Estimating a Changepoint, Boundary, or Frontier in the Presence of Observation Error

Abstract

A range of problems in economics and statistics involve calculation of the boundary, or frontier, of the support of a distribution. Several practical and attractive solutions exist if the sampled distribution has a sharp discontinuity at the frontier, but accuracy can be greatly diminished if the data are observed with error. Indeed, if the error is additive and has variance σ² then inaccuracies are usually of order σ, for small σ. In this article we suggest an elementary method for reducing the effect of error to O(σ²), and show that refinements can improve accuracy still further, to O(σ³) or less. The problem is inherently ill-posed, however, to such an extent that the frontier is generally not even identifiable unless the error distribution is known. The latter assumption is unreasonable in most practical settings, not in the least because the error is often asymmetrically distributed. For example, in the context of productivity analysis the error distribution tends to have a longer tail in the direction of underestimation of production. Nevertheless, even when the error distribution is unknown, it is often true that error variance is relatively low, and so methods for reducing systematic error in that case are useful in practice.

KEY WORDS:

• Changepoint analysis
• Data envelopment analysis
• Deconvolution
• Frontier estimation
• Kernel methods
• Nonparametric estimation
• Panel data
• Production frontier