Estimation of quantile mixtures via L-moments and trimmed L-moments

Moments or cumulants have been traditionally used to characterize a probability distribution or an observed data set. Recently, L-moments and trimmed L-moments have been noticed as appealing alternatives to the conventional moments. This paper promotes the use of L-moments proposing new parametric families of distributions that can be estimated by the method of L-moments. The theoretical L-moments are defined by the quantile function i.e. the inverse of cumulative distribution function. An approach for constructing parametric families from quantile functions is presented. Because of the analogy to mixtures of densities, this class of parametric families is called quantile mixtures. The method of L-moments is a natural way to estimate the parameters of quantile mixtures. As an example, two parametric families are introduced: the normal-polynomial quantile mixture and the Cauchypolynomial quantile mixture. The proposed quantile mixtures are applied to model monthly, weekly and daily returns of some major stock indexes.