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

Abstract

x aγi i is completely monotonic on (0, ∞). 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.

Some logarithmically completely monotonic functions and inequalities for multinomial coefficients and multivariate beta functions

“…Remark 5.3. This paper is a revised version of the preprint [33] whose first version was announced almost at the same time as the preprint [21] which has been formally published as [20]. This paper is a companion of the papers [22,23,28,31,32].…”

From inequalities involving exponential functions and sums to logarithmically complete monotonicity of ratios of gamma functions

“…In [20, Lemma 1.4] , the inequality (1.1) was generalized as one which can be reformulated as where and such that .…”

Monotonicity properties for a ratio of finite many gamma functions

“…This paper is motivated by a series of papers [2,11,12,16,19,21,27,29,32]. For detailed review and survey, please read the papers [19,27,29,32] and closely related references therein.…”