Semiparametric density estimation by local \(L_ 2\)-fitting. (English) Zbl 1091.62023

Naito, Kanta

Ann. Stat. 32, No. 3, 1162-1191 (2004).

Summary: This article examines density estimation by combining a parametric approach with a nonparametric factor. The plug-in parametric estimator is seen as a crude estimator of the true density and is adjusted by a nonparametric factor. The nonparametric factor is derived by a criterion called local \(L_2\)-fitting. A class of estimators that have multiplicative adjustment is provided, including estimators proposed by several authors as special cases, and the asymptotic theories are developed. Theoretical comparison reveals that the estimators in this class are better than, or at least competitive with, the traditional kernel estimator in a broad class of densities. The asymptotically best estimator in this class can be obtained from the elegant feature of the bias function.

MSC:

62G07	Density estimation
62G20	Asymptotic properties of nonparametric inference
62F10	Point estimation
62F12	Asymptotic properties of parametric estimators

Keywords:

adjustment; density estimation; kernel; local fitting; parametric model

Software:

KernSmooth

References:

[1]	Azzalini, A. (1985). A class of distributions which includes the normal ones. Scand. J. Statist. 12 171–178. · Zbl 0581.62014
[2]	Copas, J. B. (1995). Local likelihood based on kernel censoring. J. Roy. Statist. Soc. Ser. B 57 221–235. · Zbl 0812.62025
[3]	Eguchi, S. and Copas, J. B. (1998). A class of local likelihood methods and near-parametric asymptotics. J. R. Stat. Soc. Ser. B Stat. Methodol. 60 709–724. · Zbl 0910.62034 · doi:10.1111/1467-9868.00150
[4]	Hall, P. and Marron, J. S. (1987). Estimation of integrated squared density derivatives. Statist. Probab. Lett. 6 109–115. · Zbl 0628.62029 · doi:10.1016/0167-7152(87)90083-6
[5]	Härdle, W., Marron, J. S. and Wand, M. P. (1990). Bandwidth choice for density derivatives. J. Roy. Statist. Soc. Ser. B 52 223–232. · Zbl 0699.62036
[6]	Hjort, N. L. and Glad, I. K. (1995). Nonparametric density estimation with a parametric start. Ann. Statist. 23 882–904. JSTOR: · Zbl 0838.62027 · doi:10.1214/aos/1176324627
[7]	Hjort, N. L. and Jones, M. C. (1996). Locally parametric nonparametric density estimation. Ann. Statist. 24 1619–1647. · Zbl 0867.62030 · doi:10.1214/aos/1032298288
[8]	Izenman, A. J. (1991). Recent developments in nonparametric density estimation. J. Amer. Statist. Assoc. 86 205–224. · Zbl 0734.62040 · doi:10.2307/2289732
[9]	Jones, M. C., Linton, O. and Nielsen, J. P. (1995). A simple bias reduction method for density estimation. Biometrika 82 327–338. · Zbl 0823.62033 · doi:10.1093/biomet/82.2.327
[10]	Jones, M. C. and Signorini, D. F. (1997). A comparison of higher-order bias kernel density estimators. J. Amer. Statist. Assoc. 92 1063–1073. · Zbl 0888.62035 · doi:10.2307/2965571
[11]	Marron, J. S. and Wand, M. P. (1992). Exact mean integrated squared error. Ann. Statist. 20 712–736. JSTOR: · Zbl 0746.62040 · doi:10.1214/aos/1176348653
[12]	Naito, K. (1998). Density estimation by local \(L_2\)-fitting. Technical Report 98-7, Statistical Research Group, Hiroshima Univ.
[13]	Serfling, R. J. (1980). Approximation Theorems of Mathematical Statistics . Wiley, New York. · Zbl 0538.62002
[14]	Shao, J. (1991). Second-order differentiability and jackknife. Statist. Sinica 1 185–202. · Zbl 0820.62044
[15]	Simonoff, J. S. (1996). Smoothing Methods in Statistics . Springer, New York. · Zbl 0859.62035
[16]	Wand, M. P. and Jones, M. C. (1995). Kernel Smoothing . Chapman and Hall, London. · Zbl 0854.62043