Generalised Lambda Distributions by Method of Moments and Maximum Likelihood using the JSE-ASI Returns

Abstract

The four-parameter generalised lambda distribution provides the flexibility required to describe the key moments of any distribution as compared with the normal distribution which characterises the distribution with only two moments. As markets have increasingly become nervous, the inadequacies of the normal distribution in capturing correctly the tail events and describing fully the entire distribution of market returns have been laid bare. The focus of this paper is to compare the generalised method of moments (GMM) and maximum likelihood essential estimates (MLE) methods as subsets of the GLD for a better fit of JSE All Share Index returns data. We have demonstrated that the appropriate method of the GLD to completely describe the measures of central tendency and dispersion by additionally capturing the risk dimensions of skewness and kurtosis of the return distribution is the Generalised Method of Moments (GMM) with the Kolmogorov-Smirnoff Distance good-of-fit statistics and the quantile-quantile graph. These measures are very important to any investor in the equity markets.