UNIVERSITY OF CAMBRIDGE MATHEMATICAL TRIPOS Part III Monday, 6 June, 2016 1:30 pm to 3:30 pm PAPER 210 TOPICS IN STATISTICAL THEORY Attempt no more than THREE questions. There are FOUR questions in total. The questions carry equal weight. STATIONERY REQUIREMENTS Cover sheet Treasury Tag Script paper SPECIAL REQUIREMENTS None You may not start to read the questions printed on the subsequent pages until instructed to do so by the Invigilator.
UNIVERSITY OF CAMBRIDGE 2 1 For some positive integers n and m, we observe n balls placed in m bins, labelled from 1 to m. Let S1, ... , Sa non-empty disjoint subsets of {1, ... , m}, each with cardinality k < m. Under distribution Po, the balls are placed independently and uniformly in the m bins. Under distribution P1, an element j of {1, ... , d} is drawn uniformly at random, and conditionally on j, the balls are placed in the m bins with the following distribution Qj, independently: We flip a coin with probability of heads T € (0, 1). If it falls on heads, the ball is placed uniformly in one of the k bins with a label in Sj, and if it falls on tails, uniformly in the m bins. We denote by bi E {1, ... , m} the label of the bin in which ball i is placed. The number of balls in bins with index in Sj is cj = [{i | bi € Sj }|. We write E = k/m. (a) For any j E {1, ... , d}, give the distribution of cj, under Po and Qj using the parameter E. (b) Show without proof that if X has binomial distribution B(n, p), then X - np is sub-Gaussian with parameter n/4. Use this to derive without proof a bound on the probability that X - np > t, for all t > 0. State any theorem that you use. (c) Fix 8 € (0,1). Using the statistic C = max16j6d Cj, give a test y such that whenever we have r TT (1 - €) > log(d) r log (12/5) , +2 2n Po(=1) VP1(v=0) 68. (d) Define the chi-square divergence between two distributions. Write ×2(P1, Po) in terms of the likelihood ratios Q j/P0 . (e) Using without proof the fact that 0 x2(P1, Po) = 1 - 1 (1 - 12)" + , 1 + T2 (1/8 - 1) " - 1, In show that for v E (0, 1/2), when we have 0 1 TT E 2-1 01/2 G 6 r log(16v?d) , 1 inf Po(v = 1) V P 1() = 0) > > - v , where the infimum is taken over all tests v. Part III, Paper 210
UNIVERSITY OF CAMBRIDGE 3 2 Let d > 2 be an integer and S be an unknown subset of {1, ... , d} of k consecutive integers (modulo d). Let u(S) be the unit vector of Rd defined by u(S) = 1s/Vk (where