Nonparametric Estimation via Expected Order Statistics

Abstract: The empirical distribution function assigns mass $1/n$ to each of the $n$ observations in a sample. As these are highly variable, estimation error may be reduced by replacing them with estimated observations that are asymptotically less variable. Motivated by this idea, we introduce a nonparametric estimator obtained by assigning mass $1/m$ to $m$ estimated expected order statistics, with $m$ chosen arbitrarily. The estimator enjoys several finite-sample properties and yields a rich asymptotic theory. Its estimation error relative to its population counterpart is controlled by the $L^1$ error of the empirical distribution. Moreover, every $L$-functional of the new estimator corresponds to an $L$-functional of the empirical distribution with updated weights. We establish almost sure convergence in $L^p$ norm and Wasserstein distance as $n \to \infty$, and derive weak convergence of the associated empirical quantile process in $L^p(0,1)$, for $p\in[1,\infty)$ and $m$ fixed, and for $p=1,2$ as $n,m \to \infty$. These results yield asymptotic distributions for distance-based functionals, including $L^p$ and Wasserstein metrics. Bootstrap validity is also established. Simulations show that the estimator often improves on the empirical distribution and remains competitive with kernel methods, with more stable performance across different distributional settings.