## Exact risk improvement of bandwidth selectors for kernel density estimation with directional data.(English)Zbl 1327.62241

Summary: New bandwidth selectors for kernel density estimation with directional data are presented in this work. These selectors are based on asymptotic and exact error expressions for the kernel density estimator combined with mixtures of von Mises distributions. The performance of the proposed selectors is investigated in a simulation study and compared with other existing rules for a large variety of directional scenarios, sample sizes and dimensions. The selector based on the exact error expression turns out to have the best behaviour of the studied selectors for almost all the situations. This selector is illustrated with real data for the circular and spherical cases.

### MSC:

 62G07 Density estimation

### Software:

movMF; KernSmooth
Full Text:

### References:

 [1] Azzalini, A. (1985). A class of distributions which includes the normal ones. Scand. J. Statist. 12 171-178. · Zbl 0581.62014 [2] Bai, Z. D., Rao, C. R. and Zhao, L. C. (1988). Kernel estimators of density function of directional data. J. Multivariate Anal. 27 24-39. · Zbl 0669.62015 [3] Banerjee, A., Dhillon, I. S., Ghosh, J. and Sra, S. (2005). Clustering on the unit hypersphere using von Mises-Fisher distributions. J. Mach. Learn. Res. 6 1345-1382. · Zbl 1190.62116 [4] Bingham, C. and Mardia, K. V. (1978). A small circle distribution on the sphere. Biometrika 65 379-389. · Zbl 0388.62020 [5] Cabella, P. and Marinucci, D. (2009). Statistical challenges in the analysis of cosmic microwave background radiation. Ann. Appl. Stat. 3 61-95. · Zbl 1160.62097 [6] Cao, R., Cuevas, A. and Gonzalez Manteiga, W. (1994). A comparative study of several smoothing methods in density estimation. Comput. Statist. Data Anal. 17 153-176. · Zbl 0937.62518 [7] Chacón, J. E. and Duong, T. (2013). Data-driven density derivative estimation, with applications to nonparametric clustering and bump hunting. Electron. J. Stat. 7 499-532. · Zbl 1337.62067 [8] Chiu, S.-T. (1996). A comparative review of bandwidth selection for kernel density estimation. Statist. Sinica 6 129-145. · Zbl 0850.62359 [9] Ćwik, J. and Koronacki, J. (1997). A combined adaptive-mixtures/plug-in estimator of multivariate probability densities. Comput. Statist. Data Anal. 26 199-218. · Zbl 0915.62021 [10] Di Marzio, M., Panzera, A. and Taylor, C. C. (2009). Local polynomial regression for circular predictors. Statist. Probab. Lett. 79 2066-2075. · Zbl 1171.62327 [11] Di Marzio, M., Panzera, A. and Taylor, C. C. (2011). Kernel density estimation on the torus. J. Statist. Plann. Inference 141 2156-2173. · Zbl 1208.62065 [12] Durastanti, C., Lan, X. and Marinucci, D. (2013). Needlet-Whittle estimates on the unit sphere. Electron. J. Stat. 7 597-646. · Zbl 1337.62287 [13] Fernández-Durán, J. J. (2004). Circular distributions based on nonnegative trigonometric sums. Biometrics 60 499-503. · Zbl 1274.62352 [14] Fernández-Durán, J. J. (2007). Models for circular-linear and circular-circular data constructed from circular distributions based on nonnegative trigonometric sums. Biometrics 63 579-585. · Zbl 1134.62030 [15] Fernández-Durán, J. J. and Gregorio-Domínguez, M. M. (2010). Maximum likelihood estimation of nonnegative trigonometric sum models using a Newton-like algorithm on manifolds. Electron. J. Stat. 4 1402-1410. · Zbl 1264.49029 [16] García-Portugués, E., Crujeiras, R. M. and González-Manteiga, W. (2012). Exploring wind direction and SO$$_{2}$$ concentration by circular-linear density estimation. Stoch. Environ. Res. Risk