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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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