Hostname: page-component-cd9895bd7-gvvz8 Total loading time: 0 Render date: 2024-12-23T17:22:45.352Z Has data issue: false hasContentIssue false
  • English
  • Français

Asymptotic unbiased density estimators

Published online by Cambridge University Press:  21 February 2009

Nicolas W. Hengartner  and
Éric Matzner-Løber
Nicolas W. Hengartner
Affiliation:
Stochastics Group, Los Alamos National Laboratory, NM 87545, USA.
Éric Matzner-Løber
Affiliation:
UMR 6625, IRMAR, Université Rennes 2, 35043 France; [email protected].
Get access

Abstract

This paper introduces a computationally tractable density estimatorthat has the same asymptotic variance as the classical Nadaraya-Watsondensity estimator but whose asymptotic bias is zero. We achieve this resultusing a two stage estimator that applies a multiplicative bias correctionto an oversmooth pilot estimator. Simulations show that our asymptotic results are available for samples as low as n = 50, where we see an improvement of as much as 20% over the traditionnal estimator.

Keywords

Nonparametric density estimationkernel smootherasymptotic normalitybias reductionconfidence intervals
Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 13 , July 2009 , pp. 1 - 14
Copyright
© EDP Sciences, SMAI, 2009

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Abramson, I.S., On bandwidth variation in kernel estimates – a square root law. Ann. Statist. 10 (1982) 1217–1223. CrossRef
Abramson, I.S., Adaptive density flattening-metric distortion principle for combining bias in nearest neighbor methods. Ann. Statist. 12 (1984) 880–886. CrossRef
Barron, A.R., Györfi, L. and van der Meulen, E.C., Distribution Estimation Consistent in Total Variation and in Two Types of Information Divergence. IEEE Trans. Inf. Theory 38 (1992) 1437–1453. CrossRef
Berlinet, A., Hierarchies of higher order kernels. Prob. Theory Related Fields 94 (1993) 489–504. CrossRef
Granovsky, B.L. and Müller, H.-G., Optimizing kernel methods: a unifying variational principle. Ins. Statist. Rev. 59 (1991) 373–388. CrossRef
Hall, P., On the bias of variable bandwidth curve estimators. Biometrika 77 (1990) 529–535. CrossRef
Hall, P. and Marron, J.S., Variable window width kernel estimates of a probability density. Prob. Theory Related Fields 80 (1988) 37–49. CrossRef
Hjort, N.L. and Glad, I.K., Nonparametric density estimation with a parametric start. Ann. Statist. 23 (1995) 882–904. CrossRef
Hjort, N.L. and Jones, M.C., Locally parametric nonparametric density estimation. Ann. Statist. 24 (1996) 1619–1647.
Jones, M.C., Variable kernel density estimates variable kernel density estimates. Aust. J. Statist. 32 (1990) 361–371. Correction 33 (1991) 119. CrossRef
Jones, M.C., Linton, O.B. and Nielsen, J.P., A simple bias reduction method for density estimation. Biometrika 82 (1995) 327–38. CrossRef
Jones, M.C., McKay, I.J. and Variable, T.-C. Hu location and scale kernel density estimation. Inst. Statist. Math. 46 (1994) 521–535.
McKay, I., A note on bias reduction in variable kernel density estimates. Can. J. Statist. 21 (1993) 367–375. CrossRef
Marron, J.S. and Wand, M.P., Exact mean integrated squared error. Ann. Statist. 20 (1992) 712–736. CrossRef
Nielson, J.P. and Linton, O., A multiplicative bias reduction method for nonparametric regression. Statist. Probab. Lett. 19 (1994) 181–187.
Rosenblatt, M., Remarks on some nonparametric estimates of a density function. Ann. Math. Statist. 27 (1956) 832–837. CrossRef
Stute, W., A law of iterated logarithm for kernel density estimators. Ann. Probab. 10 (1982) 414–422. CrossRef
Terrel, G. and Scott, D., On improving convergence rates for non-negative kernel density estimators. Ann. Statist. 8 (1980) 1160–1163. CrossRef
M.P. Wand and M.C. Jones, Kernel Smoothing. Chapman and Hall, London (1995).