Multivariate Nonparametric Estimation of the Pickands Dependence Function using Bernstein Polynomials

Many applications in risk analysis, especially in environmental sciences, require the estimation of the dependence among multivariate maxima. A way to do this is by inferring the Pickands dependence function of the underlying extreme-value copula. A nonparametric estimator is constructed as the sample equivalent of a multivariate extension of the madogram. Shape constraints on the family of Pickands dependence functions are taken into account by means of a representation in terms of a specific type of Bernstein polynomials. The large-sample theory of the estimator is developed and its finite-sample performance is evaluated with a simulation study. The approach is illustrated by analyzing clusters consisting of seven weather stations that have recorded weekly maxima of hourly rainfall in France from 1993 to 2011

On the simultaneous approximation of derivatives by Lagrange and Hermite interpolation

AbstractWe investigate here, for a positive integer q, simultaneous approximation of the first q derivatives of a function by the derivatives of its Lagrange interpolant, and then we augment this procedure by Hermite interpolation at the endpoints of the interval, obtaining a great improvement in the quality of approximation. In both cases, we estimate the quality of simultaneous approximation in terms of the norm of an associated Lagrange interpolation, and the estimates are thus valid for any sequence of interpolations by polynomials of successively higher degree. This communication continues work begun by K. Balázs and generalizes a recent work of Muneer Yousif Elnour, who treats simultaneous approximation with nodes at the zeroes of the Tchebycheff polynomials. Our efforts to obtain results which are independent of the choice of nodes have also led to some interesting consequences of a theorem of Gopengauz on simultaneous approximation