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Stochastic orders in preservation properties by Bernstein-type operators

Part of: Distribution theory - Probability Approximations and expansions

Published online by Cambridge University Press:  01 July 2016

Jose A. Adell  and
Ana Perez-Palomares
Jose A. Adell*
Affiliation:
Universidad de Zaragoza
Ana Perez-Palomares*
Affiliation:
Universidad de Zaragoza
*
∗ Postal address: Departamento de Métodos Estadísticos, Facultad de Ciencias, Universidad de Zaragoza, 50009 Zaragoza, Spain.
∗ Postal address: Departamento de Métodos Estadísticos, Facultad de Ciencias, Universidad de Zaragoza, 50009 Zaragoza, Spain.
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Abstract

In this paper, we are concerned with preservation properties of first and second order by an operator L representable in terms of a stochastic process Z with non-decreasing right-continuous paths. We introduce the derived operator D of L and the derived process V of Z in order to characterize the preservation of absolute continuity and convexity. To obtain different characterizations of the preservation of convexity, we introduce two kinds of duality, the first referring to the process Z and the second to the derived process V. We illustrate the preceding results by considering some examples of interest both in probability and in approximation theory - namely, mixtures, centred subordinators, Bernstein polynomials and beta operators. In most of them, we find bidensities to describe the duality between the derived processes. A unified approach based on stochastic orders is given.

Keywords

Stochastic orderpreservation propertiesBernstein-type operatorsderived processdual processbidensitysubordinator

MSC classification

Primary: 60E15: Inequalities; stochastic orderings
Secondary: 41A36: Approximation by positive operators
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 31 , Issue 2 , June 1999 , pp. 492 - 509
Copyright
Copyright © Applied Probability Trust 1999 

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Footnotes

Research supported by the DGICYT PB95-0809 Grant.

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