STA131C, Spring 2007Prof. W. PolonikHW # 4due: May 9, 2007 in class1. An ecologist takes data (Yi, xi), i= 1, . . . , nwherexiis the size of an area andYiisthe number of moss plants in that area. We model the data by assuming theYito beindependent withYi∼Poisson(θ·xi), i= 1, . . . , n.(a) (10 pts) Show that the LSE ofθis given byhatwideθLSE=Pni=1xiYiPni=1x2i.(b) (10 pts) Show thathatwideθLSEis unbiased with Var(hatwideθLSE) =θPni=1x3i(Pni=1x2i)2.(c) (10 pts) Show that the MLE ofθishatwideθMLE=Pni=1YiPni=1xi.(d) (10 pts) Show that thehatwideθMLEis also unbiased and find its variance.(e) (10 pts) Find the UMVUE ofθand show that its variance attains the Cram´er-Raobound. Is this UMVUE also BLUE?2. Consider the one-way ANOVA modelYij=μi+ǫij,j= 1, . . . , ni, i= 1, . . . , I,withǫijiid random variables with mean 0 and varianceσ2.LetYisquaresmallsolid=1ni∑nij=1Yi,and letn=∑Ii=1nidenote the total sample size.
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