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INTEGRATED SQUARED ERROR OF KERNEL-TYPE

ESTIMATOR OF DISTRIBUTION FUNCTION

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SHINGO SHIRAHATA^{1} AND IN-SUN CHU^{2}

^{1} *Department of Statistics, School of General Education, Osaka University,*

Toyonaka, Osaka 560, Japan

^{2} *Department of Mathematical Science, Faculty of Engineering Science, Osaka University,*

Toyonaka, Osaka 560, Japan
(Received November 5, 1990; revised August 28, 1991)

**Abstract.**
Let *X*_{1},....,*X*_{n} be a random sample drawn from
distribution function *F*(*x*) with density function *f*(*x*) and suppose we want to
estimate *F*(*x*). It is already shown that kernel estimator of *F*(*x*) is better
than usual empirical distribution function in the sense of mean integrated
squared
error. In this paper we derive integrated squared error of kernel estimator
and compare the error with that of the empirical distribution function. It is
shown that the superiority of kernel estimators is not necessarily true
in the sense of integrated squared error.

*Key words and phrases*:
Nonparametric distribution function
estimator,
kernel function, mean integrated squared error, integrated squared error.

**Source**
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