随机效应零膨胀索赔次数回归模型

索赔频率预测是非寿险费率厘定的重要组成部分。最常使用的索赔频率预测模型是泊松回归和负二项回归,以及与它们相对应的零膨胀回归模型。但是,当索赔次数观察值既具有零膨胀特征,又存在组内相依结构时,上述模型都不能很好地拟合实际数据。为此,本文在泊松分布、负二项分布、广义泊松分布、P型负二项分布等条件下分别建立了随机效应零膨胀损失次数回归模型。为了改进模型的预测效果,对于连续型的解释变量,还引入了二次平滑项,并建立了结构性零比例与解释变量之间的回归关系。基于一组实际索赔次数数据的实证分析结果表明,该模型可以显著改进现有模型的拟合效果。     

New polya and inverse polya distributions of order k

New Polya and inverse Polya distributions of order k are derived by means of generalized urn models and by compounding the binomial and negative binomial distributions of order k of Philippou (1986, 1983) with the beta distribution. It i s noted that the present Polpa distribution of order k includes as special cases a new hypergeometric distribution of order k, a negative one,an inverse one, and a discrete uniform of the same order. The probability generating functions, means and variances of the new distributions are obtained, and five asymptotic results are established relating them to the abovedmentioned binomial and negative binomial distributions of order k, and to the Poisson distribution of the same order of Philippou (1983).Moment estimates are also given and applications are indicated.     

Bivariate Negative Binomial Generalized Linear Models for Environmental Count Data

We propose a new bivariate negative binomial model with constant correlation structure, which was derived from a contagious bivariate distribution of two independent Poisson mass functions, by mixing the proposed bivariate gamma type density with constantly correlated covariance structure (Iwasaki & Tsubaki, 2005), which satisfies the integrability condition of McCullagh & Nelder (1989, p. 334). The proposed bivariate gamma type density comes from a natural exponential family. Joe (1997) points out the necessity of a multivariate gamma distribution to derive a multivariate distribution with negative binomial margins, and the luck of a convenient form of multivariate gamma distribution to get a model with greater flexibility in a dependent structure with indices of dispersion. In this paper we first derive a new bivariate negative binomial distribution as well as the first two cumulants, and, secondly, formulate bivariate generalized linear models with a constantly correlated negative binomial covariance structure in addition to the moment estimator of the components of the matrix. We finally fit the bivariate negative binomial models to two correlated environmental data sets.     

Zero-inflated proportion data models applied to a biological control assay

Biological control of pests is an important branch of entomology, providing environmentally friendly forms of crop protection. Bioassays are used to find the optimal conditions for the production of parasites and strategies for application in the field. In some of these assays, proportions are measured and, often, these data have an inflated number of zeros. In this work, six models will be applied to data sets obtained from biological control assays for Diatraea saccharalis , a common pest in sugar cane production. A natural choice for modelling proportion data is the binomial model. The second model will be an overdispersed version of the binomial model, estimated by a quasi-likelihood method. This model was initially built to model overdispersion generated by individual variability in the probability of success. When interest is only in the positive proportion data, a model can be based on the truncated binomial distribution and in its overdispersed version. The last two models include the zero proportions and are based on a finite mixture model with the binomial distribution or its overdispersed version for the positive data. Here, we will present the models, discuss their estimation and compare the results.     

Modeling spatial overdispersion via the generalized Waring point process

Modeling spatial overdispersion requires point process models with finite‐dimensional distributions that are overdisperse relative to the Poisson distribution. Fitting such models usually heavily relies on the properties of stationarity, ergodicity, and orderliness. In addition, although processes based on negative binomial finite‐dimensional distributions have been widely considered, they typically fail to simultaneously satisfy the three required properties for fitting. Indeed, it has been conjectured by Diggle and Milne that no negative binomial model can satisfy all three properties. In light of this, we change perspective and construct a new process based on a different overdisperse count model, namely, the generalized Waring (GW) distribution. While comparably tractable and flexible to negative binomial processes, the GW process is shown to possess all required properties and additionally span the negative binomial and Poisson processes as limiting cases. In this sense, the GW process provides an approximate resolution to the conundrum highlighted by Diggle and Milne.     

Improved auxiliary mixture sampling for hierarchical models of non-Gaussian data

The article considers Bayesian analysis of hierarchical models for count, binomial and multinomial data using efficient MCMC sampling procedures. To this end, an improved method of auxiliary mixture sampling is proposed. In contrast to previously proposed samplers the method uses a bounded number of latent variables per observation, independent of the intensity of the underlying Poisson process in the case of count data, or of the number of experiments in the case of binomial and multinomial data. The bounded number of latent variables results in a more general error distribution, which is a negative log-Gamma distribution with arbitrary integer shape parameter. The required approximations of these distributions by Gaussian mixtures have been computed. Overall, the improvement leads to a substantial increase in efficiency of auxiliary mixture sampling for highly structured models. The method is illustrated for finite mixtures of generalized linear models and an epidemiological case study.     

