Some problems of statistics can be reduced to extremal problems of minimizing functionals of smooth functions defined on the cube [0, 1] m, m≥2. In this paper, we study a class of extremal problems that is closely connected to the problem of testing multivariate independence. By solving the extremal problem, we provide a unified approach to establishing weak convergence for a wide class of empirical processes which emerge in connection with testing independence. The use of our result is also illustrated by describing the domain of local asymptotic optimality of some nonparametric tests of independence.

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Keywords asymptotic efficiency of test statistics, boundary-value problem, local asymptotic optimality, multivariate independence, the Green function
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Journal Journal of Nonparametric Statistics
Nazarov, A. (Alexander), & Stepanova, N. (2012). An extremal problem with applications to the problem of testing multivariate independence. Journal of Nonparametric Statistics, 24(1), 3–17. doi:10.1080/10485252.2011.603831