Modern Bayesian Inference in Zero-Inflated Poisson Models (original) (raw)
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Properties of the zero-and-one inflated Poisson distribution and likelihood-based inference methods
Statistics and Its Interface, 2016
To model count data with excess zeros and excess ones, in their unpublished manuscript, Melkersson and Olsson (1999) extended the zero-inflated Poisson distribution to a zero-and-one-inflated Poisson (ZOIP) distribution. However, the distributional theory and corresponding properties of the ZOIP have not yet been explored, and likelihoodbased inference methods for parameters of interest were not well developed. In this paper, we extensively study the ZOIP distribution by first constructing five equivalent stochastic representations for the ZOIP random variable and then deriving other important distributional properties. Maximum likelihood estimates of parameters are obtained by both the Fisher scoring and expectation-maximization algorithms. Bootstrap confidence intervals for parameters of interest and testing hypotheses under large sample sizes are provided. Simulations studies are performed and five real data sets are used to illustrate the proposed methods.
A BAYESIAN APROACH FOR ZERO-INFLATED COUNT DATA IN PSYCHOLOGICAL RESEARCH
Count data in psychological research are commonly modelled using zero-inflated Poisson regression. This model can be viewed as a latent mixture of an " always-zero " component and a Poisson component. In this study we introduce a Bayesian approach for zero inflated Poisson model, and discuss model comparisons and the interpretation of their parameters. As illustrated with two real-world examples psychological research, both Bayesian and classic approach of models can easily be fitted with a great gain.
Bayesian methods for Poisson models
2000
To account for overdispersion in count data, that is variation in excess of that justified from the assumed model, one may consider an additional source of variation, by assuming that each observation, Y2 , i = 1,... , m, arises from a conditionally independent Poisson distribution, given its respective mean O, i = 1,... , M. We review various frequentist methods for the estimation of the Poisson parameters O, i = 1,... , m, which are based on the inadmissibility of the usual unbiased maximum likelihood estimator, in terms of the associated risk in dimensions greater than two. The so called shrinkage estimators adjust the maximum likelihood estimates towards a fixed or data-determined point, abandoning unbiasedness in favour of lower risk. Inferences for the parameters of interest can also be drawn employing Bayesian methods. Conjugate models are often adopted to facilitate the computational procedure. In this thesis we assume a nonconjugate log-normal prior distribution, which allo...
Bias-reduced maximum likelihood estimation of the zero-inflated Poisson distribution
Communications in Statistics - Theory and Methods, 2016
We investigate the small-sample quality of the maximum likelihood estimators (MLEs) of the parameters of the zero-inflated Poisson distribution. The finite-sample biases are determined to O(n-1) using an analytic bias reduction methodology based on the work of Cox and Snell (1968) and Cordeiro and Klein (1994). Monte Carlo simulations show that the MLEs have very small percentage biases for this distribution, but the analytic bias reduction methods essentially eliminate the bias without adversely affecting the mean squared error s of the estimators. The analytic adjustment compares favourably with the parametric bootstrap bias-corrected estimator, in terms of bias reduction itself, as well as with respect to mean squared error and Pitman's nearness measure.
arXiv (Cornell University), 2017
The main object of this article is to present an extension of the zero-inflated Poisson-Lindley distribution, called of zero-modified Poisson-Lindley. The additional parameter π of the zero-modified Poisson-Lindley has a natural interpretation in terms of either zero-deflated/inflated proportion. Inference is dealt with by using the likelihood approach. In particular the maximum likelihood estimators of the distribution's parameter are compared in small and large samples. We also consider an alternative bias-correction mechanism based on Efron's bootstrap resampling. The model is applied to real data sets and found to perform better than other competing models.
