On the Discrete POISSON‐Inverse GAUSSian Distribution

Shaban, S. A.

doi:10.1002/bimj.4710230309

Cited by 11 publications

(6 citation statements)

References 2 publications

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“…levels of overdispersion, is extensive and provides several other mixed distributions as the Poisson-Lindley (Sankaran, 1970), the Poisson-Lognormal (Bulmer, 1974), the Poisson-Inverse Gaussian (Shaban, 1981), the Negative Binomial-Lindley (Zamani and Ismail, 2010), the Poisson-Shanker (Shanker, 2016a), the Poisson-Sujatha (Shanker, 2016b), among others.…”

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confidence: 99%

Bayesian approach for the zero-modified Poisson–Lindley regression model

Bertoli¹,

Conceição²,

Andrade³

et al. 2019

Braz. J. Probab. Stat.

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The primary goal of this paper is to introduce the zero-modified Poisson-Lindley regression model as an alternative to model overdispersed count data exhibiting inflation or deflation of zeros in the presence of covariates. The zero-modification is incorporated by considering that a zerotruncated process produces positive observations and consequently, the proposed model can be fitted without any previous information about the zeromodification present in a given dataset. A fully Bayesian approach based on the g-prior method has been considered for inference concerns. An intensive Monte Carlo simulation study has been conducted to evaluate the performance of the developed methodology and the maximum likelihood estimators. The proposed model was considered for the analysis of a real dataset on the number of bids received by 126 U.S. firms between 1978-1985, and the impact of choosing different prior distributions for the regression coefficients has been studied. A sensitivity analysis to detect influential points has been performed based on the Kullback-Leibler divergence. A general comparison with some well-known regression models for discrete data has been presented. IntroductionThe standard Poisson (P) distribution is the most adopted discrete model for the analysis of count data in several research fields, mainly due to its great simplicity and by having its computational implementation available for many statistical packages. However, it is well-known that such model is not a suitable choice for the analysis of counts in which the variance-to-mean ratio is not (at least) close to 1, that is, when the equidispersion property is violated. Apart from data transformation, the most popular way to circumvent such an issue is the use of finite mixture models (McLachlan and Peel, 2000) that can accommodate, for example, overdispersion (Karlis and Xekalaki, 2005). The Negative Binomial (N B) distribution (that may arise as a P mixture model by using a Gamma distribution for the continuous part) is undoubtedly the most famous alternative to model extra-P variability. However, the literature concerning discrete models, which can handle different

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confidence: 99%

Bayesian approach for the zero-modified Poisson–Lindley regression model

Bertoli¹,

Conceição²,

Andrade³

et al. 2019

Braz. J. Probab. Stat.

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“…However, there is extensive literature regarding other discrete mixed distributions that can allow for different overdispersion levels. Among these models, we can cite the one‐parameter Poisson‐Lindley, 25 Poisson‐lognormal, 26 Poisson‐inverse Gaussian, 27 Poisson‐Janardan, 28 Poisson‐Amarendra, 29 Poisson‐Shanker, 30 Poisson‐Sujatha, 31 Poisson‐Akash, 32 Poisson‐Aradhana, 33 Poisson‐Garima, 34 and Poisson‐Ishita 35 distributions.…”

Section: Introductionmentioning

confidence: 99%

Statistical process control of overdispersed count data based on one‐parameter Poisson mixture models

Jesus

Ferreira

Boaventura

et al. 2022

Quality & Reliability Eng

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The Poisson distribution is a discrete model widely used to analyze count data. Statistical control charts based on this distribution, such as the c$c$ and u$u$ charts, are relatively well‐established in the literature. Nevertheless, many studies suggest the need for alternative approaches that allow for modeling overdispersion, a phenomenon that can be observed in several fields, including biology, ecology, healthcare, marketing, economics, and industry. The one‐parameter Poisson mixture distributions, whose literature is extensive and essential, can model extra‐Poisson variability, accommodating different overdispersion levels. The distributions belonging to this class of models, including the Poisson‐Lindley (PL), Poisson‐Shanker (PSh), and Poisson‐Sujatha (PSu) models, can thus be used as interesting alternatives to the usual Poisson and COM‐Poisson distributions for analyzing count data in several areas. In this paper, we consider the class of probabilistic models mentioned above (as well as the cited three members of such a class) to develop novel and useful statistical control charts for counting processes, monitoring count data that exhibit overdispersion. The performance of the so‐called one‐parameter Poisson mixture charts, namely the PLc$\text{PL}_c$‐PLu$\text{PL}_u$, PShc$\text{PSh}_c$‐PShu$\text{PSh}_u$, and PSuc$\text{PSu}_c$‐PSuu$\text{PSu}_u$ charts, is measured by the average run length in exhaustive numerical simulations. Some data sets are used to illustrate the applicability of the proposed methodology.

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“…The negative binomial

distribution (that may arise as a

mixture model by using a gamma distribution for the continuous part) is undoubtedly the most popular alternative to model extra-

variability. There is extensive literature regarding other discrete mixed distributions that can accommodate different levels of overdispersion, for example, the Poisson–Lindley [ 2 ], the Poisson–lognormal [ 3 ], the Poisson–inverse Gaussian [ 4 ], the negative binomial–Lindley [ 5 ], the Poisson–Janardan [ 6 ], the two-parameter Poisson–Lindley [ 7 ], the Poisson–Amarendra [ 8 ], the Poisson–Shanker [ 9 ], the Poisson–Sujatha

[ 10 ], the quasi-Poisson–Lindley [ 11 ], the weighted negative binomial–Lindley [ 12 ] the Poisson-weighted Lindley [ 13 ], the binomial-discrete Lindley [ 14 ], and the two-parameter Poisson–Sujatha [ 15 ], among many others.…”

Section: Introductionmentioning

confidence: 99%

A New Regression Model for the Analysis of Overdispersed and Zero-Modified Count Data

Bertoli

Conceição

Andrade

et al. 2021

Entropy

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Count datasets are traditionally analyzed using the ordinary Poisson distribution. However, said model has its applicability limited, as it can be somewhat restrictive to handling specific data structures. In this case, the need arises for obtaining alternative models that accommodate, for example, overdispersion and zero modification (inflation/deflation at the frequency of zeros). In practical terms, these are the most prevalent structures ruling the nature of discrete phenomena nowadays. Hence, this paper’s primary goal was to jointly address these issues by deriving a fixed-effects regression model based on the hurdle version of the Poisson–Sujatha distribution. In this framework, the zero modification is incorporated by considering that a binary probability model determines which outcomes are zero-valued, and a zero-truncated process is responsible for generating positive observations. Posterior inferences for the model parameters were obtained from a fully Bayesian approach based on the g-prior method. Intensive Monte Carlo simulation studies were performed to assess the Bayesian estimators’ empirical properties, and the obtained results have been discussed. The proposed model was considered for analyzing a real dataset, and its competitiveness regarding some well-established fixed-effects models for count data was evaluated. A sensitivity analysis to detect observations that may impact parameter estimates was performed based on standard divergence measures. The Bayesian p-value and the randomized quantile residuals were considered for the task of model validation.

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On the Discrete POISSON‐Inverse GAUSSian Distribution

Cited by 11 publications

References 2 publications

Bayesian approach for the zero-modified Poisson–Lindley regression model

Bayesian approach for the zero-modified Poisson–Lindley regression model

Statistical process control of overdispersed count data based on one‐parameter Poisson mixture models

A New Regression Model for the Analysis of Overdispersed and Zero-Modified Count Data

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