On the Power of a One-Sided Test of Fit for Continuous Probability Function

Abstract

If $F(x)$ is a continuous distribution function of a random variable $X$, and $F_n(x)$ the empirical distribution function determined by a sample $X_1, X_2, \cdots, X_n$, then the probability $\mathrm{Pr}\{F(x) \leqq F_n(x) + \epsilon \text{for all} x\}$ is known [1] to be a function $P_n(\epsilon)$, independent of $F(x)$. A closed expression for $P_n(\epsilon)$ and a table of some of its values were presented in [2]. In the present paper $P_n(\epsilon)$ is used to test a hypothesis $F(x) = H(x)$ against an alternative $F(x) = G(x)$. The power of this test is studied and sharp upper and lower bounds for it are obtained for alternatives such that $\sup_{-\infty < x < + \infty}\{H(x) - G(x)\} = \delta$, with preassigned $\delta$. The results of [2] are assumed known.