for each customer the group to which he belongs, as well his expected number of defaults. This is of particular interest for financial purposes as it allows the estimation of the expected loss from each customer and thus the ability to create rules for providing the loan on a personal basis. This classification allows the means to determine the optimal interest rate to maximize bank profit under asymmetrical information. This paper uses real data and compares various models focusing on the finite Poisson mixture regression models.

A zero frequency alternative method to the moment method of estimation in finite poisson mixtures

The method of moments (MM) has been widely used for parametric estimation, as it is often computationally simple. Our interest focuses on the case of finite Poisson mixtures. The inefficiency of the method of moments relative to the Maximum Likelihood (ML) method is studied. Both the asymptotic efficiency as well as the small sample efficiency is examined. The case of samples that fail to lead to MM estimates is also considered. The results discourage the use of the MM estimators for two reasons; the first is that they are inefficient relative to the ML estimators and the second is the high probability of failing to lead to valid estimates. Another method, which considers replacing the third'moment by the zero frequency, is examined. This method turns out to be more efficient than the moment method and not very demanding computationally.

Choosing initial values for the EM algorithm for finite mixtures

The EM algorithm is the standard tool for maximum likelihood estimation in finite mixture models. The main drawbacks of the EM algorithm are its slow convergence and the dependence of the solution on both the stopping criterion and the initial values used. The problems referring to slow convergence and the choice of a stopping criterion have been dealt with in literature and the present paper deals with the initial value problem for the EM algorithm. The aim of this paper is to compare several methods for choosing initial values for the EM algorithm in the case of finite mixtures as well as to propose some new methods based on modifications of existing ones. The cases of finite normal mixtures with common variance and finite Poisson mixtures are examined through a simulation study.

Robust inference for finite poisson mixtures

Inference for mixture models based on likelihood estimates suffers from lack of robustness. The presence of a few spurious observations may lead to incorrect decisions. In this paper we consider robust alternatives to the likelihood inference for finite Poisson mixtures based on the minimum Hellinger distance estimates. A new test, the Hellinger deviance test, is proposed for testing the Poisson hypothesis versus a Poisson mixture hypothesis. Moreover, diagnostics based on the Hellinger gradient function in order to examine for the presence of a mixture are described. Semiparametric estimation is also discussed. All these inferential procedures combine both efficiency when the model is correct and robustness when the model is incorrect, and make the minimum Hellinger distance methodology a competitive alternative to the maximum likelihood methodology.

On Testing for the Number of Components in a Mixed Poisson Model

Poisson mixtures are usually used to describe overdispersed data. Finite Poisson mixtures are used in many practical situations where often it is of interest to determine the number of components in the mixture. Identifying how many components comprise a mixture remains a difficult problem. The likelihood ratio test (LRT) is a general statistical procedure to use. Unfortunately, a number of specific problems arise and the classical theory fails to hold. In this paper a new procedure is proposed that is based on testing whether a new component can be added to a finite Poisson mixture which eventually leads to the number of components in the mixture. It is a sequential testing procedure based on the well known LRT that utilises a resampling technique to construct the distribution of the test statistic. The application of the procedure to real data reveals some interesting features of the distribution of the test statistic.

Minimum Hellinger distance estimation for Poisson mixtures

Minimum Hellinger distance (MHD) Estimation is an appealing method of estimation for discrete data as it works well in cases where the assumed model provides a poor fit to observed data and the maximum-likelihood (ML) method fails. Often, spurious observations that may cause problems to the ML method do not seem to affect the MHD method which in general performs better with such data. In this paper we derive MHD estimates for finite Poisson mixtures. The properties of these estimators are examined and a comparison is made of their performance relative to that of the ML estimators. MHD estimators are both efficient and robust. A numerical example involving data sets on environmental complaints is presented. An iterative algorithm that facilitates computation is provided. The algorithm always converges to a minimum, but several initial values are needed to ensure that the global minimum is obtained.

High-resolution inversion of the discrete Poisson and binomial transformations

The discrete Poisson transformation involved in finite Poisson mixtures arises in many applications, for example in relating observed fluctuations in photon counts to unobserved fluctuations in the number of molecules present, in the analysis of egg counts in zoology, of plankton abundances in fisheries research, of death notice frequencies or of comet frequencies to name a few. The binomial transformation, in addition to its many applications, serves to approximate the Poisson transformation. Procedures such as the maximum likelihood method (MLM) and the moment method (MM) are frequently used to invert these transforms. For the limited data case the authors consider, the low signal-to-noise ratio in the data necessitates some modification of existing methods. They propose a filtering procedure based on regularized Fourier transform estimation followed by high-resolution eigenvector-based spectral estimation. Examples are given which compare this approach with MM and MLM. They find improved performance in the estimation of parameters for the discrete model, particularly for the cases of low data set size and reduced signal-to-noise ratios.