On estimation of parameters for bivariate von Mises and toroidal wrapped Gaussian to-rus distributions: A simulation study

The bivariate von Mises sine (BvMST) and toroidal wrapped Gaussian or normal (TWNT) distributions are defined on torus and they are applicable to statistical directional analysis of orientational data in 2D. The multivariate wrapped normal distribution in -dimension has both multivariate marginal wrapped normal models and bivariate marginal wrapped normal distributions, and thus has a theoretical merit. One drawback of the wrapped normal torus (WNT) distribution is that, unlike the sine and cosine models, it does not form an exponential family and thus additional statistical and mathematical care is required to tackle the problem. The maximum likelihood (ML), maximum pseudolikelihood (MPL) and moments’ (M) methods are numerically compared with respect to their efficiency rates for estimating the corresponding parameters of the BvMST distribution. Both MPL and M methods provide good estimates relative to the estimates of the ML method under an acceptance – rejection simulation scheme with Bingham-Angular Central Gaussian (BACG) distribution as an envelope. A proposed trigonometric moments’ (TM) method is derived and utilized for the purpose of parameters estimation of the TWNT distribution as compared to the traditional maximum likelihood technique. It provides efficient estimates either in small or large simulated random samples.