Abstract
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.
Citation
Prakash C. Joshi. "Recurrence Relations for the Mixed Moments of Order Statistics." Ann. Math. Statist. 42 (3) 1096 - 1098, June, 1971. https://doi.org/10.1214/aoms/1177693339
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