Complete monotonicity of multinomial probabilities and its application to Bernstein estimators on the simplex

TL;DR

This paper proves a complete monotonicity property of multinomial probabilities, generalizing previous binomial results, and applies it to derive inequalities and asymptotic formulas for Bernstein estimators on the simplex.

Contribution

It establishes a new complete monotonicity result for multinomial probabilities, extending prior binomial findings, and demonstrates applications in statistical density estimation.

Findings

Proved complete monotonicity of a multinomial probability function.

Derived combinatorial inequalities for multinomial coefficients.

Provided asymptotic formulas for Bernstein estimators on the simplex.

Abstract

Let $d\in \mathbb{N}$ and let $\gamma_i\in [0,\infty)$, $x_i\in (0,1)$ be such that $\sum_{i=1}^{d+1} \gamma_i = M\in (0,\infty)$ and $\sum_{i=1}^{d+1} x_i = 1$. We prove that \begin{equation*} a \mapsto \frac{\Gamma(aM + 1)}{\prod_{i=1}^{d+1} \Gamma(a \gamma_i + 1)} \prod_{i=1}^{d+1} x_i^{a\gamma_i} \end{equation*} is completely monotonic on $(0,\infty)$. This result generalizes the one found by Alzer (2018) for binomial probabilities ($d=1$). As a consequence of the log-convexity, we obtain some combinatorial inequalities for multinomial coefficients. We also show how the main result can be used to derive asymptotic formulas for quantities of interest in the context of statistical density estimation based on Bernstein polynomials on the $d$-dimensional simplex.