Estimation Techniques for Regression Model with Zero-inflated Poisson Data
International Journal of Statistics and Probability, 2015
Researchers in many fields including biomedical often make statistical inferences involving the analysis of count data that exhibit a substantially large proportion of zeros. Subjects in such research are broadly categorized into low-risk group that produces only zero counts and high-risk group leading to counts that can be modeled by a standard Poisson regression model. The aim of this study is to estimate the model parameters in presence of covariates, some of which may not have significant effects on the magnitude of the counts in presence of a large proportion of zeros. The estimation procedures we propose for the study are the pretest, shrinkage, and penalty when some of the covariates may be subject to certain restrictions. Properties of the pretest and shrinkage estimators are discussed in terms of the asymptotic distributional biases and risks. We show that if the dimension of parameters exceeds two, the risk of the shrinkage estimator is strictly less than that of the maximum likelihood estimator, and the risk of the pretest estimator depends on the validity of the restrictions on parameters. A Monte Carlo simulation study shows that the mean squared errors (MSE) of shrinkage estimator are comparable to the MSE of the penalty estimators and in particular it performs better than the penalty estimators when the dimension of the restricted parameter space is large. For illustrative purposes, the methods are applied to a real life data set
Generalized fiducial inference on the mean of zero-inflated Poisson and Poisson hurdle models
Journal of Statistical Distributions and Applications, 2021
Zero-inflated and hurdle models are widely applied to count data possessing excess zeros, where they can simultaneously model the process from how the zeros were generated and potentially help mitigate the effects of overdispersion relative to the assumed count distribution. Which model to use depends on how the zeros are generated: zero-inflated models add an additional probability mass on zero, while hurdle models are two-part models comprised of a degenerate distribution for the zeros and a zero-truncated distribution. Developing confidence intervals for such models is challenging since no closed-form function is available to calculate the mean. In this study, generalized fiducial inference is used to construct confidence intervals for the means of zero-inflated Poisson and Poisson hurdle models. The proposed methods are assessed by an intensive simulation study. An illustrative example demonstrates the inference methods.
Type I multivariate zero-inflated generalized Poisson distribution with applications
Statistics and Its Interface, 2017
Excessive zeros in multivariate count data are often encountered in practice. Since the Poisson distribution only possesses the property of equi-dispersion, the existing Type I multivariate zero-inflated Poisson distribution (Liu and Tian, 2015, CSDA) [15] cannot be used to model multivariate zero-inflated count data with over-dispersion or under-dispersion. In this paper, we extend the univariate zero-inflated generalized Poisson (ZIGP) distribution to Type I multivariate ZIGP distribution via stochastic representation aiming to model positively correlated multivariate zero-inflated count data with over-dispersion or underdispersion. Its distributional theories and associated properties are derived. Due to the complexity of the ZIGP model, we provide four useful algorithms (a very fast Fisher-scoring algorithm, an expectation/conditional-maximization algorithm, a simple EM algorithm and an explicit majorizationminimization algorithm) for finding maximum likelihood estimates of parameters of interest and develop efficient statistical inference methods for the proposed model. Simulation studies for investigating the accuracy of point estimates and confidence interval estimates and comparing the likelihood ratio test with the score test are conducted. Under both AIC and BIC, our analyses of the two data sets show that Type I multivariate ZIGP model is superior over Type I multivariate zero-inflated Poisson model.
This paper takes a fresh look on point estimation of model parameters under a Zero-Inflated Poisson (ZIP) distribution. The reason is that some finer details of point estimation, if overlooked, may lead to wrong estimates as was done by the earlier researchers. In this paper we have achieved the following new results: (a) A new set of corrected method of moments estimators has been proposed; (b) We have shown how the standard technique of differentiating the log-likelihood function to find the maximum likelihood estimators may lead to wrong estimates, as well as how to avoid this problem; and (c) A new adjusted maximum likelihood estimation technique has been proposed which not only produces meaningful estimates always, but also appears to work better compared to all other estimation techniques in terms of standardized mean squared error (SMSE) when ZIP is used to model rare events. Finally, datasets on rare events have been used to demonstrate the estimation techniques, and how the...
On the zero-modified poisson model: Bayesian analysis and posterior divergence measure
Computational Statistics, 2013
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