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 is 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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
T. W. Anderson, “Asymptotic Theory for Canonical Analysis”, J. Multivar.Anal. 70, 1–29 (1999).
J. Dauxois, G. M. Nkiet, and Y. Romain, “Linear Relative Canonical Analysis of EuclideanRandomVariables, Asymptotic Study and Some Applications”, Ann. Inst. Statist. Math. 56, 279–304 (2004).
J. Dauxois and A. Pousse, “Une extension de l’analyse canonique. Quelques applications”, Ann. Inst. H. Poincaré 11, 355–379 (1975).
J. Dauxois, Y. Romain, and S. Viguier, “Tensor Product and Statistics”, Linear Algebra Appl. 270, 59–88 (1994).
S. Dossou-Gbete and A. Pousse, “Asymptotic Study of Eigenelements of a Sequence of Random Self- Adjoint Operators”, Statistics 22, 479–491 (1991).
M. L. Eaton and D. Tyler, “On Wielandt’s Inequality and Its Application to the Asymptotic Distribution of the Eigenvalues of a Random Symmetric Matrix”, Ann. Statist. 19, 260–271 (1991).
M. L. Eaton and D. Tyler, “The Asymptotic Distribution of Singular Values with Applications to Canonical Correlations and Correspondence Analysis”, J. Multivar.Anal. 70, 238–264 (1994).
L. Ferréand A. F. Yao, “Functional Sliced Inverse Regression Analysis”, Statistics 37, 475–488 (2003).
J. Fine, “Etude asymptotique de l’analyse canonique”, Publ. Inst. Statist. Univ. Paris, 44, 21–72 (2000).
S. Gardner, J. C. Gower, and N. J. Le Roux, “A Synthesis of Canonical Variate Analysis, Generalized Canonical Correlation and Procrustes Analysis”, Comput. Statist. Data Anal. 50, 107–134 (2006).
A. Gifi, Nonlinear Multivariate Analysis (Wiley, Chichester, 1991).
J. R. Kettenring, “Canonical Analysis of Several Sets of Variables”, Biometrika 58, 433–451 (1971).
R. J. Muirhead and C. M. Waternaux, “Asymptotic Distributions in Canonical Analysis and Other Multivariate Procedures for Nonnormal Populations”, Biometrika 67, 31–43 (1980).
A. Pousse, “Etudes asymptotiques”, in Modèles pour l’analyse des données multidimensionnelles, Ed. by J. J. Droesbeke, B. Fichet, and P. Tassi (Economica, Paris, 1992).
Y. Takane, H. Hwang, and H. Abdi, “Regularized Multiple-Set Canonical Correlation Analysis”, Psychometrika 73, 753–775 (2008).
A. Tenenhaus and M. Tenenhaus, “Regularized Generalized Canonical Correlation Analysis”, Psychometrika 76, 257–284 (2011).
Author information
Authors and Affiliations
Corresponding author
About this article
Cite this article
Nkiet, G.M. Asymptotic theory of multiple-set linear canonical analysis. Math. Meth. Stat. 26, 196–211 (2017). https://doi.org/10.3103/S1066530717030036
Received:
Published:
Issue Date:
DOI: https://doi.org/10.3103/S1066530717030036