An Efficient Method in Estimating Generalized Lambda Distribution Parameters and Its Application in Financial Risk Measures

Generalized Lambda Distribution (GLD) is a highly flexible distribution. With four parameters, a location, a scale and two shape parameters, it embodies many types of density curves --bell shape, L-shape, J-shape, rectangle, U-shape, etc.. Because the GLD is adaptable to various density shapes, it has been highly prized in real-life data analysis, where the density function usually varies case by case and assumptions of the underlying distribution must be made with discretion. One potential usage of the GLD is in financial analysis, where both skewness and high kurtosis can not be properly fit by a commonly assumed normal distribution. However, until the past two decades, the GLD has been underutilized due to difficulties in the fitting.

Existing mainstream fitting methods include Method of Moments, Method of Percentiles, Maximum Likelihood Estimation and Starship, all of which apply to very limited parameter domain with high computational cost. This work aims for an improved fitting algorithm to achieve wider applicability and faster computation without compromising accuracy or precision. Revolving around concepts of value at risk (VaR) and expected shortfall (ES) in financial risk analysis, the proposed algorithm selects parameter estimates by minimizing the squared euclidean distances to sample quantiles, hence the name quantile least square (qLS). The qLS fitting method is used to estimate GLD parameters and subsequently risk measures. It yields relatively accurate and precise VaR and ES estimates in comparison to reference methods, regardless of the sample origin. The application of qLS in real-life data illuminates in potential in approximating financial data and estimating risk measures. Future research direction of the qLS method is also be proposed in an attempt for a more flexible and customizable scheme.