Maximum leave-one-out likelihood method for the location parameter of variance gamma distribution with unbounded density

Abstract

Maximum likelihood estimation of a location parameter fails when the density has unbounded mode. An alternative approach is considered by leaving out a data point to avoid the unbounded density in the full likelihood. This modification gives rise to the leave-one-out (LOO) likelihood. We propose an expectation-conditional maximisation (ECM) algorithm to estimate parameters of variance gamma (VG) distribution with unbounded density by maximising the LOO likelihood. VG distribution has normal mean-variance mixtures representation that facilitates ECM algorithm. K Podgórski, J Wallin [Maximizing leave-one-out likelihood for the location parameter of unbounded densities. Ann Inst Stat Math. 2015;67(1):19–38.] showed that the location parameter estimate which maximises the LOO likelihood is consistent and super-efficient. In the case of repeated data, the LOO likelihood is still unbounded and we extend it to Weighted LOO (WLOO) likelihood. We perform simulations to investigate the accuracy of ECM method using WLOO likelihood and compare it with other likelihood methods, including the alternating ECM with fixed and adaptive caps to the unbounded density, the ghyp package using multi-cycle ECM, and the ECM using LOO likelihood. Lastly, we explore the asymptotic properties of the location parameter estimate of VG distribution such as the optimal convergence rate and asymptotic distribution.