1. In a linear regression model with n > p. the dependent variable Y : 9 ) R" is modelled as following the distribution given by Y = 35+ E, E ~ N(0,021), where :1: 6 IR\"? is a fixed. known matrix whose columns are assumed to be linearly independent and 6 6 IR\" is a fixed but unknown vector. Similarlykcr2 is assumed fixed but unknown. The estimated error (also known as residual) is E = HE, where H = I $(CETIL')1.'L'T 6 RM" is known as the hat matrix. Note: In this question, you are encouraged to bring to bear all your knowledge of STAT0005 and of linear algebra. If you do decide to use results from STAT0006, however, then you will need to rederive them from scratch instead of just quoting them. (a) Show that Ha: = 0. [1] (b) Show that HT 2 H. [1] (c) Show that H2 = H. Show that if v E R" is orthogonal to all column vectors of x. then Ho = 2). Hence obtain all the eigenvalues of H and their geometric multiplicities (i.e. how many linearly independent eigenvectors there are for each different eigenvalue). [4] (d) Show that the mgf of Z = ETE can be written in the form MZ(3) = kan exp (%uTM'1u) do. where K E R is a constant and M E RM" is a matrix. Simplify this to obtain an expression for the mgf that does not involve any integrals or determinants. You may use without proof that if the matrix A 6 RM" has eigenvalues A1, . . . , An then det(A) = H?=1Ai- Hint: If A is an eigenvalue of the matrix A, what is the corresponding eigenvalue of I A? [6] (e) Hence obtain an unbiased estimator of o2 and provide its sampling distribution. For which matrix a: does your estimator cocoincide with S2 from the lecture? [3]