Abstract
This paper first proves that the sample based Pearson’s product-moment correlation coefficient and the quotient correlation coefficient are asymptotically independent, which is a very important property as it shows that these two correlation coefficients measure completely different dependencies between two random variables, and they can be very useful if they are simultaneously applied to data analysis. Motivated from this fact, the paper introduces a new way of combining these two sample based correlation coefficients into maximal strength measures of variable association. Second, the paper introduces a new marginal distribution transformation method which is based on a rank-preserving scale regeneration procedure, and is distribution free. In testing hypothesis of independence between two continuous random variables, the limiting distributions of the combined measures are shown to follow a max-linear of two independent χ2 random variables. The new measures as test statistics are compared with several existing tests. Theoretical results and simulation examples show that the new tests are clearly superior. In real data analysis, the paper proposes to incorporate nonlinear data transformation into the rank-preserving scale regeneration procedure, and a conditional expectation test procedure whose test statistic is shown to have a non-standard limit distribution. Data analysis results suggest that this new testing procedure can detect inherent dependencies in the data and could lead to a more meaningful decision making.
Highlights
The correlation based test is more efficient when testing independence between two positive random variables
In the following sub-sections, we study asymptotic independence of order statistics, and asymptotic independence of sample correlations coefficients, which is very important in constructing a new maximal strength dependence measure and a powerful test statistic
We propose a test statistic mainly based on Proposition 3.1, which controls empirical Type I error probabilities being less than their pre-specified nominal levels and still gives high detecting powers in testing independence between two dependent random variables
Summary
Suppose X and Y are identically distributed positive random variables satisfying P(X ≥ Y ) > 0, P(X ≤ Y ) > 0. One of the most usages of the quotient correlation coefficient is to test the hypothesis of independence between two random variables X and Y using the gamma test statistic nqn. In the following sub-sections, we study asymptotic independence of order statistics, and asymptotic independence of sample correlations coefficients, which is very important in constructing a new maximal strength dependence measure and a powerful test statistic. Nqn(i, j) −L→ ζ, where ζ is a random variable with the gamma(i + j,1) density given by ((i + j − 1)!)−1xi+j−1e−x, x > 0 This theorem is very useful in constructing a robust statistical measure based on observed values. No matter how random variables between ξ and η are dependent on each other, ξand Mη are asymptotically independent, i.e. as n → ∞, where Φ is the standard multivariate normal distribution function, p and q are fixed integer numbers. With Fg(x) is a gamma(2,1) distribution function
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