Optimal rates of convergence in the Weibull model based on kernel-type estimators

Cécile Mercadier (1) , Philippe Soulier (2)

1 PSPM - Probabilités, statistique, physique mathématique
2 MODAL'X - Modélisation aléatoire de Paris X

Let $F$ be a distribution function in the maximal domain of attraction of the Gumbel distribution and such that $-\log(1-F(x)) = x^{1/\theta} L(x)$ for a positive real number $\theta$, called the Weibul tail index, and a slowly varying function~$L$. It is well known that the estimators of $\theta$ have a very slow rate of convergence. We establish here a sharp optimality result in the minimax sense, that is when $L$ is treated as an infinite dimensional nuisance parameter belonging to some functional class. We also establish the rate optimal asymptotic property of a data-driven choice of the sample fraction that is used for estimation.

Mots clés: Weibul tail index; Rates of convergence; Kernel-type estimators; Optimal sample fraction; Sequential procedure

Domaines: Statistiques [math.ST]; Théorie [stat.TH]

Cécile Mercadier, Philippe Soulier. Optimal rates of convergence in the Weibull model based on kernel-type estimators. Statistics and Probability Letters, 2012, 82 (3), pp.548-556. ⟨10.1016/j.spl.2011.11.022⟩. ⟨hal-00565628v2⟩