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

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.