Open Access
July, 1986 On the Number of Crossings of Empirical Distribution Functions
Vijayan N. Nair, Lawrence A. Shepp, Michael J. Klass
Ann. Probab. 14(3): 877-890 (July, 1986). DOI: 10.1214/aop/1176992444


Let $F$ and $G$ be two continuous distribution functions that cross at a finite number of points $-\infty \leq t_1 < \cdots < t_k \leq \infty$. We study the limiting behavior of the number of times the empirical distribution function $G_n$ crosses $F$ and the number of times $G_n$ crosses $F_n$. It is shown that these variables can be represented, as $n \rightarrow \infty$, as the sum of $k$ independent geometric random variables whose distributions depend on $F$ and $G$ only through $F'(t_i)/G'(t_i), i = 1, \ldots, k$. The technique involves approximating $F_n(t)$ and $G_n(t)$ locally by Poisson processes and using renewal-theoretic arguments. The implication of the results to an algorithm for determining stochastic dominance in finance is discussed.


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Vijayan N. Nair. Lawrence A. Shepp. Michael J. Klass. "On the Number of Crossings of Empirical Distribution Functions." Ann. Probab. 14 (3) 877 - 890, July, 1986.


Published: July, 1986
First available in Project Euclid: 19 April 2007

zbMATH: 0593.60047
MathSciNet: MR841590
Digital Object Identifier: 10.1214/aop/1176992444

Primary: 60G17
Secondary: 60E05

Keywords: asymptotic distribution , boundary crossing probability , geometric distribution , Poisson process , renewal theory , stochastic dominance algorithm , Wiener-Hopf technique

Rights: Copyright © 1986 Institute of Mathematical Statistics

Vol.14 • No. 3 • July, 1986
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