A Simple Ordered Data Estimator For Inverse Density Weighted Functions

We consider estimation of means of functions that are scaled by an unknown density, or equivalently, integrals of conditional expectations. The "ordered data" estimator we provide is root n consistent, asymptotically normal, and is numerically extremely simple, involving little more than ordering the data and summing the results. No sample size dependent smoothing is required. A similarly simple estimator is provided for the limiting variance. The proofs include new limiting distribution results for functions of nearest neighbor spacings. Potential applications include endogeneous binary choice, willingness to pay, selection, and treatment models.