COM-negative binomial distribution: modeling overdispersion and ultrahigh zero-inflated count data

We focus on the COM-type negative binomial distribution with three parameters, which belongs to COM-type (a, b, 0) class distributions and family of equilibrium distributions of arbitrary birth-death process. Besides, we show abundant distributional properties such as overdispersion and underdispersion, log-concavity, log-convexity (infinite divisibility), pseudo compound Poisson, stochastic ordering, and asymptotic approximation. Some characterizations including sum of equicorrelated geometrically distributed random variables, conditional distribution, limit distribution of COM-negative hypergeometric distribution, and Stein’s identity are given for theoretical properties. COM-negative binomial distribution was applied to overdispersion and ultrahigh zero-inflated data sets. With the aid of ratio regression, we employ maximum likelihood method to estimate the parameters and the goodness-of-fit are evaluated by the discrete Kolmogorov-Smirnov test.     

Validation tests for the innovation distribution in INAR time series models

Goodness-of-fit tests are proposed for the innovation distribution in INAR models. The test statistics incorporate the joint probability generating function of the observations. Special emphasis is given to the INAR(1) model and particular instances of the procedures which involve innovations from the general family of Poisson stopped-sum distributions. A Monte Carlo power study of a bootstrap version of the test statistic is included as well as a real data example. Generalizations of the proposed methods are also discussed.     

Maximum Likelihood Estimation for an Observation Driven Model for Poisson Counts

This paper is concerned with an observation-driven model for time series of counts whose conditional distribution given past observations follows a Poisson distribution.This class of models is capable of modeling a wide range of dependence structures and is readily estimated using an approximation to the likelihood function. Recursive formulae for carrying out maximum likelihood estimation are provided and the technical components required for establishing a central limit theorem of the maximum likelihood estimates are given in a special case.AMS 2000 Subject Classification: Primary 62M05; Secondary 62E20     

Automatic Smoothing With Wavelets for a Wide Class of Distributions

Wavelet-based denoising techniques are well suited to estimate spatially inhomogeneous signals. Waveshrink (Donoho and Johnstone) assumes independent Gaussian errors and equispaced sampling of the signal. Various articles have relaxed some of these assumptions, but a systematic generalization to distributions such as Poisson, binomial, or Bernoulli is missing. We consider a unifying l₁-penalized likelihood approach to regularize the maximum likelihood estimation by adding an l₁ penalty of the wavelet coefficients. Our approach works for all types of wavelets and for a range of noise distributions. We develop both an algorithm to solve the estimation problem and rules to select the smoothing parameter automatically. In particular, using results from Poisson processes, we give an explicit formula for the universal smoothing parameter to denoise Poisson measurements. Simulations show that the procedure is an improvement over other methods. An astronomy example is given.     

Controlling jumps in correlated processes of Poisson counts

Processes of autocorrelated Poisson counts can often be modelled by a Poisson INAR(1) model, which proved to apply well to typical tasks of SPC. Statistical properties of this model are briefly reviewed. Based on these properties, we propose a new control chart: the combined jumps chart. It monitors the counts and jumps of a Poisson INAR(1) process simultaneously. As the bivariate process of counts and jumps is a homogeneous Markov chain, average run lengths (ARLs) can be computed exactly with the well‐known Markov chain approach. Based on an investigation of such ARLs, we derive design recommendations and show that a properly designed chart can be applied nearly universally. This is also demonstrated by a real‐data example from the insurance field. Copyright © 2008 John Wiley & Sons, Ltd.     

A conditional least squares approach to PGARCH and PARMA–PGARCH time series estimation

In this Note, a conditional least squares (CLS) estimates for periodic GARCH (PGARCH) models with martingale difference centered squared innovations is developed. The approach is extended to the PARMA–PGARCH models. We establish the strong consistency and the asymptotic normality for our estimate.     

Reliability analysis of repairable systems from in–service failure count data

Point maximum likelihood estimators for parameters, mean number of failures, and failure rate in a non–homogeneous Poisson process are derived, when only count data from k identical processes are available. Approximate confidence intervals based on the parametric bootstrap technique are considered. The performances of both the point and interval estimation procedures are assessed via Monte Carlo simulation.     

Integer Valued Autoegressive Process with Katz Arrivals and Its Application in Predicting the Count of the Cases of Respiratory Disease

The traditional PAR process (Poisson autoregressive process) assumes that the arrival process is the equi-dispersed Poisson process, with its mean being equal to its variance. Whereas the arrival process in the real DGP (data generating process) could either be over-dispersed, with variance being greater than the mean, or under-dispersed, with variance being less than the mean. This paper proposes using the Katz family distributions to model the arrival process in the INAR process (integer valued autoregressive process with Katz arrivals) and deploying Monte Carlo simulations to examine the performance of maximum likelihood (ML) and method of moments (MM) estimators of INAR-Katz model. Finally, we used the INAR-Katz process to model count data of hospital emergency room visits for respiratory disease. The results show that the INAR-Katz model outperforms the Poisson model, PAR(1) model, and has great potential in empirical application.     

