2023/01/01 by Suprio Bhar, Bhar, Suprio, Subhra Sankar Dhar +1
Decision Sciences · Mathematics · #60B12 #60F05 #62G07 #62G10 #62P12 #Advanced Statistical Methods and Models #FOS: Mathematics #Multi-Criteria Decision Making #Primary: 62H20 #Probability (math.PR) #Secondary: 62P10 #Statistical Methods and Inference #Statistics Theory (math.ST)
paper · pdf · doi:10.48550/arxiv.2301.00375
openalex publication_date 2023/01/01 · openalex created_date 2023/01/06 · openalex updated_date 2026/07/28
In this article, we study the test for independence of two random elements X and Y lying in an infinite dimensional space \calH (specifically, a real separable Hilbert space equipped with the inner product ⟨ ., .⟩_\calH). 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 (⟨ l1, X ⟩_\calH, ⟨ l2, Y ⟩_\calH) and the product of marginal probability density functions of the random variables ⟨ l1, X ⟩_\calH and ⟨ l2, Y ⟩_\calH, where l1∈\calH and l2∈\calH 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.