Improved efficient approximation of concentration parameter and confidence interval for circular distribution

The von Mises distribution for circular data, also known as the natural circular analogue of the normal distribution on the real line, has two parameters, namely, the concentration parameter and the circular mean. The solution of the maximum likelihood estimate (MLE) for the concentration parameter, , however, is analytically intractable. Thus some approximations are applied instead. In this article we propose an improved efficient approximation of obtained from the MLE. Unlike other estimations that have been shown to be only applicable for either large or small , the proposed approximation is found to be suitable for all values of . The improved approximation is obtained by solving piecewise polynomial equations involving the ratio of modified Bessel functions. The results of the simulation studies show that the improved approximation has a small bias and is superior to the traditional ones. Furthermore, the results obtained have been used in constructing the confidence interval for the concentration parameter, . Based on the simulation results, it is found that the confidence interval obtained based on the circular variance population method is superior to the confidence interval based on the normal distribution as it has smaller expected length.