Bayesian estimation for mode and anti-mode preserving circular distributions

A Bayesian estimation is considered for unknown parameters of a unimodal skew circular distribution on the circle, where the underlying distribution has mode and anti-mode preserving properties. This distribution is obtained by using a transformation of the inverse monotone function, and the shape of the resulting density can be flat-topped or sharply peaked at its mode. With regard to Bayes estimates (BEs), the boundary-avoiding priors are assumed so that the skewness and peakedness parameters of the distribution do not lie on the boundary of the parameter space. In addition to the BEs, maximum likelihood estimations (MLEs) are conducted to compare the performances in small samples, and found that the BEs are more robust than the method of maximum likelihood. As the pairs of parameters between location and skewness and between concentration and peakedness are independent of each other, approximate BEs using Lindley’s methods become rather simple. Monte Carlo simulations are performed to compare the accuracy of the BE and MLE, and some circular datasets are analyzed for illustrative purposes.