Spring '97 Stats Exam: Hypothesis Testing, Distributions, Regression, Confidence Intervals, Exams of Mathematical Statistics

Download Spring '97 Stats Exam: Hypothesis Testing, Distributions, Regression, Confidence Intervals and more Exams Mathematical Statistics in PDF only on Docsity! STAT 701 FINAL EXAMINATION SPRING 1997 Directions: Work each problem, showing all steps in your solution. 1. Let Xi, . . . , Xn be i.i.d. Poisson random variables with mean fj,. Derive the large sample distribution of 2 log A, where A is the likelihood ratio statistic for testing HQ : fj, = /J,Q vs. HI : p ^ [i0 . 2. Let ,X"i, . . . ,Xn be i.i.d. normal random variables with E[X{] = 0 and Va.T Xi = T. Suppose that r has prior density where a > 1/2, r > 0 and r > 0. Find the Bayes estimator of T with respect to squared error loss. [Note: 7rar (r) is actually a density for any a > 0.] 3. Consider the linear model Ytj = at + fixij + Cjj, i = l,2, j = 1, . . . , n, where £"=1 z,-j = 0, i = 1,2, and the e,-j are i.i.d. N(0,cr2 ). (a) Derive the least squares estimators for a1; a2 and (3 and find s 2 , the unbiased estimator for a2 based on the least squares analysis. State the distributions of these estimators. (b) Derive the numerator of the F statistic for testing HO : cti = a-2- What are the degrees of freedom associated with this F test? 4. Let YH = fj. + eu-, i = l,...,ni "&j = (J. + 8+C2J, j = I,...,n2 where the Cn and e 2j are i.i.d. ./V(0,cr2 ). (a) Find a 1 a confidence interval for 5 if a is known. Explain what tables, if any, are needed to compute the confidence limits. (b) Show how the solution to (a) leads to a test of H0 : S = 0 vs. HI : 5 / 0 and compute the power of the test. (c) If a is unknown, how do your answers to (a) change?