Miscellanea |
Miscellanea Kernel-Type Density Estimation on the Unit Interval
Department of Statistics, The Open University, Walton Hall, Milton Keynes MK7 6AA, U.K. m.c.jones{at}open.ac.uk
School of Mathematics & Statistics, Newcastle University, Newcastle upon Tyne NE1 7RU, U.K. d.a.henderson{at}ncl.ac.uk
Received for publication 1 December 2006. Revision received 1 March 2007.
We consider kernel-type methods for the estimation of a density on 0,1 which eschew explicit boundary correction. We propose using kernels that are symmetric in their two arguments; these kernels are conditional densities of bivariate copulas. We give asymptotic theory for the version of the new estimator using Gaussian copula kernels and report on simulation comparisons of it with the beta-kernel density estimator of Chen ([1]). We also provide automatic bandwidth selection in the form of ‘rule-of-thumb’ bandwidths for both estimators. As well as its competitive integrated squared error performance, advantages of the new approach include its greater range of possible values at 0 and 1, the fact that it is a bona fide density and that the individual kernels and resulting estimator are comprehensible in terms of a single simple picture.
Key Words: Boundary behaviour • Copula • Kernel density estimation
References
-
Chen S. X. Beta kernel estimators for density functions. Comp. Statist. Data Anal. (1999) 31:131–45.[CrossRef]
Chen S. X. Probability density function estimation using gamma kernels. Ann. Inst. Statist. Math. (2000) 52:471–80.[CrossRef]
Chen S. X. Beta kernel estimators for regression curves. Statist. Sinica (2000) 10:73–91.
Chen S. X. Local linear smoothers using asymmetric kernels. Ann. Inst. Statist. Math. (2002) 54:312–23.[CrossRef]
Demarta S., McNeil A. J. The t copula and related copulas. Int. Statist. Rev. (2005) 73:111–29.
Joe H. Multivariate Models and Dependence Concepts (1997) London: Chapman and Hall.
Johnson N. L. Systems of frequency curves generated by methods of translation. Biometrika (1949) 36:149–76.
Jones M. C. Simple boundary correction for kernel density estimation. Statist. Comp. (1993) 3:135–46.[CrossRef]
Karunamuni R. J., Alberts T. A generalized reflection method of boundary correction in kernel density estimation. Can. J. Statist. (2005) 33:497–509.
Klaassen C. A. J., Wellner J. A. Efficient estimation in the bivariate normal copula model: normal margins are least favourable. Bernoulli (1997) 3:55–77.[CrossRef][ISI]
Müller H. G. Smooth optimum kernel estimators near endpoints. Biometrika (1991) 78:521–30.
Nelsen R. B. An Introduction to Copulas (2006) 2nd ed. New York: Springer.
Pitt M., Chan D., Kohn R. Efficient Bayesian inference for Gaussian copula regression models. Biometrika (2006) 93:537–54.
Plackett R. L. A class of bivariate distributions. J. Am. Statist. Assoc. (1965) 60:516–22.[CrossRef][ISI]
Sheather S. J., Jones M. C. A reliable data-based bandwidth selection method for kernel density estimation. J. R. Statist. Soc. B (1991) 53:683–90.
Simonoff J. S. Smoothing Methods in Statistics (1996) New York: Springer.
Sungur E. A. Dependence information in parametrized copulas. Commun. Statist. B (1990) 19:1339–60.
Wand M. P., Jones M. C. Kernel Smoothing (1995) London: Chapman and Hall.
Zhang S., Karunamuni R. J., Jones M. C. An improved estimator of the density function at the boundary. J. Am. Statist. Assoc. (1999) 94:1231–41.[CrossRef][ISI]
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