Asymptotic properties of Cramér-Smirnov statistics--A new approach

Let Y1,..., Yn be independent identically distributed random variables with distribution function F(x, [theta]), [theta] = ([theta]'1, [theta]'2), where [theta]i (i = 1, 2) is a vector of pi components, p = p1 + p2 and for [for all][theta][set membership, variant]I, an open interval in p, F(x, [theta]) is continuous. In the present paper the author shows that the asymptotic distribution of modified Cramér-Smirnov statistic under Hn: [theta]1 = [theta]10 + n-1/2[gamma], [theta]2 unspecified, where [gamma] is a given vector independent of n, is the distribution of a sum of weighted noncentral [chi]12 variables whose weights are eigenvalues of a covariance function of a Gaussian process and noncentrality parameters are Fourier coefficients of the mean function of the Gaussian process. Further, the author exploits the special form of the covariance function by using perturbation theory to obtain the noncentrality parameters and the weights. The technique is applicable to other goodness-of-fit statistics such as U2 [G. S. Watson, Biometrika 48 (1961), 109-114].