One-Sided Confidence Contours for Probability Distribution Functions

Abstract

Let $F(x)$ be the continuous distribution function of a random variable $X,$ and $F_n(x)$ the empirical distribution function determined by a sample $X_1, X_2, \cdots, X_n$. It is well known that the probability $P_n(\epsilon)$ of $F(x)$ being everywhere majorized by $F_n(x) + \epsilon$ is independent of $F(x)$. The present paper contains the derivation of an explicit expression for $P_n(\epsilon)$, and a tabulation of the 10%, 5%, 1%, and 0.1% points of $P_n(\epsilon)$ for $n =$ 5, 8, 10, 20, 40, 50. For $n =$ 50 these values agree closely with those obtained from an asymptotic expression due to N. Smirnov.