Sieve Tail Index Regression

Abstract

The tail index is of particular interest since it gauges extreme behaviors in the heavy-tailed analysis. We consider estimating the tail index 'alpha' when covariate information is available under the Pareto-type distribution. We employ an exponential link function, which is partially linear, in order to associate a response variable with explanatory variables. Semi-nonparametric models are more robust but less sensitive to any specification issue compared to parametric cases. However, the unknown nonparametric parts induce other difficulties such as noncompact parameter spaces and ill-posed criterion problems in the semi-nonparametric models. The method of sieves resolves the complications by replacing the infinitedimensional parameter spaces with the compact finite-dimensional sieve spaces. It is easy and flexible to practice. We particularly study sieve maximum likelihood estimation and show the consistency of the estimators. Several conditions should be checked to insure consistency as standard asymptotic theory for parametric approach is not applicable.