Characterizations of non-normalized discrete probability distributions and their application in statistics
This work addresses statistical challenges for researchers dealing with non-normalized discrete distributions, offering incremental tools for fit testing and parameter estimation.
The authors tackled the problem of statistical inference for discrete probability distributions without requiring normalization constants, deriving explicit mass function identities from Stein's method and applying them to fit testing and parameter estimation. They demonstrated the soundness of their methods through simulation studies for Poisson distribution testing, negative binomial estimation, and non-normalized exponential-polynomial models.
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.