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An important topic in multivariate extreme-value theory is to develop probabilistic models and statistical methods to describe and measure the strength of dependence among extreme observations. The theory is well established for data whose dependence structure is compatible with that of asymptotically dependent models. On the contrary, in many applications data do not comply with asymptotically dependent models and thus new tools are required. This article contributes to the methodological development of such a context, by considering a component-wise maxima approach. First we propose a statistical test based on the classical Pickands dependence function to verify whether asymptotic dependence or independence holds. Then, we present a new Pickands dependence function to describe the extremal dependence under asymptotic independence. Finally, we propose an estimator of the latter, we establish its main asymptotic properties and we illustrate its performance by a simulation study.

The estimation of optimal support boundaries under the monotonicity constraint is relatively unexplored and still in full development. This article examines a new extreme-value based model which provides a valid alternative for complete envelopment frontier models that often suffer from lack of precision, and for purely stochastic ones that are known to be sensitive to model misspecification. We provide different motivating applications including the estimation of the minimal cost in production activity and the assessment of the reliability of nuclear reactors.

The authors consider the construction of intrinsic estimators for the Pickands dependence function of an extreme-value copula. They show how an arbitrary initial estimator can be modified to satisfy the required shape constraints. Their solution consists in projecting this estimator in the space of Pickands functions, which forms a closed and convex subset of a Hilbert space. As the solution is not explicit, they replace this functional parameter space by a sieve of finite-dimensional subsets. They establish the asymptotic distribution of the projection estimator and its finite-dimensional approximations, from which they conclude that the projected estimator is at least as efficient as the initial one.

A new approach is suggested for choosing the threshold when fitting the Hill estimator of a tail exponent to extreme value data. Our method is based on an easily computed diagnostic, which in turn is founded directly on the Hill estimator itself, 'symmetrized' to remove the effect of the tail exponent but designed to emphasize biases in estimates of that exponent. The attractions of the method are its accuracy, its simplicity and the generality with which it applies. This generality implies that the technique has somewhat different goals from more conventional approaches, which are designed to accommodate the minor component of a postulated two-component Pareto mixture. Our approach does not rely on the second component being Pareto distributed. Nevertheless, in the conventional setting it performs competitively with recently proposed methods, and in more general cases it achieves optimal rates of convergence. A by-product of our development is a very simple and practicable exponential approximation to the distribution of the Hill estimator under departures from the Pareto distribution.