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A generalization of the von Mises distribution, which is broad enough to cover unimodality as well as multimodality, symmetry as well as asymmetry of circular data, is discussed here. We study this distribution in some detail and discuss its many features, some inferential and computational aspects, and provide some important results including characterization properties for this distribution.

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This paper provides the self-consistent estimator (SCE) and the nonparametric maximum likelihood estimator (NPMLE) for ''middle-censored'' data, in which a data value becomes unobservable if it falls within a random interval. We provide an algorithm to find the SCE and show that the NPMLE satisfies the self-consistency equation. We find a sufficient condition for the SCE to be concentrated on the uncensored observations. In addition, we find sufficient conditions for the consistency of the SCE and prove that consistency holds for the special case when one of the ends is a constant. Some simulation results and an illustrative example, using Danish melanoma data set, are provided.

A general parametric estimation method which makes use of the coverage probabilities or spacings is proposed. Under some regularity conditions, it is shown that such estimators are asymptotically normal. This method generalizes the maximum spacing method of estimation that has been discussed in the literature. Furthermore, it is shown that the maximum spacing estimator is asymptotically most e cient within the subclass of spacings-based estimators under consideration.

This paper provides an illustrative case study on how the wind direction plays an important role in determining the ozone levels, in a suburb of Houston. Circular correlation and circular regression methods are used in the analysis and the primary goal is to illustrate how circular data analytic methods help in analyzing certain environmental issues.

a b s t r a c tMost of the tractable distributions currently available for modeling circular data are symmetric around a modal direction, prominent among them the von Mises distribution. Here we discuss a method of introducing asymmetry into any such symmetric circular model and develop general classes of non-symmetric circular distributions. In particular, we introduce and study a resulting variation of the classical von Mises distribution, along with a biological application.