Abstract
The paper is concerned with the adaptive minimax problem of testing the independence of the components of a d-dimensional random vector. The functions under alternatives consist of smooth densities supported on [0, 1]d and separated away from the product of their marginals in L2-norm. We are interested in finding the adaptive minimax rate of testing and a test that attains this rate. We focus mainly on the tests for which the error of the first kind an can decrease to zero as the number of observations increases. We show also how this property of the test affects its error of the second kind.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
F. Abramovich, I. De Feis, and T. Sapatinas, “Optimal Testing Additivity in Multiple Nonparametric Regression”, Ann. Inst. Statist. Math. 61(3), 691–714 (2009).
F. Chiabrando, Risk with Random Normalizing Factor and Adaptive Test in the Additive Model, PhD thesis (Universitè de Provence, Marseille, 2008).
M. S. Ermakov, “Asymptotically Minimax Criteria for Testing Complex Nonparametric Hypotheses”, Problemy Peredachi Informatsii 32(2), 54–67 (1996); English transl. in Problems Inform. Transmission 32 (2), 184–196 (1996).
M. Fromont and B. Laurent, “Adaptive Goodness-of-Fit Tests in a Density Model”, Ann. Statist. 34(2), 1–45 (2006).
L. Gajek, “On Improving Density Estimators which are not Bonafide Functions”, Ann. Statist. 14, 1612–1618 (1986).
G. Gayraud and Ch. Pouet, “AdaptiveMinimax Testing in the DiscreteRegression Scheme”, Probab. Theory Rel. Fields 133, 531–558 (2005).
E. Giné, R. Latala, and J. Zinn, “Exponential and Moment Inequalities for U-Statistics”, High Dimensional Prob. II (2000).
Yu. I. Ingster, “Asymptotically Minimax Testing of the Hypothesis of Independence”, Zap. Nauchn. Seminar. LOMI 153, 60–72 (1986); English transl. in J. Soviet.Math. 44, 466–476 (1989).
Yu. I. Ingster, “Minimax Testing of the Hypothesis of Independence for Ellipsoids in lp”, Zap. Nauchn. Seminar. POMI 207, 77–97 (1993), English transl. in J.Math. Sci. 81, 2406–2420 (1996).
M. Hoffmann and O. V. Lepski, “Random Rates in Anisotropic Regression”, Ann. Statist. 30(2), 325–396 (2002).
O. V. Lepski, “How to Improve the Accuracy of Estimation”, Math. Methods Statist. 8, 441–486 (1999).
V. G. Spokoiny, “Adaptive Hypothesis Testing UsingWavelets”, Ann. Statist. 24(6), 2477–2498 (1996).
A. F. Yodé, “Asymptotically Minimax Test of Independence”, Math. Methods Statist. 13, 201–234 (2004).
A. F. Yodé, “Multidimensional Nonparametric Density Estimates: Minimax Risk with Random Normalizing Factor”, Afr. Diaspora J.Math. 10(2), 27–57 (2010).
Author information
Authors and Affiliations
Corresponding author
About this article
Cite this article
Yodé, A.F. Adaptive minimax test of independence. Math. Meth. Stat. 20, 246–268 (2011). https://doi.org/10.3103/S1066530711030069
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.3103/S1066530711030069
Keywords
- independence hypothesis testing
- minimax hypothesis testing
- asymptotics of errors probabilities
- density estimation