An extension of the von mises distribution

Directional distribution theory is very useful for the estimation of directional spectra needed for the analysis of time series data. A four parameter directional exponential family is discussed. Depending on the values of its parameters this distribution can be unimodal symmetric, bimodal symmetric, unimodal non-symmetric, or bimodal non-symmetric. The moments of this distribution are found, and equations leading to maximum-likelihood estimates of the parameters along with an outline on numerical procedures for solving these equations are given. FORTRAN subroutines implementing these procedures are available from the authors. Finally, some applications of the new directional density are given.