Abstract
The process of discretization of continuous distributions creates and provides a large set of discrete probabilistic models used in various statistical applications. The most common way of doing so is by considering the probability distribution of the integral part of a continuous random variable. In this note we explore the following problem related to the latter discretization process and pose the following question: If the family of distributions that is discretized is an exponential family on the real line, when the (integral) resulting discrete probability model also generates an exponential family? We give a complete answer to this question and provide necessary and sufficient conditions under which the discretized version of an exponential family is also an exponential family.
Similar content being viewed by others
References
Chakraborty S (2015) Generating discrete analogues of continuous probability distributions-A survey of methods and constructions. J Stat Distrib Appl 2(6):1–30. https://doi.org/10.1186/s40488-015-0028-6
Gupta P, Gupta RC, Ong SH, Srivastava HM (2008) A class of Hurwitz-Lerch Zeta distributions and their applications in reliability. Appl Math Comput 196:521–531
Hagmark P-E (2008) On construction and simulation of count data models. Math Comput Simul 77:72–80
Kokonendji CC, Zocchi SS (2010) Extensions of discrete triangular distributions and boundary bias in kernel estimation for discrete functions. Statist Probab Lett 80:1655–1662
Lai CD (2013) Issues concerning constructions of discrete lifetime models. Qual Technol Quantitative Manage 10(2):251–262
Letac G, Mora M (1990) Natural real exponential families with cubic variance functions. Ann Stat 18:1–37
Funding
Not applicable
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors state that there is no conflict of interest
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Bar-Lev, S.K., Letac, G. A note on the discretization of natural exponential families on the real line. Metrika 86, 83–90 (2023). https://doi.org/10.1007/s00184-022-00863-4
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00184-022-00863-4