Some rememarks on locally most powerful unbiased tests for the detection of mixtures of poisson or binomial distributions

In t h i s note mixture models are used to represent overdispersion relative to Poisson or binomial distributions. We flnd a sufflclent condition on the mixing distribution underich the detection of mixture departures from the Poisson or binomial adrnits a locally most powerful unbiased test. The conditions specify plynoria: relations between the variance and mean of Le glxing distribution.     

Goodness-of-fit tests for binomial AR(1) processes

The binomial AR(1) model describes a nonlinear process with a first-order autoregressive (AR(1)) structure and a binomial marginal distribution. To develop goodness-of-fit tests for the binomial AR(1) model, we investigate the observed marginal distribution of the binomial AR(1) process, and we tackle its autocorrelation structure. Motivated by the family of power-divergence statistics for handling discrete multivariate data, we derive the asymptotic distribution of certain categorized power-divergence statistics for the case of a binomial AR(1) process. Then we consider Bartlett's formula, which is widely used in time series analysis to provide estimates of the asymptotic covariance between sample autocorrelations, but which is not applicable when the underlying process is nonlinear. Hence, we derive a novel Bartlett-type formula for the asymptotic distribution of the sample autocorrelations of a binomial AR(1) process, which is then applied to develop tests concerning the autocorrelation structure. Simulation studies are carried out to evaluate the size and power of the proposed tests under diverse alternative process models. Several real examples are used to illustrate our methods and findings.     

AN APPROXIMATE MAXIMUM LIKELIHOOD ESTIMATOR FOR CHAIN BINOMIAL MODELS

Using the Poisson approximation to the Binomial distribution, we construct an approximate maximum likelihood estimator (MLE) for a class of chain binomial models. Our estimator proves to have properties which may make it preferable to the exact WLE.     

Mixture formulations of a bivariate negative binomial distribution with applications

This paper considers further mixture formulations of the bivariate negative binomial (BNB) distribution of Edwards and Gurland (1961) and Subrahmaniam (1966). These formulations and some known ones are applied (1) to obtain a bivariate generalized negative binomial (BGNB) distribution of Bhattacharya (1966), (2) to establish a connection between the accident-proneness models given by the BNB, BGNB and Bhattacharya's bivariate distributions, and (3) to compute the grade correlation and distribution function of the Wicksell-Kibble bivariate gamma distribution.     

Inflated modified power series distributions with applications

In certain applications involving discrete data, it is sometimes found that X = 0 is observed with a frequency significantly higher than predicted by the assumed model. Zero inflated Poisson, binomial and negative binomial models have been employed in some clinical trials and in some regression analysis problems.

In this paper, we study the zero inflated modified power series distributions (IMPSD) which include among others the generalized Poisson and the generalized negative binomial distributions and hence the Poisson, binomial and negative binomial distributions. The structural properties along with the distribution of the sum of independent IMPSD variables are studied. The maximum likelihood estimation of the parameters of the model is examined and the variance-covariance matrix of the estimators is obtained. Finally, examples are presented for the generalized Poisson distribution to illustrate the results.     

Discrete Linnik Weibull distribution

Abstract

We introduce a new family of distributions using truncated discrete Linnik distribution. This family is a rich family of distributions which includes many important families of distributions such as Marshall–Olkin family of distributions, family of distributions generated through truncated negative binomial distribution, family of distributions generated through truncated discrete Mittag–Leffler distribution etc. Some properties of the new family of distributions are derived. A particular case of the family, a five parameter generalization of Weibull distribution, namely discrete Linnik Weibull distribution is given special attention. This distribution is a generalization of many distributions, such as extended exponentiated Weibull, exponentiated Weibull, Weibull truncated negative binomial, generalized exponential truncated negative binomial, Marshall-Olkin extended Weibull, Marshall–Olkin generalized exponential, exponential truncated negative binomial, Marshall–Olkin exponential and generalized exponential. The shape properties, moments, median, distribution of order statistics, stochastic ordering and stress–strength properties of the new generalized Weibull distribution are derived. The unknown parameters of the distribution are estimated using maximum likelihood method. The discrete Linnik Weibull distribution is fitted to a survival time data set and it is shown that the distribution is more appropriate than other competitive models.     

Accurate estimation for extra-Poisson variability assuming random effect models

In this study, the components of extra-Poisson variability are estimated assuming random effect models under a Bayesian approach. A standard existing methodology to estimate extra-Poisson variability assumes a negative binomial distribution. The obtained results show that using the proposed random effect model it is possible to get more accurate estimates for the extra-Poisson variability components when compared to the use of a negative binomial distribution where it is possible to estimate only one component of extra-Poisson variability. Some illustrative examples are introduced considering real data sets.     

