Distribution-free inequalities for the deleted and holdout error estimates

Author

Devroye, Lug P. ; Wagner, Terry J.

Volume

Issue

fYear

1979

fDate

3/1/1979 12:00:00 AM

Firstpage

202

Lastpage

207

Abstract

In the discrimination problem the random variable $\\theta$ , known to take values in ${1 ,\\ldots ,M}$ , is estimated from the random vector $X$ taking values in ${\\bf R}^{d}$ . Ali that is known about the joint distribution of $(X,O)$ is that which can be inferred from a sample $(X_{1} , \\theta_{1}, \\ldots , (X_{n}, \\theta_{n})$ of size $n$ drawn from that distribution. A discrimination rule is any procedure which determines a decision $\\hat{\\theta}$ for $\\theta$ from $X$ and $(X_{1},\\theta_{1}) , \\ldots , (X_{n}, \\theta_{n})$ . The rule is called $k$ -local if the decision $\\hat{\\theta}$ depends only on $X$ and the pairs $(X_{i}, \\theta_{i})$ ,for which $X_{i}$ is one of the $k$ closest to $X$ from $X_{1} , \\ldots ,X_{n}$ . If $L_{n}$ denotes the probability of error for a $k$ -local rule given the sample, then estimates $\\hat{L}_{n}$ of $L_{n}$ , are determined for which $P {| \\hat{L}_{n} - L_{n} \\geq \\epsilon} \\exp (- Bn)$ , where $A$ and $B$ are positive constants depending only on $d$ , $M$ , and $\\epsilon$ .

Keywords

Nonparametric estimation; Pattern classification; Computer science; Frequency estimation; Gold; Helium; Information theory; Random variables; State estimation;

fLanguage

English

Journal_Title

Information Theory, IEEE Transactions on

Publisher

ieee

ISSN

0018-9448

Type

jour

DOI

10.1109/TIT.1979.1056032

Filename

1056032

Link To Document

https://search.isc.ac/dl/search/defaultta.aspx?DTC=49&DC=930005