Assess. [17] García-Portugués, E., Crujeiras, R. M. and González-Manteiga, W. (2012). Kernel density estimation for directional-linear data. · Zbl 1328.62232 [18] Hall, P., Watson, G. S. and Cabrera, J. (1987). Kernel density estimation with spherical data. Biometrika 74 751-762. · Zbl 0632.62033 [19] Hornik, K. and Grün, B. (2012). movMF: Mixtures of von Mises-Fisher Distributions R package version 0.1-0. [20] Horová, I., Koláček, J. and Vopatová, K. (2013). Full bandwidth matrix selectors for gradient kernel density estimate. Comput. Statist. Data Anal. 57 364-376. · Zbl 1365.62127 [21] Jammalamadaka, S. R. and Lund, U. J. (2006). The effect of wind direction on ozone levels: a case study. Environ. Ecol. Stat. 13 287-298. [22] Johnson, M. E. (1987). Multivariate statistical simulation . Wiley Series in Probability and Mathematical Statistics. Applied Probability and Statistics. John Wiley & Sons Ltd., New York. · Zbl 0604.62056 [23] Jones, C., Marron, J. S. and Sheather, S. J. (1996). Progress in data-based bandwidth selection for kernel density estimation. Computation. Stat. 11 337-381. · Zbl 0897.62037 [24] Jupp, P. E. and Mardia, K. V. (1989). A unified view of the theory of directional statistics, 1975-1988. Int. Stat. Rev. 57 261-294. · Zbl 0707.62095 [25] Klemelä, J. (2000). Estimation of densities and derivatives of densities with directional data. J. Multivariate Anal. 73 18-40. · Zbl 1054.62033 [26] Lebedev, V. I. and Laikov, D. N. (1995). A quadrature formula for the sphere of the 131st algebraic order of accuracy. Dokl. Math. 59 477-481. · Zbl 0960.65029 [27] Mardia, K. V. and Jupp, P. E. (2000). Directional statistics . Wiley Series in Probability and Statistics . John Wiley & Sons Ltd., Chichester. · Zbl 0935.62065 [28] Marron, J. S. and Wand, M. P. (1992). Exact mean integrated squared error. Ann. Statist. 20 712-736. · Zbl 0746.62040 [29] Oliveira, M., Crujeiras, R. M. and Rodríguez-Casal, A. (2012). A plug-in rule for bandwidth selection in circular density estimation. Comput. Statist. Data Anal. 56 3898-3908. · Zbl 1255.62106 [30] Parzen, E. (1962). On estimation of a probability density function and mode. Ann. Math. Statist. 33 1065-1076. · Zbl 0116.11302 [31] Perryman, M. A. C. et al. (1997). The Hipparcos and Tycho Catalogues . European Space Agency. [32] Pewsey, A. (2006). Modelling asymmetrically distributed circular data using the wrapped skew-normal distribution. Environ. Ecol. Stat. 13 257-269. [33] Pukkila, T. M. and Rao, C. R. (1988). Pattern recognition based on scale invariant discriminant functions. Inform. Sci. 45 379-389. · Zbl 0652.62055 [34] Rosenblatt, M. (1956). Remarks on some nonparametric estimates of a density function. Ann. Math. Statist. 27 832-837. · Zbl 0073.14602 [35] Scott, D. W. (1992). Multivariate density estimation . John Wiley & Sons, New York. · Zbl 0850.62006 [36] Silverman, B. W. (1986). Density estimation for statistics and data analysis . Monographs on Statistics and Applied Probability . Chapman & Hall, London. · Zbl 0617.62042 [37] Taylor, C. C. (2008). Automatic bandwidth selection for circular density estimation. Comput. Statist. Data Anal. 52 3493-3500. · Zbl 1452.62269 [38] Van Leeuwen, F. (2007). Hipparcos, the new reduction of the raw data . Springer. [39] Wand, M. P. and Jones, M. C. (1995). Kernel smoothing . Monographs on Statistics and Applied Probability 60 . Chapman and Hall Ltd., London. · Zbl 0854.62043 [40] Watson, G. S. (1983). Statistics on spheres . University of Arkansas Lecture Notes in the Mathematical Sciences, 6 . John Wiley & Sons Inc., New York. · Zbl 0646.62045
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