Estimation of a Time-Varying Extreme Quantile with Application to the Measurement of the Activity of Bivalves in an Environmental Context

• Activity Number: 241
• Type: Contributed
• Date/Time: Monday, August 5, 2013 : 2:00 PM to 3:50 PM
• Sponsor: Section on Statistics and the Environment
• Author(s): Ion Grama*+ and Gilles Durrieu and Jean-Charles Massabuau and Quang Khoai Pham and Jean-Marie TRICOT
• Companies: University of South Brittany and University of South Brittany and University of Bordeaux 1, CNRS UMR 5805-EPOC and University of South Brittany and University of South Brittany
• Keywords: Extreme values ; High quantiles ; Nonparametric kernel estimator ; Bio-monitoring ; Environmental statistics

Abstract

Consider a continuous time process $X(t)$ with independent increments and assume that each observation has a regularly varying distribution function $F_t$. We propose a nonparametric estimator of the high quantiles of $F_t$ from the observations $X(t_i)$ at instants $t_i$. The idea of our approach is to adjust the tail of the distribution function $F_t$ with a Pareto distribution with parameter $\theta_t$ starting from a threshold $\tau$. The parameter $\theta_t$ is estimated using a kernel estimator of bandwidth $h$ based on the observations larger than $\tau$. Under some regularity assumptions on the underlying distributions $F_t$ and for appropriately chosen threshold $\tau$ and bandwidth $h$, we prove that the proposed estimator of $\theta_t$ is consistent and we compute its rate of convergence. We also propose a sequential tests based procedure for the automatic choice of the threshold $\tau$. We discuss an application to the measurement of the closing and opening activity of bivalves considered as bio-indicators of pollution of aquatic systems.