Properties of the Sample Correlation of the Bivariate Lognormal Distribution

Most statistics students know that the sample correlation coefficient R is 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. Our simulation analysis indicates that for the bivariate lognormal, the bias in estimating ρ can be very large and it can be substantially reduced only after a large number (3-4 million) of observations. This example could serve as an exercise for the statistics students to realise some of the pitfalls in using the sample correlation coefficient to estimate ρ.