Abstract
We propose estimating equations whose unknown parameters are the values taken by a circular density and its derivatives at a point. Specifically, we solve equations which relate local versions of population trigonometric moments with their sample counterparts. Major advantages of our approach are: higher order bias without asymptotic variance inflation, closed form for the estimators, and absence of numerical tasks. We also investigate situations where the observed data are dependent. Theoretical results along with simulation experiments are provided.
Highlights
Circular data occur when the sample space is described by a circle, as opposed to the real line in standard statistics
In this paper we propose estimating equations for circular density estimation where local versions of population trigonometric moments are equated with their empirical counterparts
When the sin-polynomial degree p is odd, we consider the matching between local trigonometric moments from order 1 up to order (p + 1)/2, obtaining the following system expressed in matrix form
Summary
Circular data occur when the sample space is described by a circle, as opposed to the real line in standard statistics. Basic kernel density estimation is well known (see, for example, [6], [2], [8] and [14]), not much has been written on more sophisticated methods, aimed at bias reduction. This would be useful for efficient point estimation in the cases of heavy density tails, multi-modality or asymmetry. As a link to previous work, we note that classic method of moments has been applied for parameter estimation by [12] when data come from a mixture of von Mises populations and the end is to separate the two components of the mixture.
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