hw2 - STAT424 Spring 2010 Homework#2 Feb 9 2010 Homework 2...

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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 ·· where ¯ y i · = 1 2 y i 1 + y i 2 , ¯ y · j = 1 3 3 X i =1 y ij , ¯ y ·· = 1 6 X i,j y ij .
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