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Article overview
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Testing Independence of Infinite Dimensional Random Elements: A Sup-norm Approach | | Suprio Bhar
; Subhra Sankar Dhar
; | | Date: |
1 Jan 2023 | | Abstract: | In this article, we study the test for independence of 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 $langle .,
.
angle_{cal{H}}$). In the course of this study, a measure of association is
proposed based on the sup-norm difference between the joint probability density
function of the bivariate random vector $(langle l_{1}, X
angle_{cal{H}},
langle l_{2}, Y
angle_{cal{H}})$ and the product of marginal probability
density functions of the random variables $langle l_{1}, X
angle_{cal{H}}$
and $langle l_{2}, Y
angle_{cal{H}}$, 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 whether $X$ and $Y$ are
independent or not, the sample version of the proposed measure of association
is considered as the test statistic after appropriate normalization, 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 three real
data sets related to climatology, biological science and chemical science. | | Source: | arXiv, 2301.00375 | | Services: | Forum | Review | PDF | Favorites | |
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