## The Annals of Statistics

- Ann. Statist.
- Volume 7, Number 4 (1979), 788-794.

### Asymptotic Normality of Permutation Statistics Derived from Weighted Sums of Bivariate Functions

C. P. Shapiro and Lawrence Hubert

#### Abstract

Statistics of the form $H_n = \sum d_{ijn}h_n(X_i, X_j)$ are considered, where $X_1, X_2, \cdots$, are independent and identically distributed random variables, the diagonal terms, $d_{iin}$, are equal to zero, and $h_n(x, y)$ is a symmetric real valued function. The asymptotic normality of such statistics is proven and the result then combined with work of Jogdeo on statistics that are weighted sums of bivariate functions of ranks to find sufficient conditions for asymptotic normality of permutation statistics derived from $H_n$.

#### Article information

**Source**

Ann. Statist., Volume 7, Number 4 (1979), 788-794.

**Dates**

First available in Project Euclid: 12 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aos/1176344728

**Digital Object Identifier**

doi:10.1214/aos/1176344728

**Mathematical Reviews number (MathSciNet)**

MR532242

**Zentralblatt MATH identifier**

0423.62020

**JSTOR**

links.jstor.org

**Subjects**

Primary: 62E20: Asymptotic distribution theory

Secondary: 62E15: Exact distribution theory

**Keywords**

Nonparametric permutation distribution clustering statistics

#### Citation

Shapiro, C. P.; Hubert, Lawrence. Asymptotic Normality of Permutation Statistics Derived from Weighted Sums of Bivariate Functions. Ann. Statist. 7 (1979), no. 4, 788--794. doi:10.1214/aos/1176344728. https://projecteuclid.org/euclid.aos/1176344728