On Some Distributions Related to the Statistic $D^+_n$

Abstract

Let $X_1 < X_2 < \cdots < X_n$ be a sample of size $n$, ordered increasingly, of a one-dimensional random variable $X$ which has the continuous cumulative distribution function $F$. It is well known, [1], that the statistic \begin{equation*}\tag{1}D^+_n = \sup_{-\infty < x < + \infty} \{F_n(x) - F(x)\},\end{equation*} where $F_n(x)$ is the empirical distribution function determined by $X_1, X_2, \cdots, X_n$, has a probability distribution independent of $F$. One may, therefore, assume that $X$ has the uniform distribution in (0, 1) and, observing that the supremum in (1) must be attained at one of the sample points, write without loss of generality \begin{equation*}\tag{2}D^+_n = \max_{1 \leqq i \leqq n} (i/n - U_i),\end{equation*} where $U_1 < U_2 < \cdots < U_n$ is an ordered sample of a random variable with uniform distribution in (0, 1). For a given $n > 0$ define the random variable $i^{\ast}$ as that value of $i$, determined uniquely with probability 1, for which the maximum in (2) is reached, i.e., such that \begin{equation*}\tag{3}D^+_n = i^{\ast}/n - U_{i^{\ast}},\end{equation*} and write \begin{equation*}\tag{3.1} U_{i^{\ast}} = U^{\ast}.\end{equation*} The main object of this paper is to obtain the distribution functions of $(i^{\ast}, U^{\ast})$, of $i^{\ast}$ and of $U^{\ast}$. The asymptotic distribution of $\alpha_n = i^{\ast}/n$ is also investigated, and bounds are obtained on the difference between the exact and the asymptotic distribution. A number of general identities, which are not commonly known, have been verified and used in proving the above-mentioned results. Since these identities may be helpful in other problems of this type, they are separated from the main proofs and appear in the next section.