Stochastic non-convex envelopment of data: Applying isotonic regression to frontier estimation

• Isotonic nonparametric least squares (INLS) fits a step function to data.
• Free disposal hull (FDH) envelops all data points by step function.
• We show that FDH is a sign-constrained version of INLS regression.
• Using the result, we develop two new methods, corrected INLS and non-convex StoNED.
• We examine performance of the proposed methods by Monte Carlo simulations.

Isotonic nonparametric least squares (INLS) is a regression method for estimating a monotonic function by fitting a step function to data. In the literature of frontier estimation, the free disposal hull (FDH) method is similarly based on the minimal assumption of monotonicity. In this paper, we link these two separately developed nonparametric methods by showing that FDH is a sign-constrained variant of INLS. We also discuss the connections to related methods such as data envelopment analysis (DEA) and convex nonparametric least squares (CNLS). Further, we examine alternative ways of applying isotonic regression to frontier estimation, analogous to corrected and modified ordinary least squares (COLS/MOLS) methods known in the parametric stream of frontier literature. We find that INLS is a useful extension to the toolbox of frontier estimation both in the deterministic and stochastic settings. In the absence of noise, the corrected INLS (CINLS) has a higher discriminating power than FDH. In the case of noisy data, we propose to apply the method of non-convex stochastic envelopment of data (non-convex StoNED), which disentangles inefficiency from noise based on the skewness of the INLS residuals. The proposed methods are illustrated by means of simulated examples.