Testing random variables for independence and identity

Abstract

Given access to independent samples of a distribution A over [n] /spl times/ [m], we show how to test whether the distributions formed by projecting A to each coordinate are independent, i.e., whether A is /spl epsi/-close in the L/sub 1/ norm to the product distribution A/sub 1//spl times/A/sub 2/ for some distributions A/sub 1/ over [n] and A/sub 2/ over [m]. The sample complexity of our test is O/spl tilde/(n/sup 2/3/m/sup 1/3/poly(/spl epsi//sup -1/)), assuming without loss of generality that m/spl les/n. We also give a matching lower bound, up to poly (log n, /spl epsi//sup -1/) factors. Furthermore, given access to samples of a distribution X over [n], we show how to test if X is /spl epsi/-close in L/sub 1/ norm to an explicitly specified distribution Y. Our test uses O/spl tilde/(n/sup 1/2/poly(/spl epsi//sup -1/)) samples, which nearly matches the known tight bounds for the case when Y is uniform.