Improving on the maximum likelihood estimators of the means in Poisson decomposable graphical models

In this article we study the simultaneous estimation of the means in Poisson decomposable graphical models. We derive some classes of estimators which improve on the maximum likelihood estimator under the normalized squared losses. Our estimators are based on the argument in Chou [Simultaneous estimation in discrete multivariate exponential families, Ann. Statist. 19 (1991) 314-328.] and shrink the maximum likelihood estimator depending on the marginal frequencies of variables forming a complete subgraph of the conditional independence graph.     

Conditioning exceedances on covariate processes

This article concerns the statistical inference for the upper tail of the conditional distribution of a response variable Y given a covariate X = x based on n random vectors within the parametric extreme value framework. Pioneering work in this field was done by Smith (Stat Sci 4:367–393, 1989) and Smith and Shively (Atmos Environ 29:3489–3499, 1995). We propose to base the inference on a conditional distribution of the point process of exceedances given the point process of covariates. It is of importance that the conditional distribution merely depends on the conditional distribution of the response variable given the covariates. In the special case of Poisson processes such a result may be found in Reiss (1993). Our results are valid within the broader model where the response variables are conditionally independent given the covariates. It is numerically exemplified that the maximum likelihood principle leads to more accurate estimators within the conditional approach than in the previous one.     

Gompertz zero-inflated cure rate regression models applied to credit risk data

In the financial market, it is important to consider that there is a proportion of customers that have settled their debt in time zero, immediately recovering their ability to pay. In this context, in this paper, we propose a survival analysis methodology that allows the insertion of times equal to zero in scenarios where credit risk is observed. The proposed model addresses the survival analysis model of the zero-inflated cure rate which incorporates the heterogeneity of three subgroups (individuals having events in the initial time, and individuals not susceptible and susceptible to the event). In our proposal, all available survival data of customers are modeled considering that the number of competitive causes follows a Poisson distribution and the baseline risk function follows a Gompertz distribution. The model parameter estimation is obtained by the maximum likelihood estimation procedure and simulation studies are conducted to evaluate the estimators' performance. The studied methodology will be applied to a credit database provided by a financial institution in Brazil.     

Optimal designs for parameter estimation of the Ornstein–Uhlenbeck process

This paper deals with optimal designs for Gaussian random fields with constant trend and exponential correlation structure, widely known as the Ornstein–Uhlenbeck process. Assuming the maximum likelihood approach, we study the optimal design problem for the estimation of the trend µ and the correlation parameter θ using a criterion based on the Fisher information matrix. For the problem of trend estimation, we give a new proof of the optimality of the equispaced design for any sample size (see Statist. Probab. Lett. 2008; 78 :1388–1396). We also show that for the estimation of the correlation parameter, an optimal design does not exist. Furthermore, we show that the optimal strategy for µ conflicts with the one for θ, since the equispaced design is the worst solution for estimating the correlation. Hence, when the inferential purpose concerns both the unknown parameters we propose the geometric progression design, namely a flexible class of procedures that allow the experimenter to choose a suitable compromise regarding the estimation's precision of the two unknown parameters guaranteeing, at the same time, high efficiency for both. Copyright © 2008 John Wiley & Sons, Ltd.     

Parametric estimation of a bivariate stable Lévy process

We propose a parametric model for a bivariate stable Lévy process based on a Lévy copula as a dependence model. We estimate the parameters of the full bivariate model by maximum likelihood estimation. As an observation scheme we assume that we observe all jumps larger than some ε>0 and base our statistical analysis on the resulting compound Poisson process. We derive the Fisher information matrix and prove asymptotic normality of all estimates when the truncation point ε→0. A simulation study investigates the loss of efficiency because of the truncation.     

含右删失数据的ZIP模型及其应用

零膨胀Poisson回归(ZIP)是处理零频数过多计数资料的有效模型,而计数数据一般含有删失或不精密的特点.本文将删失数据引入到ZIP模型中来,分别建立含右删失数据的固定效应ZIP模型,随机效应ZIP模型,通过极大边际似然函数估计法对模型进行参数估计.最后,利用实例分析验证了上述模型的可行性.     

Modelling heavy-tailedness in count time series

Count data frequently exhibit overdispersion, zero inflation and even heavy-tailedness (the tail probabilities are non-negligible or decrease very slowly) in practical applications. Many models have been proposed for modelling count data with overdispersion and zero inflation, but heavy-tailedness is less considered. The proposed model, a new integer-valued autoregressive process with generalized Poisson-inverse Gaussian innovations, is capable of capturing these features. The generalized Poisson-inverse Gaussian family is very flexible, which includes Poisson distribution, Poisson inverse Gaussian distribution, discrete stable distribution and so on. Stationarity and ergodicity of this model are investigated and the expressions of marginal mean and variance are provided. Conditional maximum likelihood is used for estimating the parameters, and consistency and asymptotic normality for the estimators are presented. Further, we consider the h-step forecast and diagnostics for the proposed model. The proposed model is applied to three real data examples. In the first example, we consider the monthly number of cases of Polio, which validates that the proposed model can take into account count data with excessive zeros. Then, we illustrate the use of the proposed model through an application to the numbers of National Science Foundation fundings. Finally, we apply the proposed model to the numbers of transactions in 5-min intervals for the stock traded at Empire District Electric Company. The second and third examples show that the proposed model has a good performance in modelling heavy-tailed count data.     

