Testing overdispersion in the zero-inflated Poisson model

The zero-inflated negative binomial (ZINB) model is used to account for commonly occurring overdispersion detected in data that are initially analyzed under the zero-inflated Poisson (ZIP) model. Tests for overdispersion (Wald test, likelihood ratio test [LRT], and score test) based on ZINB model for use in ZIP regression models have been developed. Due to similarity to the ZINB model, we consider the zero-inflated generalized Poisson (ZIGP) model as an alternate model for overdispersed zero-inflated count data. The score test has an advantage over the LRT and the Wald test in that the score test only requires that the parameter of interest be estimated under the null hypothesis. This paper proposes score tests for overdispersion based on the ZIGP model and illustrates that the derived score statistics are exactly the same as the score statistics under the ZINB model. A simulation study indicates the proposed score statistics are preferred to other tests for higher empirical power. In practice, based on the approximate mean–variance relationship in the data, the ZINB or ZIGP model can be considered, and a formal score test based on asymptotic standard normal distribution can be employed for assessing overdispersion in the ZIP model. We provide an example to illustrate the procedures for data analysis.     

The minimum L2 distance estimator for Poisson mixture models

A robust estimator is developed for Poisson mixture models with a known number of components. The proposed estimator minimizes the L₂ distance between a sample of data and the model. When the component distributions are completely known, the estimators for the mixing proportions are in closed form. When the parameters for the component Poisson distributions are unknown, numerical methods are needed to calculate the estimators. Compared to the minimum Hellinger distance estimator, the minimum L₂ estimator can be less robust to extreme outliers, and often more robust to moderate outliers.     

Score tests for heterogeneity and overdispersion in zero‐inflated Poisson and binomial regression models

Hall (2000) has described zero‐inflated Poisson and binomial regression models that include random effects to account for excess zeros and additional sources of heterogeneity in the data. The authors of the present paper propose a general score test for the null hypothesis that variance components associated with these random effects are zero. For a zero‐inflated Poisson model with random intercept, the new test reduces to an alternative to the overdispersion test of Ridout, Demério & Hinde (2001). The authors also examine their general test in the special case of the zero‐inflated binomial model with random intercept and propose an overdispersion test in that context which is based on a beta‐binomial alternative.     

Modified maximum likelihood estimator for the mixing distribution in a finite mixture

Using the likelihood equations, a method based on the generalized inverse of matrices is proposed to derive closed-form estimates for the mixing proportions in a finite mixture, when the component distributions are known.     

The Almon two parameter estimator for the distributed lag models

The two parameter estimator proposed by Özkale and Kaç?ranlar [The restricted and unrestricted two parameter estimators. Comm Statist Theory Methods. 2007;36(15):2707–2725] is a general estimator which includes the ordinary least squares, the ridge and the Liu estimators as special cases. In the present paper we introduce Almon two parameter estimator based on the two parameter estimation procedure to deal with the problem of multicollinearity for the distiributed lag models. This estimator outperforms the Almon estimator according to the matrix mean square error criterion. Moreover, a numerical example and a Monte Carlo simulation experiment are presented by using different estimators of the biasing parameters.     

Monitoring parameter shift with Poisson integer-valued GARCH models

This study examines the statistical process control chart used to detect a parameter shift with Poisson integer-valued GARCH (INGARCH) models and zero-inflated Poisson INGARCH models. INGARCH models have a conditional mean structure similar to GARCH models and are well known to be appropriate to analyzing count data that feature overdispersion. Special attention is paid in this study to conditional and general likelihood ratio-based (CLR and GLR) CUSUM charts and the score function-based CUSUM (SFCUSUM) chart. The performance of each of the proposed methods is evaluated through a simulation study, by calculating their average run length. Our findings show that the proposed methods perform adequately, and that the CLR chart outperforms the GLR chart when there is an increased shift of parameters. Moreover, the use of the SFCUSUM chart in particular is found to lead to a lower false alarm rate than the use of the CLR chart.     

Connections of the Poisson weight function to overdispersion and underdispersion

In this paper, we establish several connections of the Poisson weight function to overdispersion and underdispersion. Specifically, we establish that the logconvexity (logconcavity) of the mean weight function is a necessary and sufficient condition for overdispersion (underdispersion) when the Poisson weight function does not depend on the original Poisson parameter. We also discuss some properties of the weighted Poisson distributions (WPD). We then introduce a notion of pointwise duality between two WPDs and discuss some associated properties. Next, we present some illustrative examples and provide a discussion on various Poisson weight functions used in practice. Finally, some concluding remarks are made.     

