Estimating the Pólya process

Traditionally, a Pólya process is approached from a probability point of view. No prior inference work has been done on them. In this study, we approach the continuous-time Pólya process from an estimation point of view. We construct efficient estimators for the replacement matrix of certain classes of Pólya processes.     

Approximate Confidence Limits for a Proportion of the Pólya Distribution

The problem of constructing approximate confidence limits for a proportion parameter of the Pólya distribution is discussed. Three different methods for determining approximate one-sided and two-sided confidence limits for that parameter of the Pólya distribution have been proposed and compared. Particular cases of those confidence bounds are confidence intervals for the parameter of the binomial and the hypergeometric distributions.     

Description of a subfamily of the discrete pearson system as generalized-binomial distributions

In this paper we prove that a subfamily of distributions of the discrete Pearson, system, containing the Pólya distribution without replacement and hence the hypergeometric distribution, can be described as generalized-binomial distributions, i.e., the distribution of the number of successes which occur in independent trials. It is also shown that the probability of success will necessarily be different in each trial, with the exception of deterministic ones. As a consequence, all the properties of the generalized-binomial distribution will apply to this subfamily. Thus, applications to hypothesis testing and confidence intervals in the Pólya distribution are considered.     

Pólya–Aeppli of Order k Risk Model

In this study, we define the Pólya–Aeppli process of order k as a compound Poisson process with truncated geometric compounding distribution with success probability 1 ? ρ > 0 and investigate some of its basic properties. Using simulation, we provide a comparison between the sample paths of the Pólya–Aeppli process of order k and the Poisson process. Also, we consider a risk model in which the claim counting process {N(t)} is a Pólya-Aeppli process of order k, and call it a Pólya—Aeppli of order k risk model. For the Pólya–Aeppli of order k risk model, we derive the ruin probability and the distribution of the deficit at the time of ruin. We discuss in detail the particular case of exponentially distributed claims and provide simulation results for more general cases.     

Likelihood inference for Type I bivariate Pólya–Aeppli distribution

This article discusses likelihood inference for the Type I bivariate Pólya–Aeppli distribution. The Type I bivariate Pólya–Aeppli distribution was derived by Minkova and Balakrishnan by using compounding with geometric random variables and the trivariate reduction method. They also discussed the moment estimation of the parameters of the Type I bivariate Pólya–Aeppli distribution. Here, we carry out a simulation study to compare the performance of the developed Maximum Likelihood Estimation (MLE) method with the moment estimation. The obtained results show that, through the MLEs require more computational time compared to the moment estimates (MoM), the MLEs perform better, in most of the settings, than the MoM. Finally, we apply the Type I bivariate Pólya–Aeppli model to a real dataset containing the frequencies of railway accidents in two subsequent six-year periods for the purpose of illustration. We also carry out some hypothesis tests using the Wald test statistic. From these results, we conclude that the two variables belong to the same univariate Pólya–Aeppli distribution, but are correlated.     

The probability function of a geometric Poisson distribution

The geometric Poisson (also called Pólya–Aeppli) distribution is a particular case of the compound Poisson distribution. In this study, the explicit probability function of the geometric Poisson distribution is derived and a straightforward proof for this function is given. By means of a proposed algorithm, some numerical examples and an application on traffic accidents are also given to illustrate the usage of the probability function and proposed algorithm.     

Approximate Confidence Limits for the Difference between Parameters of Two Pólya Distributions

The problem of calculating approximate confidence limits for the difference between success probability parameters of two Pólya distributions is solved for the first time. We suggest some new methods for determining these approximate confidence limits and consider their application to special cases: namely for the binomial and hypergeometric distributions. The various approximate confidence limits are evaluated and compared.     

Unification of Dirichlet Methodology

In this article we provide a unified framework for solving Dirichlet related probability and waiting time problems. We consider a Pólya sampling scheme in which each time an object is selected, it is put back into the population along with c additional objects of the same type. By considering both fixed sample size and inverse sampling procedures, we unify the Dirichlet I, J, C, and D functions with their hypergeometric counterparts by extending these functions to Pólya sampling. We then use these functions to unify and extend the corresponding expected waiting time results.     

Inference for Type II bivariate Pólya–Aeppli distribution

In this article, we discuss the estimation of model parameters of the Type II bivariate Pólya–Aeppli distribution using the method of moments and the maximum likelihood method. We also compare some interval estimation methods. We then carry out a Monte Carlo simulation study to evaluate the performance of the proposed point and interval estimation methods. Finally, we present an example to illustrate all the inferential methods developed here.     

