Estimation of order restricted concentration parameters of von Mises distributions

The von Mises distribution is a natural circular analog of the normal distribution on the real line, and is known as the circular normal distribution. This distribution has two parameters, known as the concentration parameter and the circular mean (or the mean direction). There are practical situations where it is of interest to estimate the concentration parameters of several von Mises distributions, when it is known apriori that the concentration parameters are subject to a simple order restriction. In this article, we discuss the restricted maximum likelihood estimation of the concentration parameters K-1,..., K-m. of m(greater than or equal to2) von Mises distributions, when it is known apriori that 0 less than or equal to K-1 less than or equal to K-2 less than or equal to(...less than or equal to) K-m less than or equal to infinity. Using the theory of isotonic regression, we derive the restricted maximum likelihood estimators Of the concentration parameters. Using approximations of some statistics based on a random sample from the von Mises distributions having large concentration parameters, we propose some more estimators for the order restricted concentration parameters of two von Mises distributions. Using Monte Carlo simulations, the restricted maximum likelihood estimators and the proposed estimators, based on the assumption of large concentration parameters, are compared with the usual (unrestricted) maximum likelihood estimators under the mean squared error criterion.