Testing for overdispersion in a censored Poisson regression model

In this article, we investigate the efficiency of score tests for testing a censored Poisson regression model against censored negative binomial regression alternatives. Based on the results of a simulation study, score tests using the normal approximation, underestimate the nominal significance level. To remedy this problem, bootstrap methods are proposed. We find that bootstrap methods keep the significance level close to the nominal one and have greater power uniformly than does the normal approximation for testing the hypothesis.     

Selecting the precision parameter prior in Dirichlet process mixture models

We consider Dirichlet process mixture models in which the observed clusters in any particular dataset are not viewed as belonging to a finite set of possible clusters but rather as representatives of a latent structure in which objects belong to one of a potentially infinite number of clusters. As more information is revealed the number of inferred clusters is allowed to grow. The precision parameter of the Dirichlet process is a crucial parameter that controls the number of clusters. We develop a framework for the specification of the hyperparameters associated with the prior for the precision parameter that can be used both in the presence or absence of subjective prior information about the level of clustering. Our approach is illustrated in an analysis of clustering brands at the magazine Which?. The results are compared with the approach of Dorazio (2009) via a simulation study.     

The distribution of the Liu-type estimator of the biasing parameter in elliptically contoured models

We derive the density function of the stochastic shrinkage parameters of the Liu-type estimator in elliptical models. The correctness of derivation is checked by simulations. A real data application is also provided.     

Testing overdispersion in data with censoring using the mixture of exponential families

This resean h extends the mixture of exponential family developed by Dean (1992) to accommodate oveidispeision in data with censoring. Score statistics testing the (Existence of overdispersion based on the proposed model are obtained Simulations show that the test statistics have suffcient power in detecting the existence of overdispersion when sample size are sufficiently large and the degree of censoring is mild (i. e, 40%) The test statistics are applied to real data sets and are able to detect overdispersion in those data sets.     

Tests for detecting overdispersion in poisson models

Collings and Margolin(1985) developed a locally most powerful unbiased test for detecting negative binomial departures from a Poisson model, when the variance is a quadratic function of the mean. Kim and Park(1992) developed a locally most powerful unbiased test, when the variance is a linear function of the mean. It is found that a different mean-variance structure of a negative binomial derives a different locally optimal test statistic.

In this paper Collings and Margolin's and Kim and Park's results are unified and extended by developing a test for overdispersion in Poisson model against Katz family of distributions, Our setup has two extensions: First, Katz family of distributions is employed as an extension of the negative binomial distribution. Second, the mean-variance structure of the mixed Poisson model is given by σ² = μ+cμ^r for arbitrary but fixed r. We derive a local score test for testing H₀ : c = 0. Superiority of a new test is proved by the asymtotic relative efficiency as well as the simulation study.     

Estimation of a scale parameter in mixture models with unknown location

Estimation of the scale parameter in mixture models with unknown location is considered under Stein's loss. Under certain conditions, the inadmissibility of the “usual” estimator is established by exhibiting better estimators. In addition, robust improvements are found for a specified submodel of the original model. The results are applied to mixtures of normal distributions and mixtures of exponential distributions. Improved estimators of the variance of a normal distribution are shown to be robust under any scale mixture of normals having variance greater than the variance of that normal distribution. In particular, Stein's (Ann. Inst. Statist. Math. 16 (1964) 155) and Brewster's and Zidek's (Ann. Statist. 2 (1974) 21) estimators obtained under the normal model are robust under the t model, for arbitrary degrees of freedom, and under the double-exponential model. Improved estimators for the variance of a t distribution with unknown and arbitrary degrees of freedom are also given. In addition, improved estimators for the scale parameter of the multivariate Lomax distribution (which arises as a certain mixture of exponential distributions) are derived and the robustness of Zidek's (Ann. Statist. 1 (1973) 264) and Brewster's (Ann. Statist. 2 (1974) 553) estimators of the scale parameter of an exponential distribution is established under a class of modified Lomax distributions.     

An empirical approach to determine a threshold for assessing overdispersion in Poisson and negative binomial models for count data

Overdispersion is a problem encountered in the analysis of count data that can lead to invalid inference if unaddressed. Decision about whether data are overdispersed is often reached by checking whether the ratio of the Pearson chi-square statistic to its degrees of freedom is greater than one; however, there is currently no fixed threshold for declaring the need for statistical intervention. We consider simulated cross-sectional and longitudinal datasets containing varying magnitudes of overdispersion caused by outliers or zero inflation, as well as real datasets, to determine an appropriate threshold value of this statistic which indicates when overdispersion should be addressed.     

