Testing for the shape parameter of generalized extreme value distribution based on the <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$$L_q$$</EquationSource> </InlineEquation>-likelihood ratio statistic

This paper studies the applications of extreme value theory on analysis for closing price data of the Dow-Jones industrial index and Danish fire insurance claims data. The generalized extreme value (GEV) distribution is considered in analyzing the real data, and the hypothesis testing problem for the shape parameter of GEV distribution is investigated based on a new test statistic—the <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$$L_q$$</EquationSource> </InlineEquation>-likelihood ratio (<InlineEquation ID="IEq4"> <EquationSource Format="TEX">$$L_q$$</EquationSource> </InlineEquation>R) statistic. The <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$$L_q$$</EquationSource> </InlineEquation>R statistic can be treated as a generalized form of the classical likelihood ratio (LR) statistic. We show that the asymptotic behavior of proposed statistic is characterized by the degree of distortion <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$$q$$</EquationSource> </InlineEquation>. For small and modest sample sizes, the <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$$L_q$$</EquationSource> </InlineEquation>R statistic is still available when <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$$q$$</EquationSource> </InlineEquation> is properly chosen. By simulation studies, the proposed statistic not only performs the asymptotic properties, but also outperforms the classical LR statistic as the sample sizes are modest or even small. Meanwhile, the test power based on the new statistic is also simulated by Monte Carlo methods. At last, the models are diagnosed by graphical methods. Copyright Springer-Verlag 2013