Suppose that for a sample x1,...,xn coming from a distribution with density f(x|) we can construct the maximum likelihood function L(), with R. Sup- pose further that we can calculate the MLE, that is, L() L(), for all R. Let h : R , where denotes the codomain of the function, and let = h() and = h(). For each , let U be its pre-image through h, U = { R : h() = }, and for every define the function M() = supU L(). Show that the function M() is the likelihood function for estimating = h(), and derive that the MLE for estimating is = h(). 8.7 In this problem we refer to Example 8.6. Suppose the random variables (X1,Y1),(X2,Y2),...,(Xn,Yn) are all independent and that for each i we have Xi,Yi distributed as N(i,2). a) Set = 1 2 and express the likelihood function in terms of the parameters 1,...,n, and . b) Show that the log likelihood is l(1,...,n,) = 2nlog 12 + nlog n i=1 (Xi i)2 + (Yi i)2 2 . c) Calculate the MLEs for the parameters 1,...,n,.