BAHADUR REPRESENTATION OF KERNEL QUANTILE ESTIMATORS

Let X1, X2, ..., X(n) be i.i.d. random variables with common distribut ion function F(x) and xi(p) be the pth quantile of F. Denote by F(n)-1 (P) Parzen's estimator of xi(p). In this paper, we establish a Bahadur representation for F(n)-1(p). Under suitable conditions, with probabi lity one, the remainder in this representation is of exact order O((lo g2n/n)2m/(2m+1)). As a by-product, the LIL for kernel quantile estimat or is obtained.