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Semi-parametric regression estimation of the tail index. (English) Zbl 1387.62068
Let $$X_{1},X_{2},\ldots,X_{n}$$ be a random sample with a distribution function $$F$$ satisfying $\bar{F}\left( x\right) =x^{-\alpha} L\left(x \right), \quad x\rightarrow \infty,$ where $$\bar{F}=1-F$$ and $$L\left(x \right)$$ is a slowly varying function, satisfying $$L\left(tx \right) / L\left(x \right) \rightarrow 1$$ as $$x\rightarrow \infty,$$ for any $$t>0$$. The parameter $$\alpha>0$$ is called the tail index or the extreme value index.
Using an empirical version of the real part of the characteristic function $$U\left(t \right)= E\left[\cos\left( tX\right) \right],$$ i.e. $$U_{n}\left(t \right)= n^{-1} \sum_{j=1}^{n} \cos\left( tX_{j}\right)$$, and evaluating $$U_{n}\left( t\right)$$ at points $$t_{j}=j/\sqrt{n}, \; j=1,2,\ldots,m,\, m=\left[ n^{\delta}\right], 0<\delta< 1/2,$$ the authors consider the regression equation $\log\left[\left( 1-U_{n}\left(t_{j} \right) \right) \right] \sim \log\left[g\left( \alpha,t_{j}\right) \right] + \alpha\log t_{j} + \varepsilon_{j}, \quad 0<\alpha\leqslant 2,$ where $$\varepsilon_{j} =\log \frac{1-U_{n}\left(t_j \right) }{1-U\left(t_j \right)}$$ and $$g\left(\alpha,t \right)= \pi/2 \left[ \Gamma\left( \alpha\right) \sin\left( \alpha\pi/2 \right) \right]^{-1} L\left(1/t \right)$$ if $$0< \alpha<2$$ and $$g\left(\alpha,t \right)= \int_{0}^{1/t} x \left[F\left(-x \right) + \bar{F}\left(x \right) \right] \, dx$$ if $$\alpha =2.$$
By ordinary least squares, they obtain a simple estimator $$\hat{\alpha}$$ for $$\alpha$$. The bias reduced version of $$\hat{\alpha}$$ is also developed. The authors define a procedure to obtain a reduced bias estimator of $$\alpha$$ optimized, according to generalized crossvalidation or restricted maximum likelihood, with respect to the choice of $$m,$$ i.e. $$\delta$$.
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, illustrate, show the proposed method works in practice.
##### MSC:
 62G32 Statistics of extreme values; tail inference 62J05 Linear regression; mixed models 62P25 Applications of statistics to social sciences
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