Comparison of relative efficiency of kernel density estimator with the exponential map
This paper considers the relative efficiency of two kernel estimators which have been proposed by Park [Park, H.S. (2012). Asymptotic behavior of the kernel density estimator from a geometric viewpoint. Communications in Statistics. Theory and Methods, 41(19),3479–3496.] and Hall et al. [Hall, P., W...
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| Published in | Journal of the Korean Statistical Society Vol. 42; no. 2; pp. 267 - 275 |
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| Format | Journal Article |
| Language | English |
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Singapore
Elsevier B.V
01.06.2013
Springer Singapore 한국통계학회 |
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| ISSN | 1226-3192 2005-2863 |
| DOI | 10.1016/j.jkss.2012.08.007 |
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| Abstract | This paper considers the relative efficiency of two kernel estimators which have been proposed by Park [Park, H.S. (2012). Asymptotic behavior of the kernel density estimator from a geometric viewpoint. Communications in Statistics. Theory and Methods, 41(19),3479–3496.] and Hall et al. [Hall, P., Watson, G., & Cabrera, J. (1987). Kernel density estimation with spherical data. Biometrika, 74, 751-762.], respectively. For this, we first consider the L2-minimax rate of the estimator defined by the exponential map. In order to make a numerical study between these estimators, the computing flows for the relative efficiency are introduced on the 2-dimensional unit sphere. A comparison between the two estimators with two different kernels is also investigated. |
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| AbstractList | This paper considers the relative efficiency of two kernel estimators which have been proposed by Park [Park, H.S. (2012). Asymptotic behavior of the kernel density estimator from a geometric viewpoint. Communications in Statistics. Theory and Methods, 41(19),3479–3496.]and Hall et al. [Hall, P., Watson, G., & Cabrera, J. (1987). Kernel density estimation with spherical data. Biometrika, 74, 751-762.], respectively. For this, we first consider the L2-minimax rate of the estimator defined by the exponential map. In order to make a numerical study between these estimators, the computing flows for the relative efficiency are introduced on the 2-dimensional unit sphere. A comparison between the two estimators with two different kernels is also investigated. KCI Citation Count: 1 This paper considers the relative efficiency of two kernel estimators which have been proposed by Park [Park, H.S. (2012). Asymptotic behavior of the kernel density estimator from a geometric viewpoint. Communications in Statistics. Theory and Methods , 41(19),3479–3496.] and Hall et al. [Hall, P., Watson, G., & Cabrera, J. (1987). Kernel density estimation with spherical data. Biometrika , 74, 751-762.], respectively. For this, we first consider the L2-minimax rate of the estimator defined by the exponential map. In order to make a numerical study between these estimators, the computing flows for the relative efficiency are introduced on the 2-dimensional unit sphere. A comparison between the two estimators with two different kernels is also investigated. This paper considers the relative efficiency of two kernel estimators which have been proposed by Park [Park, H.S. (2012). Asymptotic behavior of the kernel density estimator from a geometric viewpoint. Communications in Statistics. Theory and Methods, 41(19),3479–3496.] and Hall et al. [Hall, P., Watson, G., & Cabrera, J. (1987). Kernel density estimation with spherical data. Biometrika, 74, 751-762.], respectively. For this, we first consider the L2-minimax rate of the estimator defined by the exponential map. In order to make a numerical study between these estimators, the computing flows for the relative efficiency are introduced on the 2-dimensional unit sphere. A comparison between the two estimators with two different kernels is also investigated. |
| Author | Park, Hyun Suk |
