On the Number of Crossings of Empirical Distribution Functions
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Penn collection
Statistics Papers
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asymptotic distribution
boundary crossing probability
geometric distribution
Poisson process
renewal theory
stochastic dominance algorithm
Weiner-Hopf technique
Probability
boundary crossing probability
geometric distribution
Poisson process
renewal theory
stochastic dominance algorithm
Weiner-Hopf technique
Probability
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Nair, Vijayan N
Shepp, Larry A
Klass, Michael J
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Abstract
Let F and G be two continuous distribution functions that cross at a finite number of points − ∞ ≤ t1 < ⋯ < tk ≤ ∞. We study the limiting behavior of the number of times the empirical distribution function Gn crosses F and the number of times Gn crosses Fn. It is shown that these variables can be represented, as n → ∞, as the sum of k independent geometric random variables whose distributions depend on F and G only through F′(ti)/G′(ti), i = 1, …, k. The technique involves approximating Fn(t) and Gn(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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1986
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The Annals of Probability