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2022 Characterizations of non-normalized discrete probability distributions and their application in statistics
Steffen Betsch, Bruno Ebner, Franz Nestmann
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Electron. J. Statist. 16(1): 1303-1329 (2022). DOI: 10.1214/22-EJS1983

Abstract

From the distributional characterizations that lie at the heart of Stein’s method we derive explicit formulae for the mass functions of discrete probability laws that identify those distributions. These identities are applied to develop tools for the solution of statistical problems. Our characterizations, and hence the applications built on them, do not require any knowledge about normalization constants of the probability laws. To demonstrate that our statistical methods are sound, we provide comparative simulation studies for the testing of fit to the Poisson distribution and for parameter estimation of the negative binomial family when both parameters are unknown. We also consider the problem of parameter estimation for discrete exponential-polynomial models which generally are non-normalized.

Acknowledgments

The authors thank an anonymous referee for comments that have lead to an improvement of the article.

Citation

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Steffen Betsch. Bruno Ebner. Franz Nestmann. "Characterizations of non-normalized discrete probability distributions and their application in statistics." Electron. J. Statist. 16 (1) 1303 - 1329, 2022. https://doi.org/10.1214/22-EJS1983

Information

Received: 1 June 2021; Published: 2022
First available in Project Euclid: 15 February 2022

Digital Object Identifier: 10.1214/22-EJS1983

Subjects:
Primary: 62E10
Secondary: 62G10

Keywords: Discrete exponential-polynomial models , Goodness-of-fit tests , negative binomial distribution , non-normalized models , Stein characterizations

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Vol.16 • No. 1 • 2022
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