Adaptive inference procedures for the concentration parameter of a Fisher-von Mises-Langevin distribution

Abstract

This paper focuses on the estimation and testing of a shape parameter or concentration parameter κ of a directional distribution named Fisher-von Mises- Langevin distribution. The estimators for κ using standard procedures cannot be obtained in closed or explicit forms. We derive asymptotic expressions up to first order for the bias, mean squared error and variance of the maximum likelihood estimator of κ when its mean direction is unknown. Using these expressions in a selective manner, several nearly-unbiased estimators for κ are proposed. A simulation study is carried out to compare the biases and percentage risk improvement of these new proposed estimators. The proposed estimators are seen to offer substantial improvements in terms of bias and percentage risk improvement over the earlier ones. The simulation study shows that these tests achieve size quite close to their nominal values and have quite good power performances. A nonparametric bootstrap confidence intervals for κ is presented. Finally these estimators and test procedures are applied on a real data set. ‘MATLAB’ functions are developed for practical use.