Weak Convergence Analysis of Asymptotically Optimal Hypothesis Tests

Abstract

In recent years solutions to various hypothesis testing problems in the asymptotic setting have been proposed using results from large deviations theory. Such tests are optimal in terms of appropriately defined error-exponents. For the practitioner, however, error probabilities in the finite sample size setting are more important. In this paper we show how results on weak convergence of the test statistic can be used to obtain better approximations for the error probabilities in the finite sample size setting. While this technique is popular among statisticians for common tests, we demonstrate its applicability for several recently proposed asymptotically optimal tests, including tests for robust goodness of fit, homogeneity tests, outlier hypothesis testing, and graphical model estimation.

Second-order asymptotically optimal statistical classification

“…Intuitively, this is because of two reasons. Firstly, for the type-I error probabilities to be upper bounded by a non-vanishing constant ε ∈ (0, 1) for all pairs of distributions, one needs to choose λ = Θ( 1 n ) (implied by the weak convergence analysis in [9]). Consequently, the type-II exponent then satisfies…”

“…B. Proof of Proposition 4 1) Preliminaries: In this subsection, we recall a weak convergence result of Unnikrishnan and Huang [9] and present a key lemma for the analysis of Gutman's decision rule in (4).…”

“…Under H 1 , for all pairs of distributions (P 1 ,P 2 ) ∈ P(X ) 2 , the weak convergence result in Unnikrishnan and Huang [9,Lemma 5] shows that…”

Sequential Classification with Empirically Observed Statistics

“…Unnikrishnan and Naini [4] and Unnikrishnan [5] extended Gutman's proposed test for the case with multiple test sequences and obtain an optimal test rule for a certain matching task between multiple test sequences. Furthermore, Unnikrishnan and Huang in [6] showed how one can apply the results on the weak convergence of the test statistic to obtain better approximations for the error probabilities for statistical classification in the finite sample size setting. Most of the results on the problem of classification using empirically observed statistics are obtained for the case when the underlying distributions have finite alphabet.…”

Statistical Anomaly Detection via Composite Hypothesis Testing for Markov Models

“…To derive a more accurate threshold estimator, [5], [10] use a procedure commonly used by statisticians: deriving results based on Weak Convergence (WC) of the test statistic in order to approximate the error probabilities of the Hoeffding test. Under i.i.d.…”