Gamma lifetimes and associated inference for interval-censored cure rate model with COM–Poisson competing cause

In this article, we consider a competing cause scenario and assume the wider family of Conway–Maxwell–Poisson (COM–Poisson) distribution to model the number of competing causes. Assuming the type of the data to be interval censored, the main contribution is in developing the steps of the expectation maximization (EM) algorithm to determine the maximum likelihood estimates (MLEs) of the model parameters. A profile likelihood approach within the EM framework is proposed to estimate the COM–Poisson shape parameter. An extensive simulation study is conducted to evaluate the performance of the proposed EM algorithm. Model selection within the wider class of COM–Poisson distribution is carried out using likelihood ratio test and information-based criteria. A study to demonstrate the effect of model mis-specification is also carried out. Finally, the proposed estimation method is applied to a data on smoking cessation and a detailed analysis of the obtained results is presented.     

A new compounding life distribution: the Weibull–Poisson distribution

In this paper, a new compounding distribution, named the Weibull–Poisson distribution is introduced. The shape of failure rate function of the new compounding distribution is flexible, it can be decreasing, increasing, upside-down bathtub-shaped or unimodal. A comprehensive mathematical treatment of the proposed distribution and expressions of its density, cumulative distribution function, survival function, failure rate function, the kth raw moment and quantiles are provided. Maximum likelihood method using EM algorithm is developed for parameter estimation. Asymptotic properties of the maximum likelihood estimates are discussed, and intensive simulation studies are conducted for evaluating the performance of parameter estimation. The use of the proposed distribution is illustrated with examples.     

General mixed Poisson regression models with varying dispersion

A general class of mixed Poisson regression models is introduced. This class is based on a mixing between the Poisson distribution and a distribution belonging to the exponential family. With this, we unified some overdispersed models which have been studied separately, such as negative binomial and Poisson inverse gaussian models. We consider a regression structure for both the mean and dispersion parameters of the mixed Poisson models, thus extending, and in some cases correcting, some previous models considered in the literature. An expectation–maximization (EM) algorithm is proposed for estimation of the parameters and some diagnostic measures, based on the EM algorithm, are considered. We also obtain an explicit expression for the observed information matrix. An empirical illustration is presented in order to show the performance of our class of mixed Poisson models. This paper contains a Supplementary Material.     

Estimating the Burr XII Parameters in Constant-Stress Partially Accelerated Life Tests Under Multiple Censored Data

In this article, we present the performance of the maximum likelihood estimates of the Burr XII parameters for constant-stress partially accelerated life tests under multiple censored data. Two maximum likelihood estimation methods are considered. One method is based on observed-data likelihood function and the maximum likelihood estimates are obtained by using the quasi-Newton algorithm. The other method is based on complete-data likelihood function and the maximum likelihood estimates are derived by using the expectation-maximization (EM) algorithm. The variance–covariance matrices are derived to construct the confidence intervals of the parameters. The performance of these two algorithms is compared with each other by a simulation study. The simulation results show that the maximum likelihood estimation via the EM algorithm outperforms the quasi-Newton algorithm in terms of the absolute relative bias, the bias, the root mean square error and the coverage rate. Finally, a numerical example is given to illustrate the performance of the proposed methods.     

Marginal zero-inflated regression models for count data

Data sets with excess zeroes are frequently analyzed in many disciplines. A common framework used to analyze such data is the zero-inflated (ZI) regression model. It mixes a degenerate distribution with point mass at zero with a non-degenerate distribution. The estimates from ZI models quantify the effects of covariates on the means of latent random variables, which are often not the quantities of primary interest. Recently, marginal zero-inflated Poisson (MZIP; Long et al. [A marginalized zero-inflated Poisson regression model with overall exposure effects. Stat. Med. 33 (2014), pp. 5151–5165]) and negative binomial (MZINB; Preisser et al., 2016) models have been introduced that model the mean response directly. These models yield covariate effects that have simple interpretations that are, for many applications, more appealing than those available from ZI regression. This paper outlines a general framework for marginal zero-inflated models where the latent distribution is a member of the exponential dispersion family, focusing on common distributions for count data. In particular, our discussion includes the marginal zero-inflated binomial (MZIB) model, which has not been discussed previously. The details of maximum likelihood estimation via the EM algorithm are presented and the properties of the estimators as well as Wald and likelihood ratio-based inference are examined via simulation. Two examples presented illustrate the advantages of MZIP, MZINB, and MZIB models for practical data analysis.     

