22BLUP(1) - Prediction of random variables Key distinction...

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Prediction of random variables Key distinction between fixed and random effects: Estimate means of fixed effects Estimate variance of random effects But in some instances, want to predict FUTURE values of a random effect Example (from Efron and Morris, 1975, JASA 70:311-319): Baseball players. Given a player’s performance in the beginning of the season, predict performance in rest of season. c 2011 Dept. Statistics (Iowa State University) Stat 511 section 22 1 / 15 c 2011 Dept. Statistics (Iowa State University) Stat 511 section 22 2 / 15 c 2011 Dept. Statistics (Iowa State University) Stat 511 section 22 3 / 15 c 2011 Dept. Statistics (Iowa State University) Stat 511 section 22 4 / 15
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Best predictor is found by “Shrinking” obs. performance towards overall mean. So, how much shrinkage is needed? How do we compute optimal predictor? General answer using a linear mixed effects model y = X β + Zu + , where u N 0 0 , G 0 0 R Given data y , what is our best guess for values in the unobserved vector u ? c 2011 Dept. Statistics (Iowa State University) Stat 511 section 22 5 / 15 Because u is a random vector rather than a fixed parameter, we talk about predicting u rather than estimating u . We seek a Best Linear Unbiased Predictor (BLUP) for u , which we will denote by ˆ u To be a BLUP, we require 1. ˆ u is a linear function of y . 2. ˆ u is unbiased for u so that E ( ˆ u u ) = 0 . 3. Var ( ˆ u u ) is no “larger” than Var ( v u ) , where v is any other linear and unbiased predictor. It turns out that the BLUP for u is the BLUE of E ( u | y ) .
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