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1981•The Annals of StatisticsOpen access

Asymptotic Normality of Linear Combinations of Order Statistics with a Smooth Score Function

David M. Mason

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Abstract

Asymptotic normality of linear combinations of order statistics of the form $T_n = n^{-1} \sum J(i/(n + 1))X_{in}$ is investigated along with a slightly trimmed version of $T_n$. Theorem 5 of Stigler (1974) is extended to show asymptotic normality of $T_n$ for a wide class of score functions. In addition, a proof of Theorem 4 of Stigler (1974) is given.

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Asymptotic normality of linear combinations of order statistics of the form $T_n = n^{-1} \sum J(i/(n + 1))X_{in}$ is investigated along with a slightly trimmed version of $T_n$. Theorem 5 of Stigler (1974) is extended to show asymptotic normality of $T_n$ for a wide class of score functions. In addition, a proof of Theorem 4 of Stigler (1974) is given.

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Available abstract

Asymptotic normality of linear combinations of order statistics of the form $T_n = n^{-1} \sum J(i/(n + 1))X_{in}$ is investigated along with a slightly trimmed version of $T_n$. Theorem 5 of Stigler (1974) is extended to show asymptotic normality of $T_n$ for a wide class of score functions. In addition, a proof of Theorem 4 of Stigler (1974) is given.

Key concepts: Mathematics, Asymptotic distribution, Normality, Order statistic, Asymptotic analysis, Local asymptotic normality, Order (exchange), Statistics

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