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| Keywords | 62G20 Exponential map Kernel density estimator 62G07 L2-minimax rate Relative efficiency minimax rate L |
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| References | Fletcher, P., Lu, C., & Joshi, S. (2003). Statistics of shape via principal geodesic analysis on lie groups. In Healy, Hendriks, Kim (br000035) 1998; 67 Devroye, Györfi (br000015) 1985 Hall, Watson, Cabrera (br000030) 1987; 74 Kim, Y.T., & Park, H.S. (2012). Geometric structures arising from kernel density estimation on Riemannian manifolds, Journal of Multivariate Analysis (in press). Jupp, Mardia (br000045) 1989; 57 Fisher, Lewis, Embleton (br000020) 1993 Chapman, Chen, Kim (br000010) 1995; 5 Jupp, Mardia (br000050) 2000 Park (br000055) 2012; 41 Hendriks (br000040) 2003; 84 (pp. 95–101) Madison, WI. Bai, Rao, Zaho (br000005) 1988; 27 Chapman, Chen, Kim (CR2) 1995; 5 Jupp, Mardia (CR10) 2000 Hendriks (CR8) 2003; 84 Jupp, Mardia (CR9) 1989; 57 Fletcher, Lu, Joshi (CR5) 2003 Devroye, Györfi (CR3) 1985 Fisher, Lewis, Embleton (CR4) 1993 Hall, Watson, Cabrera (CR6) 1987; 74 Kim, Park (CR12) 2012 Bai, Rao, Zaho (CR1) 1988; 27 Park (CR11) 2012; 41 Healy, Hendriks, Kim (CR7) 1998; 67 Hall (10.1016/j.jkss.2012.08.007_br000030) 1987; 74 10.1016/j.jkss.2012.08.007_br000060 Devroye (10.1016/j.jkss.2012.08.007_br000015) 1985 Healy (10.1016/j.jkss.2012.08.007_br000035) 1998; 67 Jupp (10.1016/j.jkss.2012.08.007_br000045) 1989; 57 Chapman (10.1016/j.jkss.2012.08.007_br000010) 1995; 5 Jupp (10.1016/j.jkss.2012.08.007_br000050) 2000 10.1016/j.jkss.2012.08.007_br000025 Hendriks (10.1016/j.jkss.2012.08.007_br000040) 2003; 84 Bai (10.1016/j.jkss.2012.08.007_br000005) 1988; 27 Fisher (10.1016/j.jkss.2012.08.007_br000020) 1993 Park (10.1016/j.jkss.2012.08.007_br000055) 2012; 41 |
| References_xml | – volume: 84 start-page: 209 year: 2003 end-page: 221 ident: br000040 article-title: Application of fast spherical Fourier transform to density estimation publication-title: Journal of Multivariate Analysis – reference: Fletcher, P., Lu, C., & Joshi, S. (2003). Statistics of shape via principal geodesic analysis on lie groups. In – volume: 74 start-page: 751 year: 1987 end-page: 762 ident: br000030 article-title: Kernel density estimation with spherical data publication-title: Biometrika – volume: 57 start-page: 261 year: 1989 end-page: 294 ident: br000045 article-title: A unified view of the theory of directional statistics; 1975–1988 publication-title: International Statistical Review – volume: 27 start-page: 24 year: 1988 end-page: 39 ident: br000005 article-title: Kernel estimators of density function of directional data publication-title: Journal of Multivariate Analysis – reference: (pp. 95–101) Madison, WI. – reference: Kim, Y.T., & Park, H.S. (2012). Geometric structures arising from kernel density estimation on Riemannian manifolds, Journal of Multivariate Analysis (in press). – year: 1993 ident: br000020 article-title: Statistical analysis of spherical data – volume: 5 start-page: 173 year: 1995 end-page: 220 ident: br000010 article-title: Assesing geometric integrity through spherical regression techniques publication-title: Statistica Sinica – year: 2000 ident: br000050 article-title: Directional statistics – year: 1985 ident: br000015 article-title: Nonparametric density estimation: the – volume: 67 start-page: 1 year: 1998 end-page: 22 ident: br000035 article-title: Spherical deconvolution publication-title: Journal of Multivariate Analysis – volume: 41 start-page: 3479 year: 2012 end-page: 3496 ident: br000055 article-title: Asymptotic behavior of the kernel density estimator from a geometric viewpoint publication-title: Communications in Statistics. Theory and Methods – volume: 41 start-page: 3479 issue: 19 year: 2012 end-page: 3496 ident: CR11 article-title: Asymptotic behavior of the kernel density estimator from a geometric viewpoint publication-title: Communications in Statistics. Theory and Methods doi: 10.1080/03610926.2011.585009 – volume: 5 start-page: 173 year: 1995 end-page: 220 ident: CR2 article-title: Assesing geometric integrity