On the asymptotic behaviour of the ISE for automatic kernel distribution estimators

In this paper we give an asymptotic expansion in probability for the integrated square error <<(J)over tilde>(n) = integral ((F) over tilde (n)(x) - f(X))(2) dF(x) (ISE), where F is the common distribution function of the independent and identically distributed real random variables X-1, ..., X-n, and (F) over tilde (n) is the kernel estimator of F with random bandwidth. This expansion enables us to describe the asymptotic behaviour in probability of (J) over tilde (n), and to present an asymptotic comparison, in the sense of ISE, of some distribution function estimators. These results, which extend the conclusions of Shirahata and Chu (1992) to the context of automatic estimators, give us, at least from an asymptotic point of view, a theoretical justification for the natural conjecture that the choice of the bandwidth in the distribution function estimation has not the main role as in the probability density estimation context. Some numerical results, for finite sample sizes, are also presented.