Efficient nonparametric estimation of a distribution function
We consider efficient estimation of a distribution function F under several models based on the nonparametric likelihood principal. Under the symmetry model, we derive the nonparametric MLE of F and show that it coincides with the symmetrized estimator. Under the auxiliary-sample model, we discuss an estimator based on the total law of probability and show that it coincides with the nonparametric MLE of F. Assuming quadrant dependence, we show that the estimator has a minimum asymptotic relative efficiency of one with respect to the empirical distribution function. We consider the intersection of the two models and present an efficient hybrid estimator. The estimator is based on the conditional distribution function after post stratification of the sample. We show that the hybrid estimator has an asymptotic normal distribution and converges to the nonparametric MLE of F under the assumption of conditional symmetry. A Monte Carlo simulation assesses the small sample efficiency of the proposed estimators under the Plackett family of bivariate distributions. A bootstrap algorithm to obtain the power of the Wilcoxon signed rank test is presented.
distribution function symmetry nonparametric mle symmetrized estimator conditional symmetry bootstrap auxiliary samples