Abstract
The sample correlation coefficient R is almost universally used to estimate the population correlation coefficient ρ. If the pair (X,Y) has a bivariate normal distribution, this would not cause any trouble. However, if the marginals are nonnormal, particularly if they have high skewness and kurtosis, the estimated value from a sample may be quite different from the population correlation coefficient ρ.The bivariate lognormal is chosen as our case study for this robustness study. Two approaches are used: (i) by simulation and (ii) numerical computations.Our simulation analysis indicates that for the bivariate lognormal, the bias in estimating ρ can be very large if ρ≠0, and it can be substantially reduced only after a large number (three to four million) of observations. This phenomenon, though unexpected at first, was found to be consistent to our findings by our numerical analysis.
Highlights
The Pearson product-moment correlation coefficient p is a measure of linear dependence between a pair of random variables (X,Y)
While the properties of R for the bivariate normal are clearly understood, the same cannot be said about nonnormal bivariate populations
Various specific nonnormal populations have been investigated, the messages on the robustness of R are conflicting Johnson et al (1995, pp.580) remarked that "Contradictory, confusing, and uncoordinated floods of information on the ’robustness’ properties of the sample correlation coefficient R are scattered in dozens of journals." We do not intend to enter into the fray
Summary
The Pearson product-moment correlation coefficient p is a measure of linear dependence between a pair of random variables (X,Y). The sample (product-moment) correlation coefficient R, derived from n observations of the pair (X, Y), is normally used to estimate p. The size of the bias and the variance of R are still rather hazy for general bivariate nonnormal populations when p 0, since. Various specific nonnormal populations have been investigated, the messages on the robustness of R are conflicting Johnson et al (1995, pp.580) remarked that "Contradictory, confusing, and uncoordinated floods of information on the ’robustness’ properties of the sample correlation coefficient R are scattered in dozens of journals." We do not intend to enter into the fray. We compute the lower cross-cumulant ratios that contribute to the terms in n-1 and n-2 in the mean and variance of R The magnitude of these values cause difficulties in estimating the sizes of the bias and variance. They do shed some light on where the difficulties lie
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