Efficient estimation of partially linear tail index models using B-splines

The tail index is an important parameter in extreme value theory. In this paper, we consider a simple yet flexible spline estimation method for partially linear tail index models. We approximate the unknown function by B-splines and construct an approximate log-likelihood function to estimate the coefficients of the linear covariates and the B-spline basis functions. Consistency and asymptotic normality of the estimators are established. Subsequently, the proposed method is illustrated by using simulations and applications to the Fremantle annual maximum sea levels data and Chicago air pollution data.