Concentration Inequalities for Randomly Permuted Sums

Abstract

Initially motivated by the study of the non-asymptotic properties of non-parametric tests based on permutation methods, concentration inequalities for uniformly permuted sums have been largely studied in the literature. Recently, Delyon et al. proved a new Bernstein-type concentration inequality based on martingale theory. This work presents a new proof of this inequality based on the fundamental inequalities for random permutations of Talagrand. The idea is to first obtain a rough inequality for the square root of the permuted sum, and then, iterate the previous analysis and plug this first inequality to obtain a general concentration of permuted sums around their median. Then, concentration inequalities around the mean are deduced. This method allows us to obtain the Bernstein-type inequality up to constants, and, in particular, to recovers the Gaussian behavior of such permuted sums under classical conditions encountered in the literature. Then, an application to the study of the second kind error rate of permutation tests of independence is presented.Mathematics Subject Classification: 60E15, 60C05.

Minimax optimality of permutation tests

“…Recently, there has been another line of research studying the power of the permutation test from a non-asymptotic point of view (e.g. Albert, 2015Albert, , 2019Kim et al, 2018Kim et al, , 2019. This framework, based on a concentration bound for a permuted test statistic, allows us to study the power in more general and complex settings than the asymptotic approach at the expense of being less precise (mainly in terms of constant factors).…”

“…structure of the permuted test statistic. Several attempts have been made to overcome such difficulty focusing on linear-type statistics (Albert, 2019), regressor-based statistics (Kim et al, 2019), the Cramér-von Mises statistic (Kim et al, 2018) and maximum-type kernel-based statistics (Kim, 2019). Our work contributes to this line of research by developing some general tools for studying the finite-sample performance of permutation tests with a specific focus on degenerate U -statistics.…”

Detection of sparse positive dependence

“…We start with characterizing the behavior of the LRT, which provides a benchmark. We then study some other testing procedures that do not require knowledge of the model parameters: 1 • The covariance test rejects for large values of ∑ i X i Y i , and coincides with Rao's score test in the present context. This is the classical test for independence, specifically designed for the case where ε = 1 and ρ > 0 under the alternative.…”

“…where F and G are unknown distribution functions on the real line, and Φ is the standard normal distribution function, while ε ∈ [0, 1 2) is the contamination proportion and 0 ≤ ρ ≤ 1 is the correlation between Z 1 and Z 2 in the contaminated component, as before in model (1). Li et al [16] also used a copula mixture model, but they placed emphasis on the mean while we focus on the dependence.…”

“…In this section, we focus on the Gaussian mixture model (1). We start by deriving a lower bound on the performance of the likelihood ratio test, which provides a benchmark for the other (adaptive) tests, which we subsequently analyze.…”