The Poisson Birnbaum–Saunders model with long-term survivors

In this paper, we propose a cure rate survival model by assuming that the number of competing causes of the event of interest follows the Poisson distribution and the time to event has the Birnbaum–Saunders (BS) distribution. We define the Poisson BS distribution and provide two useful representations for its density function which facilitate to obtain some mathematical properties. Two closed-form expressions for the moments of the new distribution are given. We estimate the parameters of the model with cure rate using maximum likelihood. For different parameter settings, sample sizes and censoring percentages, several simulations are performed. We derive the appropriate matrices for assessing local influence on the parameter estimates under different perturbation schemes and present some ways to perform a global influence study. We analyse a real data set from the medical area.     

The truncated generalized poisson distribution and its estimation

A discrete probability model always gets truncated during the sampling process and the point of truncation depends upon the sample size. Also, the generalized Poisson distribution cannot be used with full justification when the second parameter is negative. To avoid these problems a truncated generalized Poisson distribution is defined and studied. Estimation of its parameters by moments method, maximum likelihood method and a mixed method are considered. Some examples are given to illustrate the effect on the parameters’ estimates when a non-truncated GPD is used instead of a truncated GPD.     

Gamma variates of fractional shape as functioials of a homogeneous multidimensional poisson process

If events are scattered in Rⁿ in accordance with a homogeneous Poisson process and if X is the location of the event with minimal [d]l^P norm, then in the case p = n the n^th absolute powers of the coordinates of X form a sample of size n from a gamma distribution with shape parameter 1/n. In an age of parallel computing, this fact may lead to some attractive simulation methods. One possibility is to generate R = [d]X[d] and U = Y/[d]X[d] independently, perhaps by setting U = Y/[d]Y[d] where Y has any p.d.f. which is a function only of ¦Y¦. We consider for example Y having the uniform distribution in an l^P ball.     

The analysis of multivariate recurrent events with partially missing event types

In many clinical studies, subjects are at risk of experiencing more than one type of potentially recurrent event. In some situations, however, the occurrence of an event is observed, but the specific type is not determined. We consider the analysis of this type of incomplete data when the objectives are to summarize features of conditional intensity functions and associated treatment effects, and to study the association between different types of event. Here we describe a likelihood approach based on joint models for the multi-type recurrent events where parameter estimation is obtained from a Monte-Carlo EM algorithm. Simulation studies show that the proposed method gives unbiased estimators for regression coefficients and variance–covariance parameters, and the coverage probabilities of confidence intervals for regression coefficients are close to the nominal level. When the distribution of the frailty variable is misspecified, the method still provides estimators of the regression coefficients with good properties. The proposed method is applied to a motivating data set from an asthma study in which exacerbations were to be sub-typed by cellular analysis of sputum samples as eosinophilic or non-eosinophilic.     

地震死亡人数预测与巨灾保险基金测算

巨灾保险制度在很大程度上依赖于巨灾损失的建模分析。由于巨灾损失通常存在极端值,一般的统计分布很难对其进行有效拟合。本文以我国大陆地区1950-2015年期间的地震灾害为研究样本,基于二维泊松过程建立了地震灾害死亡人数的预测模型。根据地震死亡人数的分布特征,将地震灾害分为非巨灾事件和巨灾事件,分别用右截断的负二项分布和右截断的广义帕累托分布拟合死亡人数;用齐次泊松过程描述地震灾害在给定期间的发生次数;用Panjer迭代法和快速傅里叶变换计算地震死亡人数在特定时期的分布以及风险度量值;用蒙特卡罗模拟法测算我国地震死亡保险基金的规模和纯保费水平。与传统的巨灾模型相比,本文提出的方法同时考虑了地震灾害发生的时间和地震死亡人数两个维度,并用贝叶斯方法估计模型参数,对地震死亡人数的拟合更加合理,为完善我国地震死亡保险提供了一种新的思路。     

Selection procedures for sparse data

The problem of selecting the best of k populations is studied for data which are incomplete as some of the values have been deleted randomly. This situation is met in extreme value analysis where only data exceeding a threshold are observable. For increasing sample size we study the case where the probability that a value is observed tends to zero, but the sparse condition is satisfied, so that the mean number of observable values in each population is bounded away from zero and infinity as the sample size tends to infinity. The incomplete data are described by thinned point processes which are approximated by Poisson point processes. Under weak assumptions and after suitable transformations these processes converge to a Poisson point process. Optimal selection rules for the limit model are used to construct asymptotically optimal selection rules for the original sequence of models. The results are applied to extreme value data for high thresholds data.     

