In the present paper, we prove that the probabilities of the Pólya urn distribution (with negative replacement) satisfy a monotonicity property similar to that of the binomial distribution. As a consequence, we show that the corresponding random variables 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–Stancu-type operator introduced in (Pascu et al. in Proc. Rom. Acad., Ser. A: Math. Phys. Tech. Sci. Inf. Sci. 2019, in press) 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 or estimates of the error of approximating an integral by the trapezoidal rule).
Journal of Inequalities and Applications
Digital Object Identifier (DOI)
Tripşa, Florenţa and Pascu, Nicolae R., "Stochastic ordering of Pólya random variables and monotonicity of the Bernstein–Stancu operator for a negative parameter" (2019). Faculty Publications. 4329.