Geodesic
Exact and Limiting Probability Distributions of Some Smirnov Type Statistics
Canadian mathematical bulletin, Tome 8 (1965) no. 1, pp. 93-103

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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. 
Csörgo, Miklós. Exact and Limiting Probability Distributions of Some Smirnov Type Statistics. Canadian mathematical bulletin, Tome 8 (1965) no. 1, pp. 93-103. doi: 10.4153/CMB-1965-012-5
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[1] 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] 2. Feller, William (1948), On the Kolmogorov-Smirnov limit theorems for empirical distributions, Ann. Math. Statist. 19, pp.177-189. Google Scholar

[3] 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] 4. Smirnov, N. (1939), Sur les écarts de la courbe de distribution empirique, Rec. Math. (Mat. Sbornik), 6, pp. 3-26. Google Scholar

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