Maximum Likelihood Estimation of the Shape Parameters of the Generalized Lambda Distribution: A Finite-Sample Simulation Study

Source abstract

The Generalized Lambda Distribution (GLD) is a flexible four parameter distribution that can describe a wide range of symmetric, skewed and heavy tailed data patterns. However, its parameters are difficult to estimate because the likelihood equations are nonlinear and usually do not have analytical solution. This paper considers the finite-sample performance of maximum likelihood estimators for the two shape parameters of the Ramberg–Schmeiser GLD. The location and scale parameters are fixed at and , while and are numerically determined using the Newton–Raphson algorithm. A Monte Carlo simulation of 2000 replications is performed for sample sizes . The performance of the estimator is measured in terms of mean estimate, bias, empirical variance, mean squared error, observed-fisher-information-based variance and the ratio of the latter to empirical variance. The results show that as the sample size rises, the estimates become closer to the genuine parameter values. The bias, empirical variance and mean squared error are much reduced which shows that accuracy and precision are improved. For n = 1,000, the biases of and are 0.0022 and 0.0067, respectively. The ratios of observed Fisher information to empirical variance are 0.9693 for and 0.9907 for , indicating that the asymptotic and empirical variance estimates are in good agreement. The results indicate that, for the parameter configuration and initialization scheme considered, the finite-sample performance of MLE using Newton–Raphson improves as the sample size increases. This improvement is reflected in lower bias, empirical variance, and MSE, as well as closer agreement between OFI-based and empirical variance estimates at n=1,000.