Statistics Practice Exam - Hypothesis Testing, Interval Estimation, Point Estimation

ORIE 6700 Practice Final Exam Problem 1 Suppose that X1 . . . Xn are sampled i.i.d. from N(µ, 4). You would like to estimate µ, however you do not observe Xi directly. Instead you observe a Y1 . . . Yn, where Yi = X 3 i /3. (a) Consider the hypothesis testing problem in which the null hypothesis is µ  1 and the alternate hypothesis is µ > 1. Construct a p-value as a function of the data (Y1, . . . Yn). Specify an ↵-level test constructed from the p-value. (b) Give a 1� ↵ confidence interval for estimating µ. (c) Consider the Bayesian setting where you assume a standard normal prior for µ. Give a 1� ↵ posterior credible interval for µ. Problem 2 Consider a data matrix Y 2 Zn⇥n+ , in which Yij represents the number of emails that an individual i sent to individual j within a fixed time period. Let individuals in {1, . . . n/2} belong to community 1, and let individuals in {n/2 + 1, . . . n} belong to community 2 (assume n is even). Let Yij for i 6= j be distributed as an independent random variable according to Poisson(�ij) (for i = j assume Yii = 0 as there are no self-emails). Assume the parameters satisfy �ij = 8 >< >: p for i, j belonging to the same community q for i, j belonging to di↵erent communities 0 if i = j for some unknown parameters p 2 R+ denoting rate of communication within communities, and q 2 R+ denoting rate of communication across communities. Our goal is to estimate the di↵erence, ⌧(p, q) = p� q. Let us denote the full dataset as Y = {Yij}i 6=j . (a) Show that the pdf of the data vector Y = {Yij}i 6=j parameterized by the unknown parameters (p, q) 2 R2+ is an exponential family. Specify the su�cient statistics and natural parameters. Is it minimal? Is it full rank? (b) Suppose that the unknown parameters p and q were drawn from a prior. What is the conjugate prior for this model? Compute the posterior distribution for (p, q) assuming p and q are sampled from the conjugate prior. (c) Compute the UMVUE for ⌧(p, q), and compute its mean squared error (MSE). (d) Provide a lower bound on the MSE, E(⌧̂(Y)� ⌧(p, q))2, for estimating ⌧(p, q) = p� q by reducing to hypothesis testing and using Le Cam’s method. You may use that that DKL(X1kX2)  (�1��2) 2 �2 for X1 ⇠ Poisson(�1) and X2 ⇠ Poisson(�2). 1 ' -08555883ohm Problem 3 Let X = (X1, . . . Xn) be n samples drawn iid from an unknown distribution with pdf f(x). Consider testing between the null hypothesis H0 : f(x) = (2⇡) �1/2 exp ✓ �1 2 x 2 ◆ , �1 < x < 1, and the alternate hypothesis H1 : f(x) = 1 2 exp(�|x|), �1 < x < 1. Define the statistic T (X) := 1 n nX i=1 (|Xi|� 1)2. (a) Show that the UMP test for testing H0 against H1 is given by a test that rejects the null hypothesis if T (X) � q for some q. (b) Derive an upper bound on the Type-I error rate of the above test. Your bound should be exponentially decaying in large q and n. (Hint: Find an upper bound for T (X) which is sub-exponential.) (c) Recall that a random variable Z with mean µ is sub-exponential with parameters (⌫, b) if E[e�(Z�µ)]  e� 2⌫2/2 , 8 |�| < 1 b . Show that under the alternate hypothesis H1, the statistic T (X) is not sub-exponential for any values of (⌫, b). 2