On the Asymptotic of Likelihood Ratios for Self-normalized Large Deviations

Motivated by multiple statistical hypothesis testing, we obtain the limit of likelihood ratio of large deviations for self-normalized random variables, specifically, the ratio of P ( √ n(X̄ + d/n) ≥ xnV ) to P ( √ nX̄ ≥ xnV ), as n → ∞, where X̄ and V are the sample mean and standard deviation of iid X1, . . . , Xn, respectively, d > 0 is a constant and xn → ∞. We show that the limit can have a simple form e0 , where z0 is the unique maximizer of zf(x) with f the density of Xi. The result is applied to derive the minimum sample size per test in order to control the error rate of multiple testing at a target level, when real signals are different from noise signals only by a small shift.