Kolmogorov statistic in the case of a piecewise-continuous distribution function (Q1116229)

Besides the discrete and continuous random variables occurring in practical work, on encounters, in a natural context, discrete-continuous r.v.'s. This is the situation when one is investigating such problems as the operating time of a system after the first renewal, transmission of a signal through a limiter-filter, the length of the intersection of a random interval with some fixed interval, and so on. The piecewise- continuous distribution functions of these r.v.'s may be expressed as mixed distributions. Indeed, if F(x) has no singular component, Lebesgue's theorem yields a decomposition \(F(x)=\Phi^ p(x)+\Phi^ c(x)\), where \(\Phi^ p(x)\) is a discrete component and \(\Phi^ c(x)\) is absolutely continuous, both \(\Phi^ p(x)\) and \(\Phi^ c(x)\) being nondecreasing functions. For example, if the jumps of F(x) are concentrated in a finite interval then, denoting \(F^ p(x)=\Phi^ p(x)/\alpha\), where \(\alpha =\lim_{x\to \infty}[F(x)-\Phi^ c(x)]\), we have \(F(x)-\alpha F^ p(x)=\Phi^ c(x)\). Since \[ \lim_{x\to \infty}[F(x)-\alpha F^ p(x)]=\lim_{x\to \infty}\Phi^ c(x)=1-\alpha, \] it follows, denoting \(F^ c(x)=\Phi^ c(x)/(1-\alpha)\), that \[ (1)\quad F(x)=\alpha F^ p(x)+(1-\alpha)F^ c(x),\quad 0<\alpha <1. \] In this paper we address the problem of testing the hypothesis that the observed r.v. has a distribution expressible as (1). The weights \(F^ c(x)\), \(F^ p(x)\) will be considered given while the observations \(\{x_ 1,x_ 2,...,x_ n\}\) are independent and identically distributed. We propose to use Kolmogorov's statistic as a measure of the deviation of the empirical distribution function from the hypothetical distribution; this statistic is given by \[ D_ n=\sqrt{n}\sup_{x}| F_ n(x)-F(x)|, \] whe \(F_ n(x)\) is the empirical distribution function corresponding to \(\{x_ 1,..,x_ n\}\).