INTEGRATED SQUARED ERROR OF KERNEL-TYPE ESTIMATOR OF DISTRIBUTION FUNCTION

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

1Department of Statistics, School of General Education, Osaka University, Toyonaka, Osaka 560, Japan

2Department 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.