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Title: Test for independence of infinite dimensional random elements Authors:  Suprio Bhar - Indian Institute of Technology Kanpur (India)
Subhra Sankar Dhar - IIT Kanpur (India) [presenting]
Abstract: A test for independence is studied for two random elements $X$ and $Y$ lying in an infinite dimensional space ${\cal{H}}$ (specifically, a real separable Hilbert space equipped with the inner product $\langle ., .\rangle_{\cal{H}}$). A measure of association is proposed based on the appropriate difference between the joint probability density function of the bivariate random vector $(\langle l_{1}, X \rangle_{\cal{H}}, \langle l_{2}, Y \rangle_{\cal{H}})$ and the product of marginal probability density functions of the random variables $\langle l_{1}, X \rangle_{\cal{H}}$ and $\langle l_{2}, Y \rangle_{\cal{H}}$, where $l_{1}\in{\cal{H}}$ and $l_{2}\in{\cal{H}}$ are two arbitrary elements. It is established that the proposed measure of association equals zero if and only if the random elements are independent. In order to carry out the test of whether $X$ and $Y$ are independent or not, the sample version of the proposed measure of association is considered as the test statistic, and the asymptotic distributions of the test statistic under the null and the local alternatives are derived. The performance of the new test is investigated for simulated data sets, and the practicability of the test is shown for one real data set related to climatology.