Statistica Sinica 16(2006), 1233-1253

ROBUST NONPARAMETRIC FRONTIER ESTIMATORS:

QUALITATIVE ROBUSTNESS AND INFLUENCE FUNCTION

Abdelaati Daouia and Anne Ruiz-Gazen

Université Paul Sabatier and Université des Sciences Sociales, Toulouse

Abstract: In the context of productivity analysis, the most popular nonparametric estimators of a monotone boundary are defined as lowest monotone functions covering all sample points and are very non-robust. Two alternatives have been addressed recently: one by Cazals, Florens and Simar (2002) is based on a concept of expected order-$m$ frontiers; the other by Aragon, Daouia and Thomas-Agnan (2005) is based on extreme quantiles of a nonstandard conditional probability density. Unlike usual methods, both alternatives are shown to be qualitatively robust and bias-robust. Moreover, for the quantile approach, the influence function remains bounded even when the quantile order tends to one under the conditions that the conditional density function is not null, and is continuous on its support. When these conditions do not hold, the robust behavior of the quantile approach is shown on two numerical examples. A data set is provided to show the advantage of the robust proposals and the use of gross-error sensitivity as a diagnostic tool to detect anomalous data.

Key words and phrases: Estimation of a monotone boundary, gross-error sensitivity, influence function, maximum bias, outliers detection, productivity analysis, Prohorov distance, qualitative robustness.