Optimality Of Some Two-Stage Sampling Methods

Two-stage sampling plans in which the estimator is computed from the second sample only, and has a guaranteed precision irrespective of the parameter value, are considered. We prove the following optimality property for existing strategies in binomial and Poisson populations: there exists no other

strategy, of the same form, that guarantees the same precision, without increasing the expected sample size for some parameter valucs. The method of proof is illustrnted by proving a similar result for normal populations     

Biomedical applications for a generalized linear functional Poisson model

Many medical and biological studies involve response in the form of Poisson counts which can bemodelled using explanatory variables which also arise from count data. If the explanatory variables are observable without error (also as Poisson counts) we have a generalized linear model with a logarithmic link function and Poisson error structure. If,however, some of the explanatory variables are not directly observable, but arise with superimposed errors (again of Poisson form), the model is of a new type:a generalised linear functional Poisson model. In this paper,maximum likelihood estimates of the parameters of this model are determined along with the information matrix which (on noting its particular patterned form) is amenable to inversion in explicit form. Methods are proposed of an iterative type for computing estimates of the parameters and of their variational properties (e.g. standard errors) for this model, which also has application in other fields such as road traffic studies.     

Tests for outliers in the inverse Gaussian distribution,with application to first hitting time models

The inverse Gaussian (IG) distribution is often applied in statistical modelling, especially with lifetime data. We present tests for outlying values of the parameters (μ, λ) of this distribution when data are available from a sample of independent units and possibly with more than one event per unit. Outlier tests are constructed from likelihood ratio tests for equality of parameters. The test for an outlying value of λ is based on an F-distributed statistic that is transformed to an approximate normal statistic when there are unequal numbers of events per unit. Simulation studies are used to confirm that Bonferroni tests have accurate size and to examine the powers of the tests. The application to first hitting time models, where the IG distribution is derived from an underlying Wiener process, is described. The tests are illustrated on data concerning the strength of different lots of insulating material.     

Selection and Ranking Procedures for Type I Extreme Value Populations and a Related Homogeneity Test

Consider k (≥2) independent Type I extreme value populations with unknown location parameters and common known scale parameter. With samples of same size, we study procedures based on the sample means for (1) selecting the population having the largest location parameter, (2) selecting the population having the smallest location parameter, and (3) testing for equality of all the location parameters. We use Bechhofer's indifference-zone and Gupta's subset selection formulations. Tables of constants for implemention are provided based on approximation for the distribution of the standardized sample mean by a generalized Tukey's lambda distribution. Examples are provided for all procedures.     

Markov zero-inflated Poisson regression models for a time series of counts with excess zeros

This paper discusses a class of Markov zero-inflated Poisson regression models for a time series of counts with the presence of excess zero relative to a Poisson distribution, in which the frequency distribution changes according to an underlying two-state Markov chain. Features of the proposed model, estimation method based on the EM and quasi-Newton algorithms, and other implementation issues are discussed. A Monte Carlo study shows that the estimation method is accurate and reliable as long as the sample size is reasonably large, and the choice of starting probabilities for the Markov process has little impact on the parameter estimates. The methodology is illustrated using daily numbers of phone calls reporting faults for a mainframe computer system.     

The discrete Lindley distribution: properties and applications

Modelling count data is one of the most important issues in statistical research. In this paper, a new probability mass function is introduced by discretizing the continuous failure model of the Lindley distribution. The model obtained is over-dispersed and competitive with the Poisson distribution to fit automobile claim frequency data. After revising some of its properties a compound discrete Lindley distribution is obtained in closed form. This model is suitable to be applied in the collective risk model when both number of claims and size of a single claim are implemented into the model. The new compound distribution fades away to zero much more slowly than the classical compound Poisson distribution, being therefore suitable for modelling extreme data.     

Parameter estimation of the generalized Pareto distribution—Part I

The generalized Pareto distribution (GPD) has been widely used in the extreme value framework. The success of the GPD when applied to real data sets depends substantially on the parameter estimation process. Several methods exist in the literature for estimating the GPD parameters. Mostly, the estimation is performed by maximum likelihood (ML). Alternatively, the probability weighted moments (PWM) and the method of moments (MOM) are often used, especially when the sample sizes are small. Although these three approaches are the most common and quite useful in many situations, their extensive use is also due to the lack of knowledge about other estimation methods. Actually, many other methods, besides the ones mentioned above, exist in the extreme value and hydrological literatures and as such are not widely known to practitioners in other areas. This paper is the first one of two papers that aim to fill in this gap. We shall extensively review some of the methods used for estimating the GPD parameters, focusing on those that can be applied in practical situations in a quite simple and straightforward manner.     

A Latent Process Model for Temporal Extremes

This paper presents a hierarchical approach to modelling extremes of a stationary time series. The procedure comprises two stages. In the first stage, exceedances over a high threshold are modelled through a generalized Pareto distribution, which is represented as a mixture of an exponential variable with a Gamma distributed rate parameter. In the second stage, a latent Gamma process is embedded inside the exponential distribution in order to induce temporal dependence among exceedances. Unlike other hierarchical extreme‐value models, this version has marginal distributions that belong to the generalized Pareto family, so that the classical extreme‐value paradigm is respected. In addition, analytical developments show that different choices of the underlying Gamma process can lead to different degrees of temporal dependence of extremes, including asymptotic independence. The model is tested through a simulation study in a Markov chain setting and used for the analysis of two datasets, one environmental and one financial. In both cases, a good flexibility in capturing different types of tail behaviour is obtained.     

