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March, 1973 Asymptotic Distributions for Quadratic Forms with Applications to Tests of Fit
T. de Wet, J. H. Venter
Ann. Statist. 1(2): 380-387 (March, 1973). DOI: 10.1214/aos/1176342378

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.

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T. de Wet. J. H. Venter. "Asymptotic Distributions for Quadratic Forms with Applications to Tests of Fit." Ann. Statist. 1 (2) 380 - 387, March, 1973. https://doi.org/10.1214/aos/1176342378

Information

Published: March, 1973
First available in Project Euclid: 12 April 2007

zbMATH: 0256.62018
MathSciNet: MR353543
Digital Object Identifier: 10.1214/aos/1176342378

Subjects:
Primary: 62E20
Secondary: 60F99

Rights: Copyright © 1973 Institute of Mathematical Statistics

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Vol.1 • No. 2 • March, 1973
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