GU4204/GR5204 - Statistical Inference Final Exam - Fall 2016 December 20 Please show your work in all the problems to get full credit. Time: 90 min Name: - UNI: 1. Suppose X1 , X2 , ..., Xn is a random sample from the uniform (, ) distribution. We are interested in testing H0 : = 1 versus H1 : > 1 (a) Using the Neyman-Pearson lemma, find the most powerful test at the level of significance = 0.05 . (10 pt) (b) Calculate the power function of the test found in part (a), and find its limit as +. (10 pt) 2. Suppose that X1 , . . . , Xn from a normal distribution N (, 2 ) where both and 2 are unknown. We wish to test the hypotheses H0 : = 0 versus H1 : 6= 0 at the level . Find the likelihood ratio test and show that it is equivalent to the t-test. (20 pt) 3. Suppose X1 , X2 , ..., Xn is a random sample from the pdf f (x|) = ex I(0,) (x), with > 0 an unknown parameter. We are interested in performing the likelihood ratio test (LRT) at the level of significance = 0.05 for testing H0 : = 1 vs H1 : > 1. Pn (a) Show that the rejection region of the LRT is of the form {X : i=1 Xi < c}. (10 pt) Pn (b) What is the distribution of i=1 Xi under the null hypothesis? Use that to find the critical value c. (10 pt) 4. Suppose that in a sequence of n Bernoulli trials, the probability p of success in unknown. We are interested in applying the chi-squared goodness-of-fit test for testing H0 : p = p0 vs H1 : p 6= p0 , for some 0 < p0 < 1. Denote the proportion of successes observed by X n . (a) Show that the test statistic Q can be written as Q = n(X n p0 ) p0 (1p0 ) 2 . (10 pt) (b) What is the asymptotic distribution of Q under the null hypothesis? Use that to find the rejection region. (10 pt) 5. Suppose we obtain observations Y1 , Y2 , ..., Yn which can be described by the following linear relationship Yi = x2i + \u000fi , i = 1, 2, ..., n, where x1 , x2 , ..., xn are fixed constants and \u000f1 , \u000f2 , ..., \u000fn are IID N (0, 2 ). (a) Find the least squares estimator of . (10 pt) (b) Is unbiased? (10 pt) (10 pt) (c) Find the distribution of . (d) How would you test the hypothesis H0 : = 0 versus H1 : 6= 0? Describe the test and the critical value. (10 pt) 1