## The Annals of Probability

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

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

#### 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