## The Annals of Statistics

### Asymptotic Distributions for Quadratic Forms with Applications to Tests of Fit

#### Abstract

Let $Z_1, Z_2,\cdots$, be independent and identically distributed random variables and $\{c_{ijn}\}$ real numbers; put $T_n = \sum^n_{i,j = 1} c_{ijn}Z_iZ_j$. This paper gives conditions under which the distribution of $T_n - ET_n$ converges to the distribution of $\sum \Upsilon_m(Y_m^2 - 1)$ with $\{\Upsilon_m\}$ a real sequence and $Y_1, Y_2,\cdots$ independent $N(0, 1)$ random variables. The results are applied to the calculation of the asymptotic distributions of test criteria of the form $Q_n^W = \sum \lbrack F_0(X_{kn}) - k/n + 1\rbrack^2W(k/n + 1)$ for testing the hypothesis that $X_{1n}, X_{2n},\cdots, X_{nn}$ are the order statistics of an independent sample from the distribution function $F_0$; here $W$ is a weight function.

#### Article information

Source
Ann. Statist., Volume 1, Number 2 (1973), 380-387.

Dates
First available in Project Euclid: 12 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aos/1176342378

Digital Object Identifier
doi:10.1214/aos/1176342378

Mathematical Reviews number (MathSciNet)
MR353543

Zentralblatt MATH identifier
0256.62018

JSTOR
links.jstor.org

Subjects
Primary: 62E20: Asymptotic distribution theory
Secondary: 60F99: None of the above, but in this section

#### Citation

de Wet, T.; Venter, J. H. Asymptotic Distributions for Quadratic Forms with Applications to Tests of Fit. Ann. Statist. 1 (1973), no. 2, 380--387. doi:10.1214/aos/1176342378. https://projecteuclid.org/euclid.aos/1176342378