Rates of convergence of extreme for general error distribution under power normalization (Q419250)

The authors consider the rates of convergence of extremes for a general error distribution (GED) \(F\) under power normalization. The probability density function of the GED is given by NEWLINE\[NEWLINE f_{\upsilon}\left( x\right) =\frac{\upsilon\exp\left\{ -\left( 1/2\right) \left| \text{\thinspace}x/\lambda\right| ^{\upsilon}\right\} }{\lambda 2^{1+1/\upsilon}{\Gamma}\left( 1/\upsilon\right) } NEWLINE\]NEWLINE for \(\upsilon>0\) and \(x\in\mathbb R\), where \(\lambda=\left[ 2^{-2/\upsilon }{\Gamma}\left( 1/\upsilon\right) /{\Gamma}\left( 3/\upsilon\right) \right] ^{1/2}\) and \({\Gamma}\left( \cdot\right) \) denote the Gamma function. The GED belongs to the domain of attraction of a distribution \({\Phi}_{1}\) under power normalization, where \({\Phi}_{1}\left( x\right) =\exp\left\{ -x^{-1}\right\} \), \(x>0\).NEWLINENEWLINEHe provides two main results. The first result states that the uniform convergence rate of \(F^{n}\left( \alpha_{n}x^{\beta_{n}}\right) \) to its limit \({\Phi}_{1}\left( x\right) \), where \(\alpha_{n}>0\) and \(\beta_{n}>0\) are some normalizing constants, is of order \(O\left( \frac{1}{\log n}\right)\). The second result shows that the pointwise convergence rate of \(F^{n}\left( \alpha_{n}x^{\beta_{n}}\right) \) to its limit is of order \(O\left( \exp\left\{ -x^{-1}\right\} \frac{1}{x} \beta_{n}\right) \).