hwk2sol - PSTAT 127 Winter 2017 Homework 2 Solution Throughout this homework you may state and use any results you know from pre-requisite classes(PSTAT

hwk2sol - PSTAT 127 Winter 2017 Homework 2 Solution...

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PSTAT 127, Winter 2017, Homework 2 - Solution Throughout this homework, you may state and use any results you know from pre-requisite classes (PSTAT 120A, 120B, 126) as well as lectures. Clear working must be shown to receive credit. 1. (20 points) Mean square error of a point estimator for a parameterθ:LetY1, Y2, . . . , Ynbe a random sample from some pdf/pmffY(y;θ). LetWbe a point estimatorW=h(Y1, Y2, . . . , Yn)forθ. The bias ofWas a point estimator forθis defined asBias(W)=E(W)-θ.The mean square error of a point estimatorWforθis defined asMSE(W)=E(W-θ)2Show thatMSE(W)=Var(W) + [ Bias(W) ]2.MSE(W) =E[(W-θ)2]=E[W-E(W) +E(W)-θ]2=E[W-E(W)]2+E[E(W)-θ]2+ 2E[W-E(W)][E(W)-θ]= Var(W) +E[Bias(W)]2|{z}constant+ 2[E(W)-E(W)|{z}=0][E(W)-θ]= Var(W) + [Bias(W)]2.2. (45 points) Consider the Gaussian regression model with no interceptand with one explanatory variable, given byYi=βxi+iwithiiidN(0, σ2)fori∈ {1, . . . , n}. Assume than an observation pair(xi, yi)is available for eachi∈ {1, . . . , n}. Condition on thexivaluesthroughout. Perform all of the following steps from first principles based on PSTAT 120A/B definitions (without usingmatrix results)(a) (10 points) Write down the likelihood function forβandσ2based on the observations. (b) (10 points) Find the maximum likelihood estimators ofβandσ2, clearly showing your working. 1
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