Kernel estimator of extreme value index (EVI) and high quantiles for heavy-tailed distributions under dependence serials using the Box-Cox transformation

The Box-Cox transformation is used to render the data more conducive to statistical analysis. Its application to extreme values statistics improves the convergence rate of certain classical tail index estimators and reduces bias in the context of independent and identically distributed (i.i.d) random variables. In this paper, we explore a bias reduction estimator for the extreme value index under $\beta$-mixing serial dependence. Our approach is based on kernel estimation of the extreme value index and the Box-Cox transformation methodology. Under specific regular conditions, we establish the asymptotic normality of the proposed estimator and deduce an asymptotically unbiased estimator for high quantiles. Through a simulation study, we delineate the performance of our proposals, comparing them to alternative estimators recently introduced in the literature. We further illustrate these theoretical results by applying them to real wind speed data.