Nadaraya’s Estimates for Large Quantiles and Free Disposal Support Curves
A new characterization of partial boundaries of a free disposal multivariate support, lying near the true support curve, is introduced by making use of large quantiles of a simple transformation of the underlying multivariate distribution. Pointwise empirical and smoothed estimators of the full and partial support curves are built as extreme sample and smoothed quantiles. The extreme-value theory holds then automatically for the empirical frontiers and we show that some fundamental properties of extreme order statistics carry over to Nadaraya’s estimates of upper quantile-based frontiers. The benefits of the new class of partial boundaries are illustrated through simulated examples and a real data set, and both empirical and smoothed estimates are compared via Monte Carlo experiments. When the transformed distribution is attracted to the Weibull extreme-value type distribution, the smoothed estimator of the full frontier outperforms frankly the sample estimator in terms of both bias and Mean-Squared Error, under optimal bandwidth. In this domain of attraction, Nadaraya’s estimates of extreme quantiles might be superior to the sample versions in terms of MSE although they have a higher bias. However, smoothing seems to be useless in the heavy tailed case.
KeywordsAsymptotic Normality Quantile Function Tail Index Monte Carlo Experiment Free Disposal Hull
The authors thank an anonymous reviewer for his valuable comments which led to a considerable improvement of the manuscript. This research was supported by the French “Agence Nationale pour la Recherche” under grant ANR-08-BLAN-0106-01/EPI project.
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