Stacked Grenander and rearrangement estimators of a discrete distribution

TL;DR

This paper introduces stacked Grenander and rearrangement estimators for discrete distributions with infinite support, demonstrating their strong consistency, asymptotic properties, and practical advantages over existing methods.

Contribution

It proposes novel stacked estimators combining isotonic regression and rearrangement, with proven consistency, asymptotic distributions, and improved performance for small datasets.

Findings

Stacked Grenander estimator outperforms the rearrangement estimator.

Estimators are strongly consistent with a9(n) convergence rate.

Constructed asymptotic confidence bands for the estimators.

Abstract

In this paper we consider the stacking of isotonic regression and the method of rearrangement with the empirical estimator to estimate a discrete distribution with an infinite support. The estimators are proved to be strongly consistent with $\sqrt{n}$-rate of convergence. We obtain the asymptotic distributions of the estimators and construct the asymptotically correct conservative global confidence bands. We show that stacked Grenander estimator outperforms the stacked rearrangement estimator. The new estimators behave well even for small sized data sets and provide a trade-off between goodness-of-fit and shape constraints.