Self-normalized large deviations

Large Deviations for Quasi-arithmetically Self-normalized Random Variables

We introduce a family of convex (concave) functions called sup (inf) of powers, which are used as generator functions for a special type of quasi-arithmetic means. Using these means we generalize the large deviation result that was obtained in the homogeneous case by Shao [14] on selfnormalized statistics. Furthermore, in the homogenous case, we derive the Bahadur exact slope for tests using se...

Self-normalized Moderate Deviations and Lils

Let fXn;n 1g be i.i.d. R d-valued random variables. We prove Partial Moderate Deviation Principles for self-normalized partial sums subject to minimal moment assumptions. Applications to the self-normalized law of the iterated logarithm are also discussed.

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 si...

Large Deviations

Large Deviations

This paper is based on Wald Lectures given at the annual meeting of the IMS in Minneapolis during August 2005. It is a survey of the theory of large deviations. 1. Large deviations for sums. The role of " large deviations " is best understood through an example. Suppose that X random variables, for instance, normally distributed with mean zero and variance 1. Then, E e θ(X 1 +···+X n) = E[e θX ...