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Articles

A distribution-free test of independence based on a modified mean variance index

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Pages 235-259 | Received 23 Feb 2022, Accepted 03 Apr 2023, Published online: 28 Apr 2023
 

Abstract

Cui and Zhong (2019), (Computational Statistics & Data Analysis, 139, 117–133) proposed a test based on the mean variance (MV) index to test independence between a categorical random variable Y with R categories and a continuous random variable X. They ingeniously proved the asymptotic normality of the MV test statistic when R diverges to infinity, which brings many merits to the MV test, including making it more convenient for independence testing when R is large. This paper considers a new test called the integral Pearson chi-square (IPC) test, whose test statistic can be viewed as a modified MV test statistic. A central limit theorem of the martingale difference is used to show that the asymptotic null distribution of the standardized IPC test statistic when R is diverging is also a normal distribution, rendering the IPC test sharing many merits with the MV test. As an application of such a theoretical finding, the IPC test is extended to test independence between continuous random variables. The finite sample performance of the proposed test is assessed by Monte Carlo simulations, and a real data example is presented for illustration.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work was supported by National Natural Science Foundation of China [Grant numbers 12271286, 11931001 and 11771241].