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May 2016 Parametric Stein operators and variance bounds
Christophe Ley, Yvik Swan
Braz. J. Probab. Stat. 30(2): 171-195 (May 2016). DOI: 10.1214/14-BJPS271


Stein operators are (differential/difference) operators which arise within the so-called Stein’s method for stochastic approximation. We propose a new mechanism for constructing such operators for arbitrary (continuous or discrete) parametric distributions with continuous dependence on the parameter. We provide explicit general expressions for location, scale and skewness families. We also provide a general expression for discrete distributions. We use properties of our operators to provide upper and lower variance bounds (only lower bounds in the discrete case) on functionals $h(X)$ of random variables $X$ following parametric distributions. These bounds are expressed in terms of the first two moments of the derivatives (or differences) of $h$. We provide general variance bounds for location, scale and skewness families and apply our bounds to specific examples (namely the Gaussian, exponential, gamma and Poisson distributions). The results obtained via our techniques are systematically competitive with, and sometimes improve on, the best bounds available in the literature.


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Christophe Ley. Yvik Swan. "Parametric Stein operators and variance bounds." Braz. J. Probab. Stat. 30 (2) 171 - 195, May 2016.


Received: 1 September 2013; Accepted: 1 November 2014; Published: May 2016
First available in Project Euclid: 31 March 2016

zbMATH: 1376.60044
MathSciNet: MR3481100
Digital Object Identifier: 10.1214/14-BJPS271

Keywords: Chernoff inequality , Cramér–Rao inequality , parameter of interest , Stein characterization , Stein’s method

Rights: Copyright © 2016 Brazilian Statistical Association


Vol.30 • No. 2 • May 2016
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