Some bivariate families of lagrangian probability distributions

The bivariate Lagrange expansion, given by Poincare (1986), has been explained and slightly modified which gives bivariate Lagrangian probability models. A generalized bivariate Lagrangian Poisson distribution with six parameters has been obtained and studied. Also, the bivariate Lagrangian binomial, bivariate Lagrangian negative binomial and bivariate Lagrangian logarithmic series distribution have been obtained.     

Regression for doubly inflated multivariate Poisson distributions

Dependent multivariate count data occur in several research studies. These data can be modelled by a multivariate Poisson or Negative binomial distribution constructed using copulas. However, when some of the counts are inflated, that is, the number of observations in some cells are much larger than other cells, then the copula-based multivariate Poisson (or Negative binomial) distribution may not fit well and it is not an appropriate statistical model for the data. There is a need to modify or adjust the multivariate distribution to account for the inflated frequencies. In this article, we consider the situation where the frequencies of two cells are higher compared to the other cells and develop a doubly inflated multivariate Poisson distribution function using multivariate Gaussian copula. We also discuss procedures for regression on covariates for the doubly inflated multivariate count data. For illustrating the proposed methodologies, we present real data containing bivariate count observations with inflations in two cells. Several models and linear predictors with log link functions are considered, and we discuss maximum likelihood estimation to estimate unknown parameters of the models.     

Monitoring of zero-inflated binomial processes with a DEWMA control chart

Control charts are widely used for monitoring quality characteristics of high-yield processes. In such processes where a large number of zero observations exists in count data, the zero-inflated binomial (ZIB) models are more appropriate than the ordinary binomial models. In ZIB models, random shocks occur with probability θ, and upon the occurrence of random shocks, the number of non-conforming items in a sample of size n follows the binomial distribution with proportion p. In the present article, we study in more detail the exponentially weighted moving average control chart based on ZIB distribution (ZIB-EWMA) and we also propose a new control chart based on the double exponentially weighted moving average statistic for monitoring ZIB data (ZIB-DEWMA). The two control charts are studied in detecting upward shifts in θ or p individually, as well as in both parameters simultaneously. Through a simulation study, we compare the performance of the proposed chart with the ZIB-Shewhart, ZIB-EWMA and ZIB-CUSUM charts. Finally, an illustrative example is also presented to display the practical application of the ZIB charts.     

A bivariate mixture of negative binomial distributions and its applications

Abstract

We construct a new bivariate mixture of negative binomial distributions which represents over-dispersed data more efficiently. This is an extension of a univariate mixture of beta and negative binomial distributions. Characteristics of this joint distribution are studied including conditional distributions. Some properties of the correlation coefficient are explored. We demonstrate the applicability of our proposed model by fitting to three real data sets with correlated count data. A comparison is made with some previously used models to show the effectiveness of the new model.     

On Markov-Pólya Distribution and the Katz Family of Distributions

This article is concerned with the Markov-Pólya distribution and its links with the Katz family of distributions. The Katz family is defined through a first-order recursion of remarkable form; it (only) covers the Poisson, negative binomial and binomial distributions. The Markov-Pólya distribution arises in the study of certain urn or population models that incorporate (anti)contagion effects. The present work is motivated by questions and applications in actuarial sciences. First, the Markov-Pólya distribution is presented as a claim frequency model. This distribution is then shown to satisfy a Katz-like recursion. As a consequence, a simple recursion is derived for computing a compound sum distribution that generalizes the Panjer algorithm in risk theory. The Katz family is also obtained as a limit of the Markov-Pólya distribution. Finally, an observed frequency of car accidents is fitted by a Markov-Pólya distribution.     

ESTIMATING A RATIO OF MEANS FROM BIVARIATE COUNTS WITH APPLICATIONS IN STEREOLOGY

In survey sampling and in stereology, it is often desirable to estimate the ratio of means θ= E(Y)/E(X) from bivariate count data (X, Y) with unknown joint distribution. We review methods that are available for this problem, with particular reference to stereological applications. We also develop new methods based on explicit statistical models for the data, and associated model diagnostics. The methods are tested on a stereological dataset. For point‐count data, binomial regression and bivariate binomial models are generally adequate. Intercept‐count data are often overdispersed relative to Poisson regression models, but adequately fitted by negative binomial regression.     

The bivariate logarithmic series distribution

The bivariate logarithmic series distribution was introduced by Subrahmaniam (1966) as a Fisher-limit to the bivariate negative binomial distribution. The present paper considers the properties of the distribution along with various models giving rise to it. Problems of estimation and the goodness-of-fit are examined. Methods for simulating the distribution are developed and illusuated.