## The Annals of Mathematical Statistics

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

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

Z. W. Birnbaum and Ronald Pyke

#### 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

**Permanent link to this document**

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**

links.jstor.org

#### 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