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- W2032085328 abstract "Let $X_1^n leqq X_2^n leqq cdots leqq X_n^n$ be the order statistics of a size $n$ sample from any distribution function $F$ not necessarily continuous. Let $alpha_j, beta_j, (j = 1,2, cdots, n)$ be any numbers. Let $P_n = P(alpha_j < X_j^n leqq beta_j, j = 1,2, cdots, n)$. A recursion is given which calculates $P_n$ for any $F$ and any $alpha_j, beta_j$. Suppose now that $F$ is continuous. A two-sided statistic of Kolmogorov-Smirnov type has the distribution function $P_{mathrm{KS}} = Plbracksup n^{frac{1}{2}}psi(F) cdot |F^n - F| leqq lambdarbrack$, where $F^n$ is the empirical distribution function of the sample and $psi(x)$ is any nonnegative weight function. As $P_{mathrm{KS}}$ has the form $P_n$, its calculation as a function of $lambda$ can be carried out by means of the recursion. This has been done for the case $psi(x) = lbrack x(1 - x)rbrack^{-frac{1}{2}}$. Curves are given which represent $lambda$ versus $1 - P_{mathrm{KS}}$ for $n = 1,2, 10, 100$. From additional computations, the precision of a truncated development of $1 - P_{mathrm{KS}}$ in powers of $lambda^{-2}$ has been determined." @default.
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- W2032085328 date "1972-02-01" @default.
- W2032085328 modified "2023-09-27" @default.
- W2032085328 title "The Calculation of Distributions of Two-Sided Kolmogorov-Smirnov Type Statistics" @default.
- W2032085328 doi "https://doi.org/10.1214/aoms/1177692700" @default.
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