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.