Abdelaati Daouia (UT1-GREMAQ-TSE), « Hyperbolic graph efficiency assessment via extremes », On arbitrages arising with honest times, TSE, Toulouse, 17 mars 2011, 12:30-14:00, salle MS 003.

We consider a sample of independent businesses in a sector of production activity where a vector of inputs X is used to produce multiple outputs Y.
An important question is: what are the most efficient businesses that might be useful to emulate?
Using ideas from extreme value theory in conjunction with the concept of hyperbolic graph efficiency measure, we have been able to come up with a reasonable answer.
From a statistical viewpoint, the set of all possible producers may be viewed as the joint support of the population of businesses (X,Y).
The identification of the top sample observations lying near its optimal boundary is clearly a problem belonging to statistical modeling of extremes.
The hyperbolic distance function measures the potential input decrease and output increase along a hyperbolic path to achieve the efficient support boundary.
Its free disposal hull estimator and two alternative extreme-value based estimators are discussed.
Practical guidelines to effect the necessary computations of these estimators are described and generated data sets illustrate how they work out in practice. The Dekkers-Einmahl-de Haan type of estimator appears to give frank improvement in fit to data and performs better than the free disposal hull estimator in terms of both bias and mean-squared error even in presence of outliers. Case studies include frontier and efficiency analysis of the activity of French postal services and...

Mathématiques de la Décision et Statistique