HW4 - STA131C, Spring 2007 Prof. W. Polonik HW # 4 due: May...

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Unformatted text preview: STA131C, Spring 2007 Prof. W. Polonik HW # 4 due: May 9, 2007 in class 1. An ecologist takes data ( Y i , x i ) , i = 1 , . . . , n where x i is the size of an area and Y i is the number of moss plants in that area. We model the data by assuming the Y i to be independent with Y i Poisson( x i ) , i = 1 , . . . , n. (a) (10 pts) Show that the LSE of is given by hatwide LSE = P n i =1 x i Y i P n i =1 x 2 i . (b) (10 pts) Show that hatwide LSE is unbiased with Var( hatwide LSE ) = P n i =1 x 3 i ( P n i =1 x 2 i ) 2 . (c) (10 pts) Show that the MLE of is hatwide MLE = P n i =1 Y i P n i =1 x i . (d) (10 pts) Show that the hatwide MLE is also unbiased and find its variance. (e) (10 pts) Find the UMVUE of and show that its variance attains the Cram er-Rao bound. Is this UMVUE also BLUE? 2. Consider the one-way ANOVA model Y ij = i + ij , j = 1 , . . . , n i , i = 1 , . . . , I, with ij iid random variables with mean 0 and variance 2 . Let Y...
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