Estimation of the reciprocal of the density quantile function at a point

Consistent estimators for the reciprocal of the density at a quantile point, which is the derivative of the quantile function, are considered. Rates of convergence of these estimators, depending on the smoothness properties of the density, are obtained. Two different, but natural, estimators of the reciprocal of the density at a quantile point, based on several samples from a location parameter family with unknown and possibly different location parameters are proposed. An important multivariate application is the estimation of the asymptotic dispersion matrix of several sample quantiles, as they involve reciprocals of the density at the corresponding population quantiles.