Estimation of circular-circular probability distribution


This paper aims to introduce an estimation algorithm for the joint densityof a circular-circular random variable, which is expressible in the form asdiscussed by Fernández-Durán (2007). The performance of the algorithm hasbeen checked with the help of a simulation study and it is found to performeciently even for small sample sizes. Furthermore, the performance of theproposed algorithm is compared with that of an existing method of densityestimation and is found to perform better than the existing one, which isindicated by the higher mean square error values for the estimates obtainedby the latter method. Finally, the application of the algorithm is displayedby estimating the joint density of a circular-circular random variable arisingin a real-life data set.

DOI Code: 10.1285/i20705948v11n1p155

Keywords: Joint circular-circular density; Circular-circular random variable; Estimation algorithm; Simulation study


Bhattacharjee, S. and Das, K. K. (2017). Comparison of estimation methods of the joint density of a circular and linear variable. Journal of Data Science, 15(1):129-154.

Downs, T. D. (1974). Rotational Angular correlation. In M. Ferin, F. Halberg & L. van der Wiele (eds) Biorhythms and Human Reproduction. Wiley, New York.

Fernández-Durán, J. J. (2007). Models for circular-linear and circular-circular data constructed from circular distributions based on nonnegative trigonometric sums. Biometrics, 63:575-589.

Fisher, N. I. (1993). Statistical Analysis of Circular Data. Cambridge University Press, Cambridge.

Giorgia, R. (2015). Factor copulas through a vine structure. Electronic Journal of Applied Statistical Analysis, 8(2):246-266.

Jammalamadaka, S. R. and SenGupta, A. (2001). Topics in Circular Statistics. World Scientific Publishing Co. Pte. Ltd., Singapore.

Johnson, R. A. and Wehrly, T. E. (1977). Measures and models for angular correlation and angular-linear correlation. J. Roy. Statist. Soc., 39:222-229.

Jones, M. C., Pewsey, A., and Kato, S. (2015). On a class of circulas: copulas for circular distributions. Annals of the Institute of Statistical Mathematics, 67:843-862.

Lund, U. and Agostinelli, C. (2012). CircStats: Circular Statistics, from "Topics in Circular Statistics" (2001). R package version 0.2-4, URL:

Lund, U. and Agostinelli, C. (2013). circular: Circular Statistics. R package version 0.4-7, URL:

Mardia, K. V. and Jupp, P. E. (2000). Directional Statistics. John Wiley & Sons Ltd., Chichester.

Sklar, A. (1959). Fonctions de répartition á n dimensions et leurs marges. Publications

de lInstitut de Statistique de LUniversité de Paris, 8:229-231.

Wehrly, T. E. and Johnson, R. A. (1980). Bivariate models for dependence of angular observations and a related Markov process. Biometrika, 67:255-256.

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