A general maximum likelihood analysis of overdispersion in generalized linear models

This paper presents an EM algorithm for maximum likelihood estimation in generalized linear models with overdispersion. The algorithm is initially derived as a form of Gaussian quadrature assuming a normal mixing distribution, but with only slight variation it can be used for a completely unknown mixing distribution, giving a straightforward method for the fully non-parametric ML estimation of this distribution. This is of value because the ML estimates of the GLM parameters may be sensitive to the specification of a parametric form for the mixing distribution. A listing of a GLIM4 algorithm for fitting the overdispersed binomial logit model is given in an appendix.A simple method is given for obtaining correct standard errors for parameter estimates when using the EM algorithm.Several examples are discussed.     

A useful distribution for fitting discrete data: revival of the Conway–Maxwell–Poisson distribution

Summary.  A useful discrete distribution (the Conway–Maxwell–Poisson distribution) is revived and its statistical and probabilistic properties are introduced and explored. This distribution is a two-parameter extension of the Poisson distribution that generalizes some well-known discrete distributions (Poisson, Bernoulli and geometric). It also leads to the generalization of distributions derived from these discrete distributions (i.e. the binomial and negative binomial distributions). We describe three methods for estimating the parameters of the Conway–Maxwell–Poisson distribution. The first is a fast simple weighted least squares method, which leads to estimates that are sufficiently accurate for practical purposes. The second method, using maximum likelihood, can be used to refine the initial estimates. This method requires iterations and is more computationally intensive. The third estimation method is Bayesian. Using the conjugate prior, the posterior density of the parameters of the Conway–Maxwell–Poisson distribution is easily computed. It is a flexible distribution that can account for overdispersion or underdispersion that is commonly encountered in count data. We also explore two sets of real world data demonstrating the flexibility and elegance of the Conway–Maxwell–Poisson distribution in fitting count data which do not seem to follow the Poisson distribution.     

Properties and applications of the Poisson-reciprocal inverse Gaussian distribution

The barely known continuous reciprocal inverse Gaussian distribution is used in this paper to introduce the Poisson-reciprocal inverse Gaussian discrete distribution. Several of its most relevant statistical properties are examined, some of them directly inherited from the reciprocal of the inverse Gaussian distribution. Furthermore, a mixed Poisson regression model that uses the reciprocal inverse Gaussian as mixing distribution is presented. Parameters estimation in this regression model is performed via an EM type algorithm. In light of the numerical results displayed in the paper, the distributions introduced in this work are competitive with the classical negative binomial and Poisson-inverse Gaussian distributions.     

Generalized Linear Factor Models: A New Local EM Estimation Algorithm

In this article, a general approach to latent variable models based on an underlying generalized linear model (GLM) with factor analysis observation process is introduced. We call these models Generalized Linear Factor Models (GLFM). The observations are produced from a general model framework that involves observed and latent variables that are assumed to be distributed in the exponential family. More specifically, we concentrate on situations where the observed variables are both discretely measured (e.g., binomial, Poisson) and continuously distributed (e.g., gamma). The common latent factors are assumed to be independent with a standard multivariate normal distribution. Practical details of training such models with a new local expectation-maximization (EM) algorithm, which can be considered as a generalized EM-type algorithm, are also discussed. In conjunction with an approximated version of the Fisher score algorithm (FSA), we show how to calculate maximum likelihood estimates of the model parameters, and to yield inferences about the unobservable path of the common factors. The methodology is illustrated by an extensive Monte Carlo simulation study and the results show promising performance.     

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.     

A robust multivariate Birnbaum–Saunders distribution: EM estimation

We propose here a robust multivariate extension of the bivariate Birnbaum–Saunders (BS) distribution derived by Kundu et al. [Bivariate Birnbaum–Saunders distribution and associated inference. J Multivariate Anal. 2010;101:113–125], based on scale mixtures of normal (SMN) distributions that are used for modelling symmetric data. This resulting multivariate BS-type distribution is an absolutely continuous distribution whose marginal and conditional distributions are of BS-type distribution of Balakrishnan et al. [Estimation in the Birnbaum–Saunders distribution based on scalemixture of normals and the EM algorithm. Stat Oper Res Trans. 2009;33:171–192]. Due to the complexity of the likelihood function, parameter estimation by direct maximization is very difficult to achieve. For this reason, we exploit the nice hierarchical representation of the proposed distribution to propose a fast and accurate EM algorithm for computing the maximum likelihood (ML) estimates of the model parameters. We then evaluate the finite-sample performance of the developed EM algorithm and the asymptotic properties of the ML estimates through empirical experiments. Finally, we illustrate the obtained results with a real data and display the robustness feature of the estimation procedure developed here.     

Estimation and Model Selection for Left-truncated and Right-censored Lifetime Data with Application to Electric Power Transformers Analysis

In lifetime analysis of electric transformers, the maximum likelihood estimation has been proposed with the EM algorithm. However, it is not clear whether the EM algorithm offers a better solution compared to the simpler Newton-Raphson (NR) algorithm. In this article, the first objective is a systematic comparison of the EM algorithm with the NR algorithm in terms of convergence performance. The second objective is to examine the performance of Akaike's information criterion (AIC) for selecting a suitable distribution among candidate models via simulations. These methods are illustrated through the electric power transformer dataset.     

