The Annals of Mathematical Statistics

Recurrence Relations for the Mixed Moments of Order Statistics

Prakash C. Joshi

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Let $X_1, X_2, \cdots, X_n$ be a random sample of size $n$ from a continuous distribution with cdf $P(x)$ and pdf $p(x)$. Let $X_{1:n} \leqq X_{2:n} \leqq \cdots \leqq X_{n:n}$ be the corresponding order statistics. Denote the first moment $E(X_{r:n})$ by $\mu_{r:n} (1 \leqq r \leqq n)$ and the mixed moment $E(X_{r:n}, X_{s:n})$ by $\mu_{r,s:n} (1 \leqq r \leqq s \leqq n)$. We assume that all these moments exist. Several recurrence relations between these moments are summarized by Govindarajulu [1]. In this note, we give a simple argument which generalizes some of the results given in [1]. These generalizations then lead to some modifications in the theorems given by Govindarajulu.

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Ann. Math. Statist., Volume 42, Number 3 (1971), 1096-1098.

First available in Project Euclid: 27 April 2007

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Joshi, Prakash C. Recurrence Relations for the Mixed Moments of Order Statistics. Ann. Math. Statist. 42 (1971), no. 3, 1096--1098. doi:10.1214/aoms/1177693339.

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