Smooth estimation of circular cumulative distribution functions and quantiles

Abstract

Smooth nonparametric estimators based on a kernel method are proposed for cumulative distribution functions (CDFs) and quantiles of circular data. A sound motivation for this is that although for euclidean data similar estimators have been widely studied, for circular data nothing similar seems to exist; albeit, remarkably, in the circular-setting local methods are implemented more easily because of the absence of boundaries on the circle. The only alternative to our method seems to be the empirical CDF, that does not take into account circularity of data when the estimate is near the cut-point, as our local method naturally does. The definition of circular CDF is different from its euclidean counterpart in many respects, and this will give rise to estimators exhibiting some ‘unusual’ features such as, for example, global efficiency measures containing a location parameter and a covariance term. Simulations along with real data case studies illustrate the findings.

Keywords:

• angular risk
• circular kernels
• circular order
• Nadaraya-type quantile estimator
• Parzen-type quantile estimator

AMS Subject Classification :

• 62G07
• 62G08
• 62G20

Acknowledgements

We are grateful to the Associate Editor and two anonymous Referees for both a careful reading of the manuscript and helpful comments, which led to an improved version of this article.

Notes

www.ndbc.noaa.gov/data/historical/stdmet/41010h2009.txt.gz.