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

Let d is an element of N and let gamma(i) is an element of [0, infinity), x(i) is an element of (0,1) be such that Sigma(d+1)(i=1) gamma(i) = M is an element of (0, infinity) and Sigma(d+1)(i=1) x(i) = 1. We prove that a bar right arrow Gamma(aM + 1)/Pi(d+1)(i=1) Gamma(a gamma(i) + 1) Pi(d+1)(i=1) x(i)(a gamma i) is completely monotonic on (0, infinity). This result generalizes the one found by Alzer [2] 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. (C) 2018 Elsevier Inc. All rights reserved.