Bayesian mixture modeling for spatial Poisson process intensities,with applications to extreme value analysis

We propose a method for the analysis of a spatial point pattern, which is assumed to arise as a set of observations from a spatial nonhomogeneous Poisson process. The spatial point pattern is observed in a bounded region, which, for most applications, is taken to be a rectangle in the space where the process is defined. The method is based on modeling a density function, defined on this bounded region, that is directly related with the intensity function of the Poisson process. We develop a flexible nonparametric mixture model for this density using a bivariate Beta distribution for the mixture kernel and a Dirichlet process prior for the mixing distribution. Using posterior simulation methods, we obtain full inference for the intensity function and any other functional of the process that might be of interest. We discuss applications to problems where inference for clustering in the spatial point pattern is of interest. Moreover, we consider applications of the methodology to extreme value analysis problems. We illustrate the modeling approach with three previously published data sets. Two of the data sets are from forestry and consist of locations of trees. The third data set consists of extremes from the Dow Jones index over a period of 1303 days.     

Modelling Dependence within Joint Tail Regions

Standard approaches for modelling dependence within joint tail regions are based on extreme value methods which assume max-stability, a particular form of joint tail dependence. We develop joint tail models based on a broader class of dependence structure which provides a natural link between max-stable models and weaker forms of dependence including independence and negative association. This approach overcomes many of the problems that are encountered with standard methods and is the basis for a Poisson process representation that generalizes existing bivariate results. We apply the new techniques to simulated and environmental data, and demonstrate the marked advantage that the new approach offers for joint tail extrapolation.     

The Poisson Generalized Linear Failure Rate Model

We formulate a new cure rate survival model by assuming that the number of competing causes of the event of interest has the Poisson distribution, and the time to this event has the generalized linear failure rate distribution. A new distribution to analyze lifetime data is defined from the proposed cure rate model, and its quantile function as well as a general expansion for the moments is derived. We estimate the parameters of the model with cure rate in the presence of covariates for censored observations using maximum likelihood and derive the observed information matrix. We obtain the appropriate matrices for assessing local influence on the parameter estimates under different perturbation schemes and present some ways to perform global influence analysis. The usefulness of the proposed cure rate survival model is illustrated in an application to real data.     

The odd log-logistic Lindley Poisson model for lifetime data

We define two new lifetime models called the odd log-logistic Lindley (OLL-L) and odd log-logistic Lindley Poisson (OLL-LP) distributions with various hazard rate shapes such as increasing, decreasing, upside-down bathtub, and bathtub. Various structural properties are derived. Certain characterizations of OLL-L distribution are presented. The maximum likelihood estimators of the unknown parameters are obtained. We propose a flexible cure rate survival model by assuming that the number of competing causes of the event of interest has a Poisson distribution and the time to event has an OLL-L distribution. The applicability of the new models is illustrated by means real datasets.     

Marginal Means/Rates Models for Multiple Type Recurrent Event Data

Recurrent events are frequently observed in biomedical studies, and often more than one type of event is of interest. Follow-up time may be censored due to loss to follow-up or administrative censoring. We propose a class of semi-parametric marginal means/rates models, with a general relative risk form, for assessing the effect of covariates on the censored event processes of interest. We formulate estimating equations for the model parameters, and examine asymptotic properties of the parameter estimators. Finite sample properties of the regression coefficients are examined through simulations. The proposed methods are applied to a retrospective cohort study of risk factors for preschool asthma.     

Bayesian inference for Poisson and multinomial log-linear models

Categorical data frequently arise in applications in the Social Sciences. In such applications, the class of log-linear models, based on either a Poisson or (product) multinomial response distribution, is a flexible model class for inference and prediction. In this paper we consider the Bayesian analysis of both Poisson and multinomial log-linear models. It is often convenient to model multinomial or product multinomial data as observations of independent Poisson variables. For multinomial data, Lindley (1964) [20] showed that this approach leads to valid Bayesian posterior inferences when the prior density for the Poisson cell means factorises in a particular way. We develop this result to provide a general framework for the analysis of multinomial or product multinomial data using a Poisson log-linear model. Valid finite population inferences are also available, which can be particularly important in modelling social data. We then focus particular attention on multivariate normal prior distributions for the log-linear model parameters. Here, an improper prior distribution for certain Poisson model parameters is required for valid multinomial analysis, and we derive conditions under which the resulting posterior distribution is proper. We also consider the construction of prior distributions across models, and for model parameters, when uncertainty exists about the appropriate form of the model. We present classes of Poisson and multinomial models, invariant under certain natural groups of permutations of the cells. We demonstrate that, if prior belief concerning the model parameters is also invariant, as is the case in a ‘reference’ analysis, then the choice of prior distribution is considerably restricted. The analysis of multivariate categorical data in the form of a contingency table is considered in detail. We illustrate the methods with two examples.