## The Annals of Statistics

### An Efron-Stein Inequality for Nonsymmetric Statistics

J. Michael Steele

#### Abstract

If $S(x_1, x_2,\cdots, x_n)$ is any function of $n$ variables and if $X_i, \hat{X}_i, 1 \leq i \leq n$ are $2n$ i.i.d. random variables then $\operatorname{var} S \leq \frac{1}{2} E \sum^n_{i=1} (S - S_i)^2$ where $S = S(X_1, X_2,\cdots, X_n)$ and $S_i$ is given by replacing the $i$th observation with $\hat{X}_i$, so $S_i = S(X_1, X_2,\cdots, \hat{X}_i,\cdots, X_n)$. This is applied to sharpen known variance bounds in the long common subsequence problem.

#### Article information

Source
Ann. Statist., Volume 14, Number 2 (1986), 753-758.

Dates
First available in Project Euclid: 12 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aos/1176349952

Digital Object Identifier
doi:10.1214/aos/1176349952

Mathematical Reviews number (MathSciNet)
MR840528

Zentralblatt MATH identifier
0604.62017

JSTOR
links.jstor.org

#### Citation

Steele, J. Michael. An Efron-Stein Inequality for Nonsymmetric Statistics. Ann. Statist. 14 (1986), no. 2, 753--758. doi:10.1214/aos/1176349952. https://projecteuclid.org/euclid.aos/1176349952