×

Asymptotic unbiased density estimators. (English) Zbl 1180.62046

Summary: This paper introduces a computationally tractable density estimator that has the same asymptotic variance as the classical Nadaraya-Watson density estimator but whose asymptotic bias is zero. We achieve this result using a two stage estimator that applies a multiplicative bias correction to 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.

MSC:

62G07 Density estimation
62G20 Asymptotic properties of nonparametric inference
65C60 Computational problems in statistics (MSC2010)

Software:

KernSmooth
PDFBibTeX XMLCite
Full Text: DOI EuDML

References:

[1] I.S. Abramson, On bandwidth variation in kernel estimates - a square root law. Ann. Statist.10 (1982) 1217-1223. Zbl0507.62040 · Zbl 0507.62040 · doi:10.1214/aos/1176345986
[2] I.S. Abramson, Adaptive density flattening-metric distortion principle for combining bias in nearest neighbor methods. Ann. Statist.12 (1984) 880-886. Zbl0539.62045 · Zbl 0539.62045 · doi:10.1214/aos/1176346708
[3] A.R. Barron, L. Györfi and E.C. van der Meulen, Distribution Estimation Consistent in Total Variation and in Two Types of Information Divergence. IEEE Trans. Inf. Theory38 (1992) 1437-1453. · Zbl 0765.62007 · doi:10.1109/18.149496
[4] A. Berlinet, Hierarchies of higher order kernels. Prob. Theory Related Fields94 (1993) 489-504. · Zbl 0795.62032 · doi:10.1007/BF01192560
[5] B.L. Granovsky and H.-G. Müller, Optimizing kernel methods: a unifying variational principle. Ins. Statist. Rev.59 (1991) 373-388. · Zbl 0749.62024 · doi:10.2307/1403693
[6] P. Hall, On the bias of variable bandwidth curve estimators. Biometrika77 (1990) 529-535. · Zbl 0733.62046 · doi:10.1093/biomet/77.3.529
[7] P. Hall and J.S. Marron, Variable window width kernel estimates of a probability density. Prob. Theory Related Fields80 (1988) 37-49. · Zbl 0637.62036 · doi:10.1007/BF00348751
[8] N.L. Hjort and I.K. Glad, Nonparametric density estimation with a parametric start. Ann. Statist.23 (1995) 882-904. Zbl0838.62027 · Zbl 0838.62027 · doi:10.1214/aos/1176324627
[9] N.L. Hjort and M.C. Jones, Locally parametric nonparametric density estimation. Ann. Statist.24 (1996) 1619-1647. Zbl0867.62030 · Zbl 0867.62030 · doi:10.1214/aos/1032298288
[10] M.C. Jones, Variable kernel density estimates variable kernel density estimates. Aust. J. Statist.32 (1990) 361-371. Correction 33 (1991) 119.
[11] M.C. Jones, O.B. Linton and J.P. Nielsen, A simple bias reduction method for density estimation. Biometrika82 (1995) 327-38. · Zbl 0823.62033 · doi:10.1093/biomet/82.2.327
[12] M.C. Jones, I.J. McKay and T.-C. Hu, Variable location and scale kernel density estimation. Inst. Statist. Math.46 (1994) 521-535. · Zbl 0818.62039
[13] I. McKay, A note on bias reduction in variable kernel density estimates. Can. J. Statist.21 (1993) 367-375. · Zbl 0799.62037 · doi:10.2307/3315701
[14] J.S. Marron and M.P. Wand, Exact mean integrated squared error. Ann. Statist.20 (1992) 712-736. · Zbl 0746.62040 · doi:10.1214/aos/1176348653
[15] J.P. Nielson and O. Linton, A multiplicative bias reduction method for nonparametric regression. Statist. Probab. Lett.19 (1994) 181-187. · Zbl 0791.62043 · doi:10.1016/0167-7152(94)90102-3
[16] M. Rosenblatt, Remarks on some nonparametric estimates of a density function. Ann. Math. Statist.27 (1956) 832-837. · Zbl 0073.14602 · doi:10.1214/aoms/1177728190
[17] W. Stute, A law of iterated logarithm for kernel density estimators. Ann. Probab.10 (1982) 414-422. · Zbl 0493.62040 · doi:10.1214/aop/1176993866
[18] G. Terrel and D. Scott, On improving convergence rates for non-negative kernel density estimators. Ann. Statist.8 (1980) 1160-1163. · Zbl 0459.62031 · doi:10.1214/aos/1176345153
[19] M.P. Wand and M.C. Jones, Kernel Smoothing. Chapman and Hall, London (1995).
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.