Covariance between variables and their order statistics for multivariate normal variables

Yosef Rinott; Ester Samuel-Cahn

doi:10.1016/0167-7152(94)90223-2

Covariance between variables and their order statistics for multivariate normal variables

Yosef Rinott^*
, Ester Samuel-Cahn

^*Corresponding author for this work

Research output: Contribution to journal › Article › peer-review

12 Scopus citations

Abstract

Siegel (1993, J. Amer. Statist. Assoc. 88, 77-80) showed that when (X₁, ..., Xn) have a multivariate normal distribution then Cov(X₁, X₍₁₎) = Σⁿ_{i = 1} Cov(X₁, X_i)P(X_i = X₍₁₎), where X₍₁₎ is the minimum of (X₁, ..., X_n). We show that a similar result holds for any order statistic. Thus X₍₁₎ can be replaced by X_(r), the rth order statistic, everywhere in the above formula. Normality is essentially also necessary for this result to hold.

Original language	English
Pages (from-to)	153-155
Number of pages	3
Journal	Statistics and Probability Letters
Volume	21
Issue number	2
DOIs	https://doi.org/10.1016/0167-7152(94)90223-2
State	Published - 23 Sep 1994
Externally published	Yes

Keywords

Multivariate normal
Order statistics

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10.1016/0167-7152(94)90223-2

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title = "Covariance between variables and their order statistics for multivariate normal variables",

abstract = "Siegel (1993, J. Amer. Statist. Assoc. 88, 77-80) showed that when (X1, ..., Xn) have a multivariate normal distribution then Cov(X1, X(1)) = Σni = 1 Cov(X1, Xi)P(Xi = X(1)), where X(1) is the minimum of (X1, ..., Xn). We show that a similar result holds for any order statistic. Thus X(1) can be replaced by X(r), the rth order statistic, everywhere in the above formula. Normality is essentially also necessary for this result to hold.",

keywords = "Multivariate normal, Order statistics",

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T1 - Covariance between variables and their order statistics for multivariate normal variables

AU - Rinott, Yosef

AU - Samuel-Cahn, Ester

PY - 1994/9/23

Y1 - 1994/9/23

N2 - Siegel (1993, J. Amer. Statist. Assoc. 88, 77-80) showed that when (X1, ..., Xn) have a multivariate normal distribution then Cov(X1, X(1)) = Σni = 1 Cov(X1, Xi)P(Xi = X(1)), where X(1) is the minimum of (X1, ..., Xn). We show that a similar result holds for any order statistic. Thus X(1) can be replaced by X(r), the rth order statistic, everywhere in the above formula. Normality is essentially also necessary for this result to hold.

AB - Siegel (1993, J. Amer. Statist. Assoc. 88, 77-80) showed that when (X1, ..., Xn) have a multivariate normal distribution then Cov(X1, X(1)) = Σni = 1 Cov(X1, Xi)P(Xi = X(1)), where X(1) is the minimum of (X1, ..., Xn). We show that a similar result holds for any order statistic. Thus X(1) can be replaced by X(r), the rth order statistic, everywhere in the above formula. Normality is essentially also necessary for this result to hold.

KW - Multivariate normal

KW - Order statistics

UR - https://www.scopus.com/pages/publications/0000389665

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SP - 153

EP - 155

JO - Statistics and Probability Letters

JF - Statistics and Probability Letters

IS - 2

ER -

Covariance between variables and their order statistics for multivariate normal variables

Abstract

Keywords

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