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Exact and Limiting Probability Distributions of Some Smirnov Type Statistics

Published online by Cambridge University Press:  20 November 2018

Miklós Csörgo
Miklós Csörgo*
Affiliation:
Princeton University
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Let F(x) be the continuous distribution function of a random variable X and Fn(x) be the empirical distribution function determined by a random sample X1, …, Xn taken on X. Using the method of Birnbaum and Tingey [1] we are going to derive the exact distributions of the random variables

and and where the indicated sup' s are taken over all x' s such that -∞ < x < xb and xa ≤ x < + ∞ with F(xb) = b, F(xa) = a in the first two cases and over all x' s so that Fn(x) ≤ b and a ≤ Fn(x) in the last two cases.

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 8 , Issue 1 , February 1965 , pp. 93 - 103
Copyright
Copyright © Canadian Mathematical Society 1965

References

1. Birnbaum, Z. W. and Tingey, Fred H. (1951), One-sided confidence contours for probability distribution functions, Ann. Math. Statist. 22, pp. 592-596.Google Scholar
2. Feller, William (1948), On the Kolmogorov-Smirnov limit theorems for empirical distributions, Ann. Math. Statist. 19, pp.177-189.Google Scholar
3. Manija, G. M. (1949), Obobschenije kriterija A.N. Kolmogorova dlja otcenki zakona racpredelenija po empiricheskim dannym, Dokl. Akad. Nauk. SSSR 69, pp. 495-497.Google Scholar
4. Smirnov, N. (1939), Sur les écarts de la courbe de distribution empirique, Rec. Math. (Mat. Sbornik), 6, pp. 3-26.Google Scholar