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Unformatted text preview: STAT424 Spring 2010 Homework #2 Feb 9, 2010 Homework 2 Due: Tuesday, Feb 16, 2010 1) Consider the linear model y 1 y 2 y 3 y 4 = μ + α + β 1 μ- α + β 1 μ + α + β 2 μ- α + β 2 + , E ( ) = 0 , Cov( ) = σ 2 I . (a) Write down the design matrix X . Find a basis for C ( X ). (b) Find the projection matrix M . Is the trace of M equal to the dimension of C ( X )? (c) Find the LS projection ˆ y simplified as much as possible. (d) Decide which of the following parameters are estimable, and for the ones that are, find an unbiased estimate. (i) α (ii) μ + β 1 (iii) β 1- β 2 (iv) β 1 + β 2 (e) Find the LS estimate of θ = E [( y 1 + y 4 ) / 2]. 2) Consider a two-way ANOVA model (without interaction) y ij = μ + α i + β j + ij where i = 1 : 3, j = 1 : 2, and ij are i.i.d. random variables with mean 0 and variance σ 2 . The projection of y (that you’ll learn) is given by ˆ y ij = ¯ y i · + ¯ y · j- ¯ y ··...
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This note was uploaded on 10/24/2010 for the course STAT 424 taught by Professor Liang,f during the Spring '08 term at University of Illinois, Urbana Champaign.
- Spring '08