## Electronic Journal of Statistics

### Semi-parametric regression estimation of the tail index

#### Abstract

Consider a distribution $F$ with regularly varying tails of index $-\alpha$. An estimation strategy for $\alpha$, exploiting the relation between the behavior of the tail at infinity and of the characteristic function at the origin, is proposed. A semi-parametric regression model does the job: a nonparametric component controls the bias and a parametric one produces the actual estimate. Implementation of the estimation strategy is quite simple as it can rely on standard software packages for generalized additive models. A generalized cross validation procedure is suggested in order to handle the bias-variance trade-off. Theoretical properties of the proposed method are derived and simulations show the performance of this estimator in a wide range of cases. An application to data sets on city sizes, facing the debated issue of distinguishing Pareto-type tails from Log-normal tails, illustrates how the proposed method works in practice.

#### Article information

Source
Electron. J. Statist., Volume 12, Number 1 (2018), 224-248.

Dates
First available in Project Euclid: 12 February 2018

https://projecteuclid.org/euclid.ejs/1518426109

Digital Object Identifier
doi:10.1214/18-EJS1394

Mathematical Reviews number (MathSciNet)
MR3763071

Zentralblatt MATH identifier
1387.62068

Subjects
Primary: 62G32: Statistics of extreme values; tail inference
Secondary: 62J05: Linear regression

#### Citation

Jia, Mofei; Taufer, Emanuele; Dickson, Maria Michela. Semi-parametric regression estimation of the tail index. Electron. J. Statist. 12 (2018), no. 1, 224--248. doi:10.1214/18-EJS1394. https://projecteuclid.org/euclid.ejs/1518426109

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