Maximizing the probability of correctly ordering random variables using linear predictors |
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Authors: | Stephen Portnoy |
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Institution: | Department of Mathematics, University of Illinois, Urbana, Illinois 61801 USA |
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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(Ti|xi)}. 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. |
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Keywords: | 62J05 Linear predictors selection index multivariate normal |
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