On goodness of fit tests for the Poisson, negative binomial and binomial distributions

Beltrán-Beltrán, J. I.; O'Reilly, Federico J.

doi:10.1007/s00362-016-0820-5

Cited by 13 publications

(8 citation statements)

References 16 publications

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“…Thus, to test the null hypothesis of a Poisson distribution, one may focus on testing for equidispersion. This is the idea behind the widely used Fisher's index of dispersion (Beltrán-Beltrán & O'Reilly, 2019;Gürtler & Henze, 2000;Kyriakoussis et al, 1998), which has also been used for some types of Poi-INARMA processes, see Aleksandrov and Weiß (2020a) and the references therein. The dispersion index Î is the quotient of the empirical variance to the mean, which can be rewritten as…”

Section: Stein-chen Gof Testsmentioning

confidence: 99%

“…Besides its original application within Poisson approximations, Weiß and Aleksandrov (2020) demonstrated that Equation (1) also constitutes a useful tool for computing diverse Poisson moments. In the present work, however, our aim is different: we use the Stein-Chen identity (1) to derive novel goodness-of-fit (GoF) tests for the Poisson distribution; see Beltrán-Beltrán and O'Reilly (2019) and Gürtler and Henze (2000) for references on Poi-GoF tests. Generally, there are two strategies for defining a GoF test statistic: one can try to capture (almost) the whole distribution, or one can focus on specific characteristic properties thereof.…”

Section: Introductionmentioning

confidence: 99%

“…Generally, there are two strategies for defining a GoF test statistic: one can try to capture (almost) the whole distribution, or one can focus on specific characteristic properties thereof. The first group comprises, for example, the famous Pearson statistic, while the second group includes the widely used Fisher's index of dispersion (Beltrán-Beltrán & O'Reilly, 2019;Gürtler & Henze, 2000;Kyriakoussis, Li, & Papadopoulos, 1998). Tests from the first group offer the potential of being powerful to nearly any violation of the null hypothesis, but they are often difficult to implement in practice, especially if not being concerned with independent and identically distributed (i.i.d.)…”

Section: Introductionmentioning

confidence: 99%

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Goodness‐of‐fit tests for Poisson count time series based on the Stein–Chen identity

Aleksandrov

Weiß

Jentsch

2021

Statistica Neerlandica

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To test the null hypothesis of a Poisson marginal distribution, test statistics based on the Stein-Chen identity are proposed. For a wide class of Poisson count time series, the asymptotic distribution of different types of Stein-Chen statistics is derived, also if multiple statistics are jointly applied. The performance of the tests is analyzed with simulations, as well as the question which Stein-Chen functions should be used for which alternative. Illustrative data examples are presented, and possible extensions of the novel Stein-Chen approach are discussed as well.

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Section: Stein-chen Gof Testsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Goodness‐of‐fit tests for Poisson count time series based on the Stein–Chen identity

Aleksandrov

Weiß

Jentsch

2021

Statistica Neerlandica

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“…The formal idea of a CSS test seems to date back to at least Bartlett (1937), although the value of sufficient statistics for GoF testing in the presence of nuisance parameters has also been used in many other ways, e.g., Durbin (1961); Kumar and Pathak (1977); Bell (1984) decompose the data into a minimal sufficient statistic and an ancillary statistic and construct a GoF test based on the parameter-free distribution of the ancillary statistic. CSS testing has gained substantial interest in the last 25 years, though with a focus on non-degenerate hypothesis testing settings (Engen and Lillegård, 1997;O'Reilly and Gracia-Medrano, 2006;Lockhart et al, 2007Lockhart et al, , 2009Lindqvist and Rannestad, 2011;Broniatowski and Caron, 2012;Lockhart, 2012;Stephens, 2012;Lindqvist and Taraldsen, 2013;Hazra, 2013;Beltrán-Beltrán and O'Reilly, 2019;Santos and Filho, 2019;Contreras-Cristán et al, 2019). Similar techniques have been used to obtain exact confidence intervals in the presence of nuisance parameters (Lillegård and Engen, 1999), again in non-degenerate settings.…”

Section: Related Workmentioning

confidence: 99%

Testing goodness-of-fit and conditional independence with approximate co-sufficient sampling

Barber¹,

Janson²

2020

Preprint

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Goodness-of-fit (GoF) testing is ubiquitous in statistics, with direct ties to model selection, confidence interval construction, conditional independence testing, and multiple testing, just to name a few applications. While testing the GoF of a simple (point) null hypothesis provides an analyst great flexibility in the choice of test statistic while still ensuring validity, most GoF tests for composite null hypotheses are far more constrained, as the test statistic must have a tractable distribution over the entire null model space. A notable exception is co-sufficient sampling (CSS): resampling the data conditional on a sufficient statistic for the null model guarantees valid GoF testing using any test statistic the analyst chooses. But CSS testing requires the null model to have a compact (in an information-theoretic sense) sufficient statistic, which only holds for a very limited class of models; even for a null model as simple as logistic regression, CSS testing is powerless. In this paper, we leverage the concept of approximate sufficiency to generalize CSS testing to essentially any parametric model with an asymptotically-efficient estimator; we call our extension "approximate CSS" (aCSS) testing. We quantify the finite-sample Type I error inflation of aCSS testing and show that it is vanishing under standard maximum likelihood asymptotics, for any choice of test statistic. We apply our proposed procedure both theoretically and in simulation to a number of models of interest to demonstrate its finite-sample Type I error and power.

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“…As a follow-up, Rao and Chakravarti (1956) used the same idea to derive an exact test for the Poisson case based on a likelihood ratio statistic (see Section 3.3.1). Conditioning on sufficient statistics has also been used recently in Beltrán-Beltrán and O'Reilly (2019) and Puig and Weiß (2020). While the just cited papers have considered models with one unknown parameter under the null hypothesis, Heller (1986) did goodness-of-fit testing for the two-parameter negative binomial distribution, assuming both parameters are unknown.…”

Section: Introductionmentioning

confidence: 99%

Conditional Goodness-of-Fit Tests for Discrete Distributions

Erlemann¹,

Lindqvist²

2020

Preprint

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In this paper, we address the problem of testing goodness-of-fit for discrete distributions, where we focus on the geometric distribution. We define new likelihoodbased goodness-of-fit tests using the beta-geometric distribution and the type I discrete Weibull distribution as alternative distributions. The tests are compared in a simulation study, where also the classical goodness-of-fit tests are considered for comparison. Throughout the paper we consider conditional testing given a minimal sufficient statistic under the null hypothesis, which enables the calculation of exact pvalues. For this purpose, a new method is developed for drawing conditional samples from the geometric distribution and the negative binomial distribution. We also explain briefly how the conditional approach can be modified for the binomial, negative binomial and Poisson distributions. It is finally noted that the simulation method may be extended to other discrete distributions having the same sufficient statistic, by using the Metropolis-Hastings algorithm.

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On goodness of fit tests for the Poisson, negative binomial and binomial distributions

Cited by 13 publications

References 16 publications

Goodness‐of‐fit tests for Poisson count time series based on the Stein–Chen identity

Goodness‐of‐fit tests for Poisson count time series based on the Stein–Chen identity

Testing goodness-of-fit and conditional independence with approximate co-sufficient sampling

Conditional Goodness-of-Fit Tests for Discrete Distributions

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