Summary: This paper presents a general framework for estimating smooth functionals of the probability distribution functions, such as the density, the hazard rate function, the mean residual time, the Lorenz curve, the spectral density, the tail index, the quantile function and many others. This framework is based on maximizing a local asymptotic pseudolikelihood associated to the empirical distribution function. Weak and strong convergence and a central limit theorem of the obtained estimators are investigated.