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The asymptotic distributions of statistics based on logarithms of spacings. (English) Zbl 0805.62019

Let \(V_ 1, V_ 2,\dots, V_{n+1}\) denote the \(n+1\) spacings of a random sample of size \(n\) from a density \(f\) on the interval \([0,1]\). The asymptotic distributions of \(\Sigma\log V_ i\) and \(\Sigma V_ i\log V_ i\) are obtained when \(f\) is a step function. The author’s results thus generalize those of D. A. Darling [Ann. Math. Statistics 24, 239-253 (1953; Zbl 0053.099)] and J. R. Gebert and B. K. Kale [Stat. Hefte, N. F. 10, 192-200 (1969; Zbl 0179.241)] who were concerned with the case of uniform \(f\).

MSC:

62E20 Asymptotic distribution theory in statistics
62G30 Order statistics; empirical distribution functions
62G20 Asymptotic properties of nonparametric inference
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