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.
翻译:从Stein方法核心的分布特征中,我们为识别这些分布的离散概率法的质量函数得出明确的公式。这些特性用于开发解决统计问题的工具。我们的特征特征,以及基于这些特性的应用,并不要求了解概率法的正常化常数。为了证明我们的统计方法是健全的,我们提供了比较模拟研究,以测试是否适合Poisson的分布,并在这两个参数都未知时对负二元系进行参数估计。我们还考虑了对通常不具有常态的离散指数-球系模型的参数估计问题。