Abstract: We propose a local linear estimator of a smooth distribution function. This estimator applies local linear techniques to observations from a regression model in which the value of the empirical distribution function equals the value of true distribution plus an error term. We show that, for most commonly used kernel functions, our local linear estimator has a smaller asymptotic mean integrated squared error than the conventional kernel distribution estimator. Importantly, since this MISE reduction occurs through a constant factor of a second order term, any bandwidth selection procedures for kernel distribution estimator can be easily adapted for our estimator. For the estimation of a smooth quantile function, we establish a regression model of the empirical quantile function and obtain a local quadratic estimator. It has better asymptotic performance than the kernel quantile estimator in both interior and boundary cases.
Key words and phrases: Distribution function, empirical quantiles, kernel, local polynomial estimation, nonparametric estimation, quantile, smoothing.