The Annals of Mathematical Statistics

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.

Article information

Source
Ann. Math. Statist., Volume 29, Number 1 (1958), 179-187.

Dates
First available in Project Euclid: 27 April 2007

https://projecteuclid.org/euclid.aoms/1177706714

Digital Object Identifier
doi:10.1214/aoms/1177706714

Mathematical Reviews number (MathSciNet)
MR93873

Zentralblatt MATH identifier
0089.14803

JSTOR

Citation

Birnbaum, Z. W.; Pyke, Ronald. On Some Distributions Related to the Statistic $D^+_n$. Ann. Math. Statist. 29 (1958), no. 1, 179--187. doi:10.1214/aoms/1177706714. https://projecteuclid.org/euclid.aoms/1177706714