Help me .to solve the following attachments.
Q2. (a) Show that N(u, (?), is not a member of the Regular Exponential Class. [Marks 4] (b) Consider a random sample of size n from a Normal distribution, N ((1, g?) where u is known. (i) Find a sufficient statistic for of using the Factorization Criterion. [Marks 4] (ii) Show that the statistic found in (i) is also sufficient using the conditional joint density method. [Marks 4] (iii) Show that the statistic found in (i) is also complete from definition and without using membership in Regular Exponential Class. [Marks 6] (iv) Construct a Uniformly minimum variance unbiased estimator for or. [Marks 8] (v) Derive the Cramer Rao Lower Bound for Unbiased Estimators of or. [Marks 4] (vi) Derive the asymptotic distribution of the Maximum Likelihood Estimator for (7. [Marks 4]1. This question is about some basic properties of projection operators. Remember from class that if W is a subspace of R" which has the orthonormal basis {1'51\" . . ,ii'P}, then the (orthogonal) projection onto W is given by multiplication by the matrix P = U UT, where U = [t'il . . . 113,] is the n x p matrix whose columns are the basis vectors of W. Also remember that UTU = I, and that any vector :i." can be written in exactly one way in the form =t+5 oemsewi. By the way, be careful of the terminology here. The matrix P of an orthogonal projection is not generally an orthogonal matrix. Remember that an orthogonal matrix S is one such that S\"1 = ST. In particular, it must be invertible. The only projection which is invertible is I. (Don't blame me; I didn't invent the terminology!) (a) Show that P2 = P and PT = P. (b) Dene Q = I P. Show that Q2 = Q, QT = Q, and P0 = QP = 0, the zero matrix. (c) Explain why Col(P) = Coi(U). (For this, remember that the column space of a matrix A is the same as the range of the linear transformation whose matrix is A. That is, the vectors in Col(A) are precisely the vectors of the form A5.) (d) Show that if a e Col(P) and 5' 6 (301(0), then a. i; = o. (Hint: Remember that a. 5' = 5T3.) (e) Explain why :i.' = Pf + inf. This means that 001(0) = Col(P)-". (Remark: It is true that if P is any n x 11. matrix such that P2 = P and PT = P, then P is the matrix of an orthogonal projection. If all you have is the P2 = P, then P is still a projection, but it doesn't have to be an orthogonal projection. That is, instead mapping vectors onto a subsPace in a direction perpendicular to the subspace, it will instead map in some oblique direction.) (1') Let A be any n x p matrix with linearly independent columns. The p x p matrix ATA must then be invertible. (You don't need to explain why this is true, but you should at least think about it and try to gure out a reason why it's true.) Dene R = A(ATA)'1AT. Show that 32 = R and RT = R. (Remark: In light of the previous remark, then, R is an orthogonal projection. In fact, it's the projection onto the column space of A. The difference here is that while the columns of A are a basis for the column space, they aren't necessarily an orthonormal basis.) (3) Show that S = I - 2P is an orthogonal matrix. (Not an orthogonal projection!). 1. (a) Consider the function given for x, y E R by f (x + iy) = 2x y+i(x2 - y+). Determine where f is complex differentiable and where f is analytic. [8 marks] Solution: Set u(x, y) = 2x'y, v(x, y) = x2 - y. Then u, = 4xy, u = 2x, vx = 2x, v. = -4y'. As all partial derivatives are continuous everywhere, the function is complex differentiable at exactly the points where the Cauchy- Riemann equations are satisfied, so if 4xy = -4y' and 2x- = -2x. The second equation is equivalent to x(x + 1) = 0, so the solutions are x = 0 and x = -1. Plugging x = 0 into the first equation gives y' = 0 so y = 0. With x = -I we obtain 0 = y - y' = >(1+ >)(1 ->), which has solutions 0, 1, -1. Hence / is complex differentiable at 0), -1, -1 +i, -1 -i. There is no nonempty open set on which f is differentiable, so f is not analytic anywhere. (b) i) Determine all z E C with el = e-x/2. [2 marks] i) Using the definition of complex powers used in the module, find all WED, = C \\ (x : x ER, x
