Biased and unbiased estimation of the circular mean resultant length and its variance

Abstract

The mean resultant length (MRL) is the length of the average of random vectors on the unit circle. It is used to measure the concentration of unimodal circular distributions. The sample MRL, as an estimator for the population MRL, has not been investigated thoroughly yet. This work examines the bias, variance and mean- squared error (MSE) of the MRL. Unbiased or near unbiased estimators are developed wherever possible for the squared and non-squared MRL, as well as their variances and MSE. All estimators are tested numerically on four representative circular distributions.

Keywords:

• circular statistics
• unbiased estimator
• resultant length