Asymptotic theory of multiple-set linear canonical analysis

Abstract

This paper deals with asymptotics for multiple-set linear canonical analysis (MSLCA). A definition of this analysis, that adapts the classical one to the context of Euclidean random variables, is given and properties of the related canonical coefficients are derived. Then, estimators of the MSLCA's elements, based on empirical covariance operators, are proposed and asymptotics for these estimators are obtained. More precisely, we prove their consistency and we obtain asymptotic normality for the estimator of the operator that gives MSLCA, and also for the estimator of the vector of canonical coefficients. These results are then used to obtain a test for mutual non-correlation between the involved Euclidean random variables.

Robust multiple-set linear canonical analysis based on minimum covariance determinant estimator

“…In this section we recall the notion of multiple-set linear canonical analysis (MSLCA) of Euclidean random variables as introduced by Nkiet [18], and also its estimation based on empirical covariance operators. Then, the robustness properties of this analysis are studied through derivation of the influence functions that correspond to the functionals related to it.…”

“…It has been introduced for many years (e.g., [12]) and has been studied since then under different aspects (e.g., [15], [23], [25]). A formulation of MSLCA within the context of Euclidean random variables has been made recently ( [18]) and permitted to obtain an asymptotic theory for this analysis when it is estimated by using empirical covariance operators. To the best of our knowledge, such estimation of MSLCA is the one that have been tackled in the literature, despite the fact that it is known to be nonrobust as it is sensitive to outliers.…”

Some correlation tests for vectors of large dimension

“…(2) Thus, under normality, (1) () (2), where in general ( 1) ) (2). There is extensive literature on the tests of H 0 in the classical set up, that is, n > p i : Anderson (1999) and Eaton and Tyler (1994) established basic asymptotic theory with an extension for nonnormal case in Muirhead and Waternaux (1980), where Nkiet (2017) discussed the case of multiple blocks. A nonparametric test is considered in Gretton and Gy€ orfi (2010), Pfister et al (2018) provide a kernel based test, Horv ath, Hu skov a, and Rice (2013) treat the case for functional data and Albert et al (2015) give permutation tests.…”

Robustifying multiple-set linear canonical analysis with S-estimator

“…, X K ) is a statistical method that allows us to analyse the relationships among these variables. It has been known for many years (e.g., [6]) and has been extensively studied (e.g., [5,8,11,12,14]). Formally, MSLCA is the search of a sequence α ( j ) 1≤ j ≤q of vectors of…”