Robustified Expected Maximum Production Frontiers

Abstract

The aim of this paper is to construct a robust nonparametric estimator for the production frontier. We study this problem under a regression model with one-sided errors, where the regression function defines the achievable maximum output, for a given level of inputs-usage, and the regression error defines the inefficiency term. The main tool is a concept of partial regression boundary defined as a special probability-weighted moment. This concept motivates a robustified unconditional alternative to the pioneering class of nonparametric conditional expected maximum production functions. We prove that both the resulting benchmark partial frontier and its estimator share the desirable monotonicity of the true full frontier. We derive the asymptotic properties of the partial and full frontier estimators, and unravel their behavior from a robustness theory point of view. We provide numerical illustrations and Monte Carlo evidence that the presented concept of unconditional expected maximum production functions is more efficient and reliable in filtering out noise than the original conditional version. The methodology is very easy and fast to implement. Its usefulness is discussed through two concrete datasets from the sector of Delivery Services, where outliers are likely to affect the traditional conditional approach.

Robust frontier estimation from noisy data: A Tikhonov regularization approach

“…x. In all of these aspects, Daouia et al (2018) have recently suggested a better alternative by transforming first the pp`1q-dimensional vector pX, Y q and its independent copies pX i , Y i q into the dimensionless random variables Y x " Y 1 IpX ď xq and Y x i " Y i 1 I pX i ď xq , i " 1, . .…”

“…where p F Y x pyq " p1{nq ř n i"1 1 IpY x i ď yq. Then, Daouia et al (2018) define the new concept of partial m-frontier ϕ m pxq " E " maxpY x 1 , . .…”

“…Consider now the Hilbert space F " L 2 pR, wq, where w stands for a probability measure. It is easily seen that S Z x p¨q´S ε x p¨q P F by using (10) and applying the fact that…”

“…We consider the same datasets from the sector of Delivery Services as in the study of Daouia et al (2018).…”

“…In contrast to Daouia et al (2018), here we consider that some noise may perturb the data of delivery post offices. We assume that the noise ε j given X j ď x has a normal distribution with zero mean and unknown variance σ 2 ε x .…”