On linear failure rates : Anthony A. Salvia. IEEE Trans. Reliab.R-27, (2) 145 (June 1978)

Abstract

Let Y1,…, Yn be independent identically distributed random variables with distribution function F(x, θ), θ = (θ′1, θ′2), where θi (i = 1, 2) is a vector of pi components, p = p1 + p2 and for ∀θ∈I, an open interval in Rp, F(x, θ) is continuous. In the present paper the author shows that the asymptotic distribution of modified Cramér-Smirnov statistic under Hn: θ1 = θ10 + n−1/2γ, θ2 unspecified, where γ is a given vector independent of n, is the distribution of a sum of weighted noncentral χ12 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].