In this paper, we propose a new transformation of circular random variables
based on circular distribution functions, which we shall call inverse
distribution function (id f ) transformation. We show that M?bius
transformation is a special case of our id f transformation. Very general
results are provided for the properties of the proposed family of id f
transformations, including their trigonometric moments, maximum entropy,
random variate generation, finite mixture and modality properties. In
particular, we shall focus our attention on a subfamily of the general
family when id f transformation is based on the cardioid circular
distribution function. Modality and shape properties are investigated for
this subfamily. In addition, we obtain further statistical properties for
the resulting distribution by applying the id f transformation to a random
variable following a von Mises distribution. In fact, we shall introduce the
Cardioid-von Mises (CvM) distribution and estimate its parameters by the
maximum likelihood method. Finally, an application of CvM family and its
inferential methods are illustrated using a real data set containing times
of gun crimes in Pittsburgh, Pennsylvania.
Bootstrap for dependent Hilbert space-valued random variables with application to von Mises statistics
