Question

4. (24 marks) Suppose that the random variables Yi,..., Yn satisfy Y-B BX,+ Ei, 1-1, , n, where βο and βι are parameters, X1, ,X, are con- stants, and e1,... ,en are independent and identically distributed ran- dom variables with Ei ~ N (0,02), where σ2 is a third unknown pa- rameter. This is the familiar form for a simple linear regression model, where the parameters A, β, and σ2 explain the relationship between a dependent (or response) variable Y and an independent (or explana- tory) variable x (a) Prove that Y N (Bo B,X,,o2). (Keep in mind that Xi,... , X are constants, not random variables, whereas e1,..., en are random variables, not constants.) (4 marks) (b) Using the fact that Yâ ~ N(80+AXi, σ*), show that the maxi- mum likelihood estimators of β0, β1, and σ2 are (8 marks) (c) Are βο and βι unbiased estimators of β0 and A, respectively? If either estimator is a biased estimator, use that derivation of the bias to find an unbiased estimator. (6 marks) (d) Calculate the mean squared errors for β0 and β1. (Keep in mind that σ2 is unknown, so it must be replaced with a suitable esti- mator.) (6 marks)

Answer 1

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