Exploring Asymptotic Normality in Multinomial Models

ABSTRACT Among the methods for analyzing categorical outcomes, the multinomial model offers a robust framework for examining the dependence between a multi‐category response variable and a set of explanatory variables. Its flexibility, versatility, and broad applicability across diverse fields make it a valuable tool, as it does not impose strict assumptions. In this work, we focus on deriving a standardized asymptotic distribution for multinomial models, significantly advancing the theoretical framework for categorical data analysis. Building on the concept of smooth statistics—characterized by having components with continuous second‐order partial derivatives in a neighborhood of a location parameter—we extend a key theorem on asymptotic normality, originally developed for Wishart matrices, to multinomial models with both finite and countable sets of possible outcomes. A main contribution of this study lies in the standardization process, as it allows addressing the challenges arising from non‐invertible covariance matrices, enabling application even starting from singular covariance matrices. This approach significantly advances the analysis of multinomial models by producing a simpler structure through standardized asymptotic distributions, thus broadening the applicability of smooth statistics. These theoretical developments are particularly relevant to the mathematical modeling of categorical processes with spatial and temporal dependence, where evolving states encounter complex dependency structures. The robustness to singular covariance matrices directly addresses challenges common in biomathematical models and others, thereby broadening the mathematical methodology for analyzing structured categorical data in such applied scientific contexts.