Optimal Choice of Nonparametric Estimates of a Density and of
its Derivatives

J 2002

Optimal Choice of Nonparametric Estimates of a Density and of its Derivatives

HOROVÁ, Ivanka, Philippe VIEU and Jiří ZELINKA

Basic information

Original name

Optimal Choice of Nonparametric Estimates of a Density and of its Derivatives

Name in Czech

Optimální volba parametrů pro jádrové odhady hustoty a jijích derivací

Authors

HOROVÁ, Ivanka (203 Czech Republic, guarantor), Philippe VIEU (250 France) and Jiří ZELINKA (203 Czech Republic)

Edition

Statistics & Decisions, Mnichov, R. Oldenbourg Verlage, 2002, 0721-2631

Other information

Language

English

Type of outcome

Článek v odborném periodiku

Field of Study

10101 Pure mathematics

Country of publisher

Germany

Confidentiality degree

není předmětem státního či obchodního tajemství

References:

URL Odkaz na text článku na stránkách časopisu - zatím prázdný

RIV identification code

RIV/00216224:14310/02:00007553

Organization unit

Faculty of Science

Keywords (in Czech)

asymptpticky optimální odhad, vyhlazovací parametr, kanonický faktor, odhad hustoty

Keywords in English

asymptotic optimal estimate; bandwidth choice; canonical kernel; density estimates; derivatives estimation; kernel order choice; polynomial kernels

Abstract

V originále

Kernel smoothers are one of the most popular nonparametric functional estimates. These smoothers depend on three parameters: the bandwidth which controls the smoothness of the estimate, the form of the kernel weight function and the order of the kernel which is related to the number of derivatives assumed to exist in the nonparametric model. Because these three problems are closely related one to each other it is necessary to address them all together. In this paper we concentrate on the estimation of a density function and of its derivatives. We propose to use polynomial kernels and we construct data-driven choices for the bandwidth and the order of the kernel. We show a~theorem stating that this method for solving simultaneously the three selection problems mentioned before is asymptotically optimal in terms of Mean Integrated Squared Errors. As a by-product of our result we show an asymptotic optimality property for a~new bandwidth selector for density derivative which is quite appealing because of the simplicity of its implementation.

Links

MSM 143100001, plan (intention)

Name: Funkcionální diferenciální rovnice a matematicko-statistické modely

Investor: Ministry of Education, Youth and Sports of the CR, Functional-differential equations and mathematical-statistical models

Citovat

HOROVÁ, Ivanka, Philippe VIEU and Jiří ZELINKA. Optimal Choice of Nonparametric Estimates of a Density and of its Derivatives. Statistics & Decisions. Mnichov: R. Oldenbourg Verlage, 2002, vol. 20, No 4, p. 355-378. ISSN 0721-2631.

@article{484464,
   author = {Horová, Ivanka and Vieu, Philippe and Zelinka, Jiří},
   article_location = {Mnichov},
   article_number = {4},
   keywords = {asymptotic optimal estimate; bandwidth choice; canonical kernel; density estimates; derivatives estimation; kernel order choice; polynomial kernels},
   language = {eng},
   issn = {0721-2631},
   journal = {Statistics & Decisions},
   title = {Optimal Choice of Nonparametric Estimates of a Density and of its Derivatives},
   url = {http://www.statistics-international.de},
   volume = {20},
   year = {2002}
}

TY  - JOUR
ID  - 484464
AU  - Horová, Ivanka - Vieu, Philippe - Zelinka, Jiří
PY  - 2002
TI  - Optimal Choice of Nonparametric Estimates of a Density and of its Derivatives
JF  - Statistics & Decisions
VL  - 20
IS  - 4
SP  - 355-378
EP  - 355-378
PB  - R. Oldenbourg Verlage
SN  - 07212631
KW  - asymptotic optimal estimate
KW  - bandwidth choice
KW  - canonical kernel
KW  - density estimates
KW  - derivatives estimation
KW  - kernel order choice
KW  - polynomial kernels
UR  - http://www.statistics-international.de
N2  - Kernel smoothers are one of the most popular nonparametric functional estimates. These smoothers depend on three parameters: the bandwidth which controls the smoothness of the estimate, the form of the kernel weight function and the order of the kernel which is related to the number of derivatives assumed to exist in the nonparametric model. Because these three problems are closely related one to each other it is necessary to address them all together. In this paper we concentrate on the estimation of a density function and of its derivatives. We propose to use polynomial kernels and we construct data-driven choices for the bandwidth and the order of the kernel. We show a~theorem stating that this method for solving simultaneously the three selection problems mentioned before is asymptotically optimal in terms of Mean Integrated Squared Errors. As a by-product of our result we show an asymptotic optimality property for a~new bandwidth selector for density derivative which is quite appealing because of the simplicity of its implementation.
ER  -

HOROVÁ, Ivanka, Philippe VIEU and Jiří ZELINKA. Optimal Choice of Nonparametric Estimates of a Density and of its Derivatives. \textit{Statistics \&{} Decisions}. Mnichov: R. Oldenbourg Verlage, 2002, vol.~20, No~4, p.~355-378. ISSN~0721-2631.

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