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Maximizing the probability of correctly ordering random variables using linear predictors

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  • Portnoy, Stephen

Abstract

Let (T1, x1), (T2, x2), ..., (Tn, xn) be a sample from a multivariate normal distribution where Ti are (unobservable) random variables and xi are random vectors in Rk. If the sample is either independent and identically distributed or satisfies a multivariate components of variance model, then the probability of correctly ordering {Ti} is maximized by ranking according to the order of the best linear predictors {E(Tixi)}. Furthermore, it orderings are chosen according to linear functions {b'xi} then the conditional probability of correct order given (Ti = t1; I = 1, ..., n) is maximized when b'xi is the best linear predictor. Examples are given to show that linear predictors may not be optimal and that using a linear combination other that the best linear predictor may give a greater probability of correctly ordering {Ti} if {(Ti, xi)} are independent but not identically distributed, or if the distributions are not normal.

Suggested Citation

  • Portnoy, Stephen, 1982. "Maximizing the probability of correctly ordering random variables using linear predictors," Journal of Multivariate Analysis, Elsevier, vol. 12(2), pages 256-269, June.
    • Handle: RePEc:eee:jmvana:v:12:y:1982:i:2:p:256-269
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    Cited by:

    1. Jiaying Gu & Roger Koenker, 2023. "Invidious Comparisons: Ranking and Selection as Compound Decisions," Econometrica, Econometric Society, vol. 91(1), pages 1-41, January.
    2. Jiaying Gu & Roger Koenker, 2020. "Invidious Comparisons: Ranking and Selection as Compound Decisions," Papers 2012.12550, arXiv.org, revised Sep 2021.

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