Nonparametric "regression" when errors are positioned at end-points

TL;DR

This paper explores nonparametric regression methods where errors are centered at their end-points, revealing unique properties and faster convergence rates compared to traditional mean-centered errors, with implications for estimating error characteristics.

Contribution

It introduces nonparametric regression techniques tailored for end-point error centering, highlighting their distinct features and super-efficient convergence rates.

Findings

Errors centered at end-points lead to super-efficient convergence rates.

Estimating error distribution characteristics becomes more important.

Contrasts with traditional mean-centered regression methods.

Abstract

Increasing practical interest has been shown in regression problems where the errors, or disturbances, are centred in a way that reflects particular characteristics of the mechanism that generated the data. In economics this occurs in problems involving data on markets, productivity and auctions, where it can be natural to centre at an end-point of the error distribution rather than at the distribution's mean. Often these cases have an extreme-value character, and in that broader context, examples involving meteorological, record-value and production-frontier data have been discussed in the literature. We shall discuss nonparametric methods for estimating regression curves in these settings, showing that they have features that contrast so starkly with those in better understood problems that they lead to apparent contradictions. For example, merely by centring errors at their end-points rather than their means the problem can change from one with a familiar nonparametric character, where the optimal convergence rate is slower than $n^{-1/2}$, to one in the super-efficient class, where the optimal rate is faster than $n^{-1/2}$. Moreover, when the errors are centred in a non-standard way there is greater intrinsic interest in estimating characteristics of the error distribution, as well as of the regression mean itself. The paper will also address this aspect of the problem.