Final Exam - Multivariate Analysis | STAT 750, Exams of Descriptive statistics

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STAT 750 FINAL EXAM December 12, 2002 1. Let P = {p(x, y, 2:9}, @ € ©} be a family of probability densities on a measurable space (Vx x Z, A@B@C). Assuming p(x, y, 2:4) > 0 prove that if Th(x, y, z) = (2, y) is sufficient for @ and (separately) Ta(z, ¥, 2) = (x,z) is sufficient: for 0, then T(z, y, z) = 2 is sufficient for 6, (Hint: Use the factorization theorem.) 2. Let (X1,¥1),---, (Xn, ¥n) be a sample from a bivariate normal population with E(Xi) = 6, E(¥)) = 6/2, var(X,)) = var(¥) = 1, corr(X;, ¥;) = p. Here p is known and @ is to be estimated. (i) Find the least favorable and the most favorable values of p for estimat- ing @. (ii) Calculate the minimum variance unbiased estimators of @ in cases of the least and most favorable values of p and their variances. 3. Let (X1,...,%m) be a sample from N,(u1,V) and let (4,...,¥,) bea sample from N,(t2,V), 1, 42, V all unknown. Assuming the samples inde- pendent, develop the likelihood ratio test of Ho : 1 = fe. 4, Let X1,...,X;, be independent s-variate normal random vectors, EX;=p unknown, var(X;) = V; known. Prove that (V>'+...+ V1)" Dy", V>LX; is an unbiased and efficient estimator of j. Kagan