The Annals of Probability

On Upper and Lower Bounds for the Variance of a Function of a Random Variable

Theophilos Cacoullos

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Abstract

Chernoff (1981) obtained an upper bound for the variance of a function of a standard normal random variable, using Hermite polynomials. Chen (1980) gave a different proof, using the Cauchy-Schwarz inequality, and extended the inequality to the case of a multivariate normal. Here it is shown how similar upper bounds can be obtained for other distributions, including discrete ones. Moreover, by using a variation of the Cramer-Rao inequality, analogous lower bounds are given for the variance of a function of a random variable which satisfies the usual regularity conditions. Matrix inequalities are also obtained. All these bounds involve the first two moments of derivatives or differences of the function.

Article information

Source
Ann. Probab., Volume 10, Number 3 (1982), 799-809.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1176993788

Digital Object Identifier
doi:10.1214/aop/1176993788

Mathematical Reviews number (MathSciNet)
MR659549

Zentralblatt MATH identifier
0492.60021

JSTOR
links.jstor.org

Subjects
Primary: 60E05: Distributions: general theory
Secondary: 62F10: Point estimation 62H99: None of the above, but in this section

Keywords
Variance bounds Cramer-Rao inequality

Citation

Cacoullos, Theophilos. On Upper and Lower Bounds for the Variance of a Function of a Random Variable. Ann. Probab. 10 (1982), no. 3, 799--809. doi:10.1214/aop/1176993788. https://projecteuclid.org/euclid.aop/1176993788


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