负二项分布的两种近似分布及其比较

负二项分布是一个重要的离散型随机变量的分布,可以用泊松分布和正态分布作为其近似分布,通过对两种近似分布进行比较分析,结果表明：在参数q很小时,泊松近似的精度好于正态近似,而且在参数q很小时,即便r不是很大,用泊松分布也能获得负二项分布较好的近似;当参数q较大时,泊松近似效果不好,相比之下,正态近似的结果不错。     

Construction of a Markov Chain with Given Stationary Distribution

Given a rational, finite probability vector, a Markov chain is constructed having the given vector as its stationary distribution.     

An Extended Binomial Distribution with Applications

Based on Skellam (Poisson difference) distribution, an extended binomial distribution is introduced as a byproduct of extending Moran's characterization of Poisson distribution to the Skellam distribution. Basic properties of the distribution are investigated. Also, estimation of the distribution parameters is obtained. Applications with real data are also described.     

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.     

A Bayesian Inference and Stochastic Dynamic Programming Approach to Determine the Best Binomial Distribution

In this research, we employ Bayesian inference and stochastic dynamic programming approaches to select the binomial population with the largest probability of success from n independent Bernoulli populations based upon the sample information. To do this, we first define a probability measure called belief for the event of selecting the best population. Second, we explain the way to model the selection problem using Bayesian inference. Third, we clarify the model by which we improve the beliefs and prove that it converges to select the best population. In this iterative approach, we update the beliefs by taking new observations on the populations under study. This is performed using Bayesian rule and prior beliefs. Fourth, we model the problem of making the decision in a predetermined number of decision stages using the stochastic dynamic programming approach. Finally, in order to understand and to evaluate the proposed methodology, we provide two numerical examples and a comparison study by simulation. The results of the comparison study show that the proposed method performs better than that of Levin and Robbins (1981 Levin , B. , Robbins , H. ( 1981 ). Selecting the highest probability in Binomial or multinomial trials . Proc. Nat. Acad. Sci. USA 78 : 4663 – 4666 .[Crossref], [PubMed], [Web of Science ®] , [Google Scholar]) for some values of estimated probability of making a correct selection.     

A New Model for Multivariate Markov Chains

We propose a new model for multivariate Markov chains of order one or higher on the basis of the mixture transition distribution (MTD) model. We call it the MTD‐Probit. The proposed model presents two attractive features: it is completely free of constraints, thereby facilitating the estimation procedure, and it is more precise at estimating the transition probabilities of a multivariate or higher‐order Markov chain than the standard MTD model.     

Regenerative Markov Chain Monte Carlo for Any Distribution

While Markov chain Monte Carlo (MCMC) methods are frequently used for difficult calculations in a wide range of scientific disciplines, they suffer from a serious limitation: their samples are not independent and identically distributed. Consequently, estimates of expectations are biased if the initial value of the chain is not drawn from the target distribution. Regenerative simulation provides an elegant solution to this problem. In this article, we propose a simple regenerative MCMC algorithm to generate variates for any distribution.     

A Bivariate Generalization of the Noncentral Negative Binomial Distribution

This article proposes a bivariate generalization of the noncentral negative binomial distribution which arises as a model in photon and neural counting. This bivariate generalization is derived as a mixed shifted bivariate negative binomial distribution. Various properties and parameter estimation, especially by a minimum distance method based on the probability generating function, are considered. To show the practical usefulness of the bivariate distribution proposed, an application to model low-flux astronomical images is discussed and a real data set has been analyzed.     

Comparison of Shared Frailty Models for Kidney Infection Data under Exponential Power Baseline Distribution

Shared frailty models are often used to model heterogeneity in survival analysis. There are certain assumptions about the baseline distribution and distribution of frailty. In this paper, four shared frailty models with frailty distribution gamma, inverse Gaussian, compound Poisson, and compound negative binomial with exponential power as baseline distribution are proposed. These models are fitted using Markov Chain Monte Carlo methods. These models are illustrated with a real life bivariate survival data set of McGilchrist and Aisbett (1991) related to kidney infection, and the best model is suggested for the data using different model comparison criteria.     

Efficient Estimation of Parameters of the Negative Binomial Distribution

In this article we investigate a class of moment-based estimators, called power method estimators, which can be almost as efficient as maximum likelihood estimators and achieve a lower asymptotic variance than the standard zero term method and method of moments estimators. We investigate different methods of implementing the power method in practice and examine the robustness and efficiency of the power method estimators.     

On the Markov property of order statistics

The order statistics from a sample of size n≥3 from a discrete distribution form a Markov chain if and only if the parent distribution is supported by one or two points. More generally, a necessary and sufficient condition for the order statistics to form a Markov chain for (n≥3) is that there does not exist any atom x₀ of the parent distribution F satisfying F(x₀-)>0 and F(x₀)<1. To derive this result a formula for the joint distribution of order statistics is proved, which is of an interest on its own. Many exponential characterizations implicitly assume the Markov property. The corresponding putative geometric characterizations cannot then be reasonably expected to obtain. Some illustrative geometric characterizations are discussed.     

Mobile safety cameras: estimating casualty reductions and the demand for secondary healthcare

We consider a fully Bayesian analysis of road casualty data at 56 designated mobile safety camera sites in the Northumbria Police Force area in the UK. It is well documented that regression to the mean (RTM) can exaggerate the effectiveness of road safety measures and, since the 1980s, an empirical Bayes (EB) estimation framework has become the gold standard for separating real treatment effects from those of RTM. In this paper we suggest some diagnostics to check the assumptions underpinning the standard estimation framework. We also show that, relative to a fully Bayesian treatment, the EB method is over-optimistic when quantifying the variability of estimates of casualty frequency. Implementing a fully Bayesian analysis via Markov chain Monte Carlo also provides a more flexible and complete inferential procedure. We assess the sensitivity of estimates of treatment effectiveness, as well as the expected monetary value of prevention owing to the implementation of the safety cameras, to different model specifications, which include the estimation of trend and the construction of informative priors for some parameters.     