Characterizing variation of nonparametric random probability measures using the Kullback–Leibler divergence

This work characterizes the dispersion of some popular random probability measures, including the bootstrap, the Bayesian bootstrap, and the Pólya tree prior. This dispersion is measured in terms of the variation of the Kullback–Leibler divergence of a random draw from the process to that of its baseline centring measure. By providing a quantitative expression of this dispersion around the baseline distribution, our work provides insight for comparing different parameterizations of the models and for the setting of prior parameters in applied Bayesian settings. This highlights some limitations of the existing canonical choice of parameter settings in the Pólya tree process.     

The Generalized Waring Process and Its Application

The generalized Waring distribution is a discrete distribution with a wide spectrum of applications in areas such as accident statistics, income analysis, environmental statistics, etc. It has been used as a model that better describes such practical situations as opposed to the Poisson distribution or the negative binomial distribution. Associated to both the Poisson and negative binomial distributions are the well-known Poisson and Pólya processes. In this article, the generalized Waring process is defined. Two models have been shown to lead to the generalized Waring process. One is related to a Cox process, while the other is a compound Poisson process. The defined generalized Waring process is shown to be a stationary, but non homogenous Markov process. Several properties are studied and the intensity, individual intensity, and Chapman–Kolmogorov differential equations of it are obtained. Moreover, the Poisson and Pólya processes are shown to arise as special cases of the generalized Waring process. Using this fact, some known results and some properties of them are obtained.     

A nonparametric Bayesian approach to copula estimation

We propose a novel Dirichlet-based Pólya tree (D-P tree) prior on the copula and based on the D-P tree prior, a nonparametric Bayesian inference procedure. Through theoretical analysis and simulations, we are able to show that the flexibility of the D-P tree prior ensures its consistency in copula estimation, thus able to detect more subtle and complex copula structures than earlier nonparametric Bayesian models, such as a Gaussian copula mixture. Furthermore, the continuity of the imposed D-P tree prior leads to a more favourable smoothing effect in copula estimation over classic frequentist methods, especially with small sets of observations. We also apply our method to the copula prediction between the S&P 500 index and the IBM stock prices during the 2007–08 financial crisis, finding that D-P tree-based methods enjoy strong robustness and flexibility over classic methods under such irregular market behaviours.     

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.     

Non-homogeneous Pólya-Aeppli process

Abstract

In this paper we suppose that the intensity parameter of the Pólya-Aeppli process is a function of time t and call the resulting process a non-homogeneous Pólya-Aeppli process (NHPAP). The NHPAP can be represented as a compound non-homogeneous Poisson process with geometric compounding distribution as well as a pure birth process. For this process we give two definitions and show their equivalence. Also, we derive some interesting properties of NHPAP and use simulation the illustrate the process for particular intensity functions. In addition, we introduce the standard risk model based on NHPAP, analyze the ruin probability for this model and include an example of the process under exponentially distributed claims.     

A non uniform bound on geometric approximation with w-functions

The aim of this paper is a use of Stein’s method and w-functions to determine a non uniform bound on the geometric approximation for a non negative integer-valued random variable. Some applications of the obtained results are provided to approximate the negative hypergeometric, Pólya and negative Pólya distributions.     

Bayesian inference for zero-and-one-inflated geometric distribution regression model using Pólya-Gamma latent variables

Abstract

In the fields of internet financial transactions and reliability engineering, there could be more zero and one observations simultaneously. In this paper, considering that it is beyond the range where the conventional model can fit, zero-and-one-inflated geometric distribution regression model is proposed. Ingeniously introducing Pólya-Gamma latent variables in the Bayesian inference, posterior sampling with high-dimensional parameters is converted to latent variables sampling and posterior sampling with lower-dimensional parameters, respectively. Circumventing the need for Metropolis-Hastings sampling, the sample with higher sampling efficiency is obtained. A simulation study is conducted to assess the performance of the proposed estimation for various sample sizes. Finally, a doctoral dissertation data set is analyzed to illustrate the practicability of the proposed method, research shows that zero-and-one-inflated geometric distribution regression model using Pólya-Gamma latent variables can achieve better fitting results.     

A Bayesian nonparametric estimator of a multivariate survival function

Using reinforced processes related to beta-Stacy process and generalized Pólya urn scheme jointly with a structure assumption about dependence, a Bayesian nonparametric prior and a predictive estimator for a multivariate survival function are provided. This estimator can be computed through an easy implementation of a Gibbs sampler algorithm. Moreover consistency of the estimator is studied.     

Asymptotic Analysis of Some Discrete Distributions by the Saddle Point Method

We present asymptotic formulas for the probability mass functions of three discrete distributions: the Neyman type A, the compound Poisson–Katz, and the convolution of negative binomial and Pólya–Aeppli. An approximation of the moments of the Neyman type A distribution is also given. All of these results are found by Hayman's encapsulation of the saddle point method.