WEAK AND STRONG UNIFORM CONSISTENCY RATES OF KERNEL DENSITY ESTIMATES FOR RANDOMLY CENSORED-DATA

Let X1,...,X(n) be i.i.d. random variables with distribution function F, and let Y1,...,Y(n) be i.i.d. with distribution function G. For i = 1,2,...,n set delta(i) = 1 if X(i) less-than-or-equal-to Y(i) and 0 otherwise, and X(i) approximately = min{X(i), Y(i)}. A kernel-type density estimate of f, the density function of F w.r.t. Lebesgue measure on the Borel sigma-field, based on the censored data (X(i) approximately, delta(i)), i = 1,...,n, is considered. Weak and strong uniform consistency properties over the whole real line are studied. Rates of convergence results are established under higher-order differentiability assumption on f. A procedure for relaxing such assumptions is also proposed.