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# HW4 - STA131C Spring 2007 Prof W Polonik HW 4 due May 9...

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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( θ · x i ) , i = 1 , . . . , n. (a) (10 pts) Show that the LSE of θ is given by hatwide θ LSE = P n i =1 x i Y i P n i =1 x 2 i . (b) (10 pts) Show that hatwide θ LSE is unbiased with Var( hatwide θ LSE ) = θ P n i =1 x 3 i ( P n i =1 x 2 i ) 2 . (c) (10 pts) Show that the MLE of θ is hatwide θ MLE = P n i =1 Y i P n i =1 x i . (d) (10 pts) Show that the hatwide θ MLE is also unbiased and find its variance. (e) (10 pts) Find the UMVUE of θ 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 = μ i + ǫ ij , j = 1 , . . . , n i , i = 1 , . . . , I, with ǫ ij iid random variables with mean 0 and variance σ 2 . Let Y i squaresmallsolid = 1 n i n i j =1 Y i , and let n = I i =1 n i denote the total sample size.
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