We previously showed that c w c x thus it follows that

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Unformatted text preview: (X). Thus, it follows that C (W ) = C (X). Copyright c 2012 Dan Nettleton (Iowa State University) Statistics 511 8 / 27 Estimation of E(y) A fundamental goal of linear model analysis is to estimate E(y). We could, of course, use y to estimate E(y). y is obviously an unbiased estimator of E(y), but it is often not a very sensible estimator. For example, suppose y1 y2 = 1 1 µ+ Should we estimate E(y) = Copyright c 2012 Dan Nettleton (Iowa State University) 1 , and we observe y = 2 µ µ by y = 6.1 2.3 . 6.1 ? 2.3 Statistics 511 9 / 27 Estimation of E(y) The Gauss-Markov linear models says that E(y) ∈ C (X), so we should use that information when estimating E(y). Consider estimating E(y) by the point in C (X) that is closest to y (as measured by the usual Euclidean distance). This unique point is called the orthogonal projection of y onto C (X) and denoted by ˆ (although it could be argued that E(y) might be y better notation). By definition, ||y − ˆ|| = minz∈C (X) ||y − z||, where ||a|| ≡ y Copyright c 2012 Dan Nettleton (Iowa State University) n 2 i=1 ai . Statistics 511 10 / 27 Orthogonal Projection Matrices In Homework Assignment 2, we will formally prove the following: 1 ∀ y ∈ IRn , ˆ = PX y, where PX is a unique n × n matrix known as an y orthogonal projection matrix. 2 PX is idempotent: PX PX = PX . 3 PX is symmetric: PX = PX . 4 PX X = X and X PX = X . 5 PX = X(X X)− X , where (X X)− is any generalized inverse of X X. Copyright c 2012 Dan Nettleton (Iowa State University) Statistics 511 11 / 27 Why Does PX X = X? PX X = PX [x1 , . . . , xp ] = [ P X x1 , . . . , P X xp ] = [ x1 , . . . , xp ] = X. Copyright c 2012 Dan Nettleton (Iowa State University) Statistics 511 12 / 27 Generalized Inverses G is a generalized inverse of a matrix A if AGA = A. We usually denote a generalized inverse of A by A− . If A is nonsingular, i.e., if A−1 exists, then A−1 is the one and only generalized inverse of A. AA−1 A = AI = IA = A If A is singular, i.e., if A−1 does not...
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This document was uploaded on 03/27/2014.

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