Vitesse de convergence presque sûre de l'estimateur à noyau du maximum de vraisemblance local

We consider the local maximum likelihood estimation of $\theta (x)$, unknown parameter of the conditional distribution of Y given $X=x$. The aim of this Note is the study of strong uniform consistency rates of the local maximum likelihood kernel estimator. Under suitable regularity conditions, we establish a uniform law of the logarithm for the maximal deviation of this estimator. The method of proof is based upon functional limit laws derived by modern empirical process theory. To cite this article: D. Blondin, C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

A Poisson process reparameterisation for Bayesian inference for extremes

A common approach to modelling extreme values is to consider the excesses above a high threshold as realisations of a non-homogeneous Poisson process. While this method offers the advantage of modelling using threshold-invariant extreme value parameters, the dependence between these parameters makes estimation more difficult. We present a novel approach for Bayesian estimation of the Poisson process model parameters by reparameterising in terms of a tuning parameter m. This paper presents a method for choosing the optimal value of m that near-orthogonalises the parameters, which is achieved by minimising the correlation between the asymptotic posterior distribution of the parameters. This choice of m ensures more rapid convergence and efficient sampling from the joint posterior distribution using Markov Chain Monte Carlo methods. Samples from the parameterisation of interest are then obtained by a simple transform. Results are presented in the cases of identically and non-identically distributed models for extreme rainfall in Cumbria, UK.     

An efficient ECM algorithm for maximum likelihood estimation in mixtures of t-factor analyzers

Mixture of t factor analyzers (MtFA) have been shown to be a sound model-based tool for robust clustering of high-dimensional data. This approach, which is deemed to be one of natural parametric extensions with respect to normal-theory models, allows for accommodation of potential noise components, atypical observations or data with longer-than-normal tails. In this paper, we propose an efficient expectation conditional maximization (ECM) algorithm for fast maximum likelihood estimation of MtFA. The proposed algorithm inherits all appealing properties of the ordinary EM algorithm such as its stability and monotonicity, but has a faster convergence rate since its CM steps are governed by a much smaller fraction of missing information. Numerical experiments based on simulated and real data show that the new procedure outperforms the commonly used EM and AECM algorithms substantially in most of the situations, regardless of how the convergence speed is assessed by the computing time or number of iterations.     

Conditional Logistic Regression With Longitudinal Follow-up and Individual-Level Random Coefficients: A Stable and Efficient Two-Step Estimation Method

The analysis of data generated by animal habitat selection studies, by family studies of genetic diseases, or by longitudinal follow-up of households often involves fitting a mixed conditional logistic regression model to longitudinal data composed of clusters of matched case-control strata. The estimation of model parameters by maximum likelihood is especially difficult when the number of cases per stratum is greater than one. In this case, the denominator of each cluster contribution to the conditional likelihood involves a complex integral in high dimension, which leads to convergence problems in the numerical maximization. In this article we show how these computational complexities can be bypassed using a global two-step analysis for nonlinear mixed effects models. The first step estimates the cluster-specific parameters and can be achieved with standard statistical methods and software based on maximum likelihood for independent data. The second step uses the EM-algorithm in conjunction with conditional restricted maximum likelihood to estimate the population parameters. We use simulations to demonstrate that the method works well when the analysis is based on a large number of strata per cluster, as in many ecological studies. We apply the proposed two-step approach to evaluate habitat selection by pairs of bison roaming freely in their natural environment. This article has supplementary material online.     

Parameter Estimation of Nonlinear Stochastic Differential Equations: Simulated Maximum Likelihood versus Extended Kalman Filter and Itô-Taylor Expansion

This article compares several estimation methods for nonlinear stochastic differential equations with discrete time measurements. The likelihood function is computed by Monte Carlo simulations of the transition probability (simulated maximum likelihood SML) using kernel density estimators and functional integrals and by using the extended Kalman filter (EKF and second-order nonlinear filter SNF). The relation with a local linearization method is discussed. A simulation study for a diffusion process in a double well potential (Ginzburg–Landau equation) shows that, for large sampling intervals, the SML methods lead to better estimation results than the likelihood approach via EKF and SNF. A second study using a nonlinear diffusion coefficient (generalized Cox–Ingersoll–Ross model) demonstrates that the EKF type estimators may serve as efficient alternatives to simple maximum quasilikelihood approaches and Monte Carlo methods.