A Monotonicity Property of a New Bernstein Type Operator

TL;DR

This paper proves a monotonicity property of a new Bernstein-type operator by analyzing the Pólya urn distribution with negative replacement, showing stochastic ordering and monotonicity of the operator.

Contribution

It introduces a monotonicity property for a new Bernstein operator based on Pólya urn distribution with negative replacement, extending known properties.

Findings

Probabilistic proof of monotonicity for Pólya urn distribution with negative replacement.

Establishment of stochastic ordering with respect to initial distribution parameter.

Identification of a refined reversed Cauchy-Bunyakovsky-Schwarz inequality.

Abstract

In the present paper we prove that the probabilities of the P\'olya urn distribution (with negative replacement) satisfy a monotonicity property similar to that of the binomial distribution (P\'olya urn distribution with no replacement). As a consequence, we show that the random variables with P\'olya urn distribution (with negative replacement) are stochastically ordered with respect to the parameter giving the initial distribution of the urn. An equivalent formulation of this result shows that the new Bernstein operator recently introduced in [3] is a monotone operator. The proofs are probabilistic in spirit and rely on various inequalities, some of which are of independent interest (e.g. a refined version of the reversed Cauchy-Bunyakovsky-Schwarz inequality).