Testing Extreme Value Models

AbstractWe consider the full statistical families of extreme value distributions $$G_{\beta ,\sigma ,\mu } $$ and generalized Pareto distributions $$H_{\beta ,\sigma ,\mu } $$ , where $$\beta \in \mathbb{R},\sigma > 0$$ , and $$\mu \in \mathbb{R}$$ denote the shape, scale and location parameters, respectively. We consider the testing problems $$\beta \geqslant 0$$ against $$\beta < 0$$ and β = 0 against β ≠ 0, where σ and μ are treated as nuisance parameters. Showing local asymptotic normality (LAN), we derive asymptotic envelope power functions for test sequences and establish tests which attain these upper bounds. The finite sample size behavior is studied by simulations.

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