Robustness of the sample correlation - The bivariate lognormal case

Abstract

The sample correlation coefficient R is almost universally used to estimate the population correlation coefficient p. 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 pρ 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. © Journal of Applied Mathematics & Decision Sciences.