On Generalized Binomial and Negative Binomial Distributions for Dependent Bernoulli Variables

We study the distributions of the random variables S_n and V_r related to a sequence of dependent Bernoulli variables, where S_n denotes the number of successes in n trials and V_r the number of trials necessary to obtain r successes. The purpose of this article is twofold: (1) Generalizing some results on the “nature” of the binomial and negative binomial distributions we show that S_n and V_r can follow any prescribed discrete distribution. The corresponding joint distributions of the Bernoulli variables are characterized as the solutions of systems of linear equations. (2) We consider a specific type of dependence of the Bernoulli variables, where the probability of a success depends only on the number of previous successes. We develop some theory based on new closed-form representations for the probability mass functions of S_n and V_r which enable direct computations of the probabilities.     

Estimating Abundance from Presence–Absence Maps via a Paired Negative‐Binomial Model

The estimation of abundance from presence–absence data is an intriguing problem in applied statistics. The classical Poisson model makes strong independence and homogeneity assumptions and in practice generally underestimates the true abundance. A controversial ad hoc method based on negative‐binomial counts (Am. Nat.) has been empirically successful but lacks theoretical justification. We first present an alternative estimator of abundance based on a paired negative binomial model that is consistent and asymptotically normally distributed. A quadruple negative binomial extension is also developed, which yields the previous ad hoc approach and resolves the controversy in the literature. We examine the performance of the estimators in a simulation study and estimate the abundance of 44 tree species in a permanent forest plot.     

A distance-based diagnostic for trans-dimensional Markov chains

Over the last decade the use of trans-dimensional sampling algorithms has become endemic in the statistical literature. In spite of their application however, there are few reliable methods to assess whether the underlying Markov chains have reached their stationary distribution. In this article we present a distance-based method for the comparison of trans-dimensional Markov chain sample output for a broad class of models. This diagnostic will simultaneously assess deviations between and within chains. Illustration of the analysis of Markov chain sample-paths is presented in simulated examples and in two common modelling situations: a finite mixture analysis and a change-point problem.     

A continuous-time Markov model for estimating readmission risk for hospital inpatients

Research concerning hospital readmissions has mostly focused on statistical and machine learning models that attempt to predict this unfortunate outcome for individual patients. These models are useful in certain settings, but their performance in many cases is insufficient for implementation in practice, and the dynamics of how readmission risk changes over time is often ignored. Our objective is to develop a model for aggregated readmission risk over time – using a continuous-time Markov chain – beginning at the point of discharge. We derive point and interval estimators for readmission risk, and find the asymptotic distributions for these probabilities. Finally, we validate our derived estimators using simulation, and apply our methods to estimate readmission risk over time using discharge and readmission data for surgical patients.     

Estimation of group means using Bayesian generalized linear mixed models

Generalized linear mixed models (GLMM) are commonly used to model the treatment effect over time while controlling for important clinical covariates. Standard software procedures often provide estimates of the outcome based on the mean of the covariates; however, these estimates will be biased for the true group means in the GLMM. Implementing GLMM in the frequentist framework can lead to issues of convergence. A simulation study demonstrating the use of fully Bayesian GLMM for providing unbiased estimates of group means is shown. These models are very straightforward to implement and can be used for a broad variety of outcomes (eg, binary, categorical, and count data) that arise in clinical trials. We demonstrate the proposed method on a data set from a clinical trial in diabetes.     

A Lagrangian Non Central Negative Binomial Distribution of the First Kind

A Lagrangian probability distribution of the first kind is proposed. Its probability mass function is expressed in terms of generalized Laguerre polynomials or, equivalently, a generalized hypergeometric function. The distribution may also be formulated as a Charlier series distribution generalized by the generalizing Consul distribution and a non central negative binomial distribution generalized by the generalizing Geeta distribution. This article studies formulation and properties of the distribution such as mixture, dispersion, recursive formulas, conditional distribution and the relationship with queuing theory. Two illustrative examples of application to fitting are given.     

A Novel Estimation Approach for Mixture Transition Distribution Model in High-Order Markov Chains

A transformation is proposed to convert the nonlinear constraints of the parameters in the mixture transition distribution (MTD) model into box-constraints. The proposed transformation removes the difficulties associated with the maximum likelihood estimation (MLE) process in the MTD modeling so that the MLEs of the parameters can be easily obtained via a hybrid algorithm from the evolutionary algorithms and/or quasi-Newton algorithms for global optimization. Simulation studies are conducted to demonstrate MTD modeling by the proposed novel approach through a global search algorithm in R environment. Finally, the proposed approach is used for the MTD modelings of three real data sets.     

Interval Estimations for the Mean of the Generalized Poisson and Generalized Negative Binomial Distributions

In this article, we extend the Wald, score, skewness-corrected score, likelihood ratio, and mid-P intervals for the means of the generalized Poisson and generalized negative binomial distributions. These distributions are the members of the discrete version of the natural exponential family (NEF) with cubic variance function (CVF). Also, the coverage probabilities, the distal and mesial noncoverage probabilities, and the lengths of the proposed confidence intervals are estimated by means of a Monte Carlo simulation study. Finally, some practical examples are provided to show the applicability of the proposed intervals in applied studies.