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.     

Inference for Log-Gamma Distribution Based on Progressively Type-II Censored Data

We discuss the maximum likelihood estimates (MLEs) of the parameters of the log-gamma distribution based on progressively Type-II censored samples. We use the profile likelihood approach to tackle the problem of the estimation of the shape parameter κ. We derive approximate maximum likelihood estimators of the parameters μ and σ and use them as initial values in the determination of the MLEs through the Newton–Raphson method. Next, we discuss the EM algorithm and propose a modified EM algorithm for the determination of the MLEs. A simulation study is conducted to evaluate the bias and mean square error of these estimators and examine their behavior as the progressive censoring scheme and the shape parameter vary. We also discuss the interval estimation of the parameters μ and σ and show that the intervals based on the asymptotic normality of MLEs have very poor probability coverages for small values of m. Finally, we present two examples to illustrate all the methods of inference discussed in this paper.     

Estimation for the scaled half-logistic distribution under Type-I progressively hybrid censoring scheme

This paper addresses the estimation for the unknown scale parameter of the half-logistic distribution based on a Type-I progressively hybrid censoring scheme. We evaluate the maximum likelihood estimate (MLE) via numerical method, and EM algorithm, and also the approximate maximum likelihood estimate (AMLE). We use a modified acceptance rejection method to obtain the Bayes estimate and corresponding highest posterior confidence intervals. We perform Monte Carlo simulations to compare the performances of the different methods, and we analyze one dataset for illustrative purposes.     

Mixture models for capture-recapture count data

The contribution investigates the problem of estimating the size of a population, also known as the missing cases problem. Suppose a registration system is targeting to identify all cases having a certain characteristic such as a specific disease (cancer, heart disease, ...), disease related condition (HIV, heroin use, ...) or a specific behavior (driving a car without license). Every case in such a registration system has a certain notification history in that it might have been identified several times (at least once) which can be understood as a particular capture-recapture situation. Typically, cases are left out which have never been listed at any occasion, and it is this frequency one wants to estimate. In this paper modelling is concentrating on the counting distribution, e.g. the distribution of the variable that counts how often a given case has been identified by the registration system. Besides very simple models like the binomial or Poisson distribution, finite (nonparametric) mixtures of these are considered providing rather flexible modelling tools. Estimation is done using maximum likelihood by means of the EM algorithm. A case study on heroin users in Bangkok in the year 2001 is completing the contribution.     

Nonparametric maximum likelihood estimation of population size based on the counting distribution

Summary.  The paper discusses the estimation of an unknown population size n . Suppose that an identification mechanism can identify n _obs cases. The Horvitz–Thompson estimator of n adjusts this number by the inverse of 1− p ₀, where the latter is the probability of not identifying a case. When repeated counts of identifying the same case are available, we can use the counting distribution for estimating p ₀ to solve the problem. Frequently, the Poisson distribution is used and, more recently, mixtures of Poisson distributions. Maximum likelihood estimation is discussed by means of the EM algorithm. For truncated Poisson mixtures, a nested EM algorithm is suggested and illustrated for several application cases. The algorithmic principles are used to show an inequality, stating that the Horvitz–Thompson estimator of n by using the mixed Poisson model is always at least as large as the estimator by using a homogeneous Poisson model. In turn, if the homogeneous Poisson model is misspecified it will, potentially strongly, underestimate the true population size. Examples from various areas illustrate this finding.     

A mixture latent variable model for modeling mixed data in heterogeneous populations and its applications

Latent variable models are widely used for jointly modeling of mixed data including nominal, ordinal, count and continuous data. In this paper, we consider a latent variable model for jointly modeling relationships between mixed binary, count and continuous variables with some observed covariates. We assume that, given a latent variable, mixed variables of interest are independent and count and continuous variables have Poisson distribution and normal distribution, respectively. As such data may be extracted from different subpopulations, consideration of an unobserved heterogeneity has to be taken into account. A mixture distribution is considered (for the distribution of the latent variable) which accounts the heterogeneity. The generalized EM algorithm which uses the Newton–Raphson algorithm inside the EM algorithm is used to compute the maximum likelihood estimates of parameters. The standard errors of the maximum likelihood estimates are computed by using the supplemented EM algorithm. Analysis of the primary biliary cirrhosis data is presented as an application of the proposed model.     

Another generalization of the logarithmic series and the geometric distributions

A new generalization of the logarithmic series distribution is presented based on a generalized negative binomial distribution obtained from a generalized Poisson distribution compounded with the truncated gamma distribution. By length biasing this generalized log-series distribution, another generalized geometric distribution is uresented. For the generalized log-series distribution, maximum likelihood estimators are developed and an example is presented for illustration.