through spherical regression techniques publication-title: Statistica Sinica – volume: 67 start-page: 1 year: 1998 end-page: 22 ident: CR7 article-title: Spherical deconvolution publication-title: Journal of Multivariate Analysis doi: 10.1006/jmva.1998.1757 – volume: 74 start-page: 751 year: 1987 end-page: 762 ident: CR6 article-title: Kernel density estimation with spherical data publication-title: Biometrika doi: 10.1093/biomet/74.4.751 – year: 2012 ident: CR12 publication-title: Geometric structures arising from kernel density estimation on Riemannian manifolds, Journal of Multivariate Analysis – start-page: 95 year: 2003 end-page: 101 ident: CR5 article-title: Statistics of shape via principal geodesic analysis on lie groups publication-title: Proc. IEEE Conf. on computer vision and pattern recodnition, vol. 1 – year: 1993 ident: CR4 publication-title: Statistical analysis of spherical data – volume: 57 start-page: 261 year: 1989 end-page: 294 ident: CR9 article-title: A unified view of the theory of directional statistics; 1975–1988 publication-title: International Statistical Review doi: 10.2307/1403799 – volume: 27 start-page: 24 year: 1988 end-page: 39 ident: CR1 article-title: Kernel estimators of density function of directional data publication-title: Journal of Multivariate Analysis doi: 10.1016/0047-259X(88)90113-3 – year: 2000 ident: CR10 publication-title: Directional statistics – year: 1985 ident: CR3 publication-title: Nonparametric density estimation: the L view – volume: 84 start-page: 209 year: 2003 end-page: 221 ident: CR8 article-title: Application of fast spherical Fourier transform to density estimation publication-title: Journal of Multivariate Analysis doi: 10.1016/S0047-259X(02)00041-6 – volume: 5 start-page: 173 year: 1995 ident: 10.1016/j.jkss.2012.08.007_br000010 article-title: Assesing geometric integrity through spherical regression techniques publication-title: Statistica Sinica – volume: 74 start-page: 751 year: 1987 ident: 10.1016/j.jkss.2012.08.007_br000030 article-title: Kernel density estimation with spherical data publication-title: Biometrika doi: 10.1093/biomet/74.4.751 – year: 1993 ident: 10.1016/j.jkss.2012.08.007_br000020 – ident: 10.1016/j.jkss.2012.08.007_br000025 doi: 10.1109/CVPR.2003.1211342 – volume: 67 start-page: 1 year: 1998 ident: 10.1016/j.jkss.2012.08.007_br000035 article-title: Spherical deconvolution publication-title: Journal of Multivariate Analysis doi: 10.1006/jmva.1998.1757 – volume: 84 start-page: 209 year: 2003 ident: 10.1016/j.jkss.2012.08.007_br000040 article-title: Application of fast spherical Fourier transform to density estimation publication-title: Journal of Multivariate Analysis doi: 10.1016/S0047-259X(02)00041-6 – volume: 57 start-page: 261 year: 1989 ident: 10.1016/j.jkss.2012.08.007_br000045 article-title: A unified view of the theory of directional statistics; 1975–1988 publication-title: International Statistical Review doi: 10.2307/1403799 – ident: 10.1016/j.jkss.2012.08.007_br000060 doi: 10.1016/j.jmva.2012.07.006 – volume: 41 start-page: 3479 issue: 19 year: 2012 ident: 10.1016/j.jkss.2012.08.007_br000055 article-title: Asymptotic behavior of the kernel density estimator from a geometric viewpoint publication-title: Communications in Statistics. Theory and Methods doi: 10.1080/03610926.2011.585009 – volume: 27 start-page: 24 year: 1988 ident: 10.1016/j.jkss.2012.08.007_br000005 article-title: Kernel estimators of density function of directional data publication-title: Journal of Multivariate Analysis doi: 10.1016/0047-259X(88)90113-3 – year: 1985 ident: 10.1016/j.jkss.2012.08.007_br000015 – year: 2000 ident: 10.1016/j.jkss.2012.08.007_br000050 |
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| SubjectTerms | [formula omitted]-minimax rate Applied Statistics Bayesian Inference Exponential map Kernel density estimator Relative efficiency Statistical Theory and Methods Statistics Statistics and Computing/Statistics Programs 통계학 |
| Title | Comparison of relative efficiency of kernel density estimator with the exponential map |
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