A new linearized ridge Poisson estimator in the presence of multicollinearity

Poisson regression is a very commonly used technique for modeling the count data in applied sciences, in which the model parameters are usually estimated by the maximum likelihood method. However, the presence of multicollinearity inflates the variance of maximum likelihood (ML) estimator and the estimated parameters give unstable results. In this article, a new linearized ridge Poisson estimator is introduced to deal with the problem of multicollinearity. Based on the asymptotic properties of ML estimator, the bias, covariance and mean squared error of the proposed estimator are obtained and the optimal choice of shrinkage parameter is derived. The performance of the existing estimators and proposed estimator is evaluated through Monte Carlo simulations and two real data applications. The results clearly reveal that the proposed estimator outperforms the existing estimators in the mean squared error sense.KEYWORDS: Poisson regression, multicollinearity, ridge Poisson estimator, linearized ridge regression estimator, mean squared errorMathematics Subject Classifications: 62J07, 62F10     

On selecting a prior for the precision parameter of Dirichlet process mixture models

In hierarchical mixture models the Dirichlet process is used to specify latent patterns of heterogeneity, particularly when the distribution of latent parameters is thought to be clustered (multimodal). The parameters of a Dirichlet process include a precision parameter α$\alpha$ and a base probability measure _G0$G_0$. In problems where α$\alpha$ is unknown and must be estimated, inferences about the level of clustering can be sensitive to the choice of prior assumed for α$\alpha$. In this paper an approach is developed for computing a prior for the precision parameter α$\alpha$ that can be used in the presence or absence of prior information about the level of clustering. This approach is illustrated in an analysis of counts of stream fishes. The results of this fully Bayesian analysis are compared with an empirical Bayes analysis of the same data and with a Bayesian analysis based on an alternative commonly used prior.     

Orthogonal Stiefel manifold optimization for eigen-decomposed covariance parameter estimation in mixture models

Within the mixture model-based clustering literature, parsimonious models with eigen-decomposed component covariance matrices have dominated for over a decade. Although originally introduced as a fourteen-member family of models, the current state-of-the-art is to utilize just ten of these models; the rationale for not using the other four models usually centers around parameter estimation difficulties. Following close examination of these four models, we find that two are actually easily implemented using existing algorithms but that two benefit from a novel approach. We present and implement algorithms that use an accelerated line search for optimization on the orthogonal Stiefel manifold. Furthermore, we show that the ‘extra’ models that these decompositions facilitate outperform the current state-of-the art when applied to two benchmark data sets.     

A score test for overdispersion in Poisson regression based on the generalized Poisson-2 model

Overdispersion is a common phenomenon in Poisson modeling. The generalized Poisson (GP) regression model accommodates both overdispersion and underdispersion in count data modeling, and is an increasingly popular platform for modeling overdispersed count data. The Poisson model is one of the special cases in the collection of models which may be specified by GP regression. Thus, we may derive a test of overdispersion which compares the equi-dispersion Poisson model within the context of the more general GP regression model. The score test has an advantage over the likelihood ratio test (LRT) and over the Wald test in that the score test only requires that the parameter of interest be estimated under the null hypothesis (the Poisson model). Herein, we propose a score test for overdispersion based on the GP model (specifically the GP-2 model) and compare the power of the test with the LRT and Wald tests. A simulation study indicates the proposed score test based on asymptotic standard normal distribution is more appropriate in practical applications.     

Analysis of the human sex ratio by using overdispersion models

For study of the human sex ratio, one of the most important data sets was collected in Saxony in the 19th century by Geissler. The data contain the sizes of families, with the sex of all children, at the time of registration of the birth of a child. These data are reanalysed to determine how the probability for each sex changes with family size. Three models for overdispersion are fitted: the beta–binomial model of Skellam, the 'multiplicative' binomial model of Altham and the double-binomial model of Efron. For each distribution, both the probability and the dispersion parameters are allowed to vary simultaneously with family size according to two separate regression equations. A finite mixture model is also fitted. The models are fitted using non-linear Poisson regression. They are compared using direct likelihood methods based on the Akaike information criterion. The multiplicative and beta–binomial models provide similar fits, substantially better than that of the double-binomial model. All models show that both the probability that the child is a boy and the dispersion are greater in larger families. There is also some indication that a point probability mass is needed for families containing children uniquely of one sex.