A global optimisation approach for parameter estimation of a mixture of double Pareto lognormal and lognormal distributions

The double Pareto Lognormal (dPlN) statistical distribution, defined in terms of both an exponentiated skewed Laplace distribution and a lognormal distribution, has proven suitable for fitting heavy tailed data. In this work we investigate inference for the mixture of a dPlN   component and (k−1)(k−1) lognormal components for k fixed, a model for extreme and skewed data which additionally captures multimodality.The optimisation criterion based on the likelihood maximisation is considered, which yields a global optimisation problem with an objective function difficult to evaluate and optimise. Variable Neighbourhood Search (VNS) is proven to be a powerful tool to overcome such difficulties. Our approach is illustrated with both simulated and real data, in which our VNS and a standard multistart are compared. The computational experience shows that the VNS is more stable numerically and provides slightly better objective values.