Testing whether the shape parameter is zero in the generalized extreme-value distribution

SUMMARY When a sequence of extreme values of a physical process has been observed it is often of interest to test whether the observations are distributed according to a type I FisherTippett extreme-value distribution rather than one of types II or III. This is equivalent to testing whether the shape parameter is zero in the generalized extreme-value distribution. Thirteen tests of this hypothesis are here compared in terms of their empirical significance levels and power in small samples. A-modified likelihood ratio test is found to give the best overall performance, and a test due to van Montfort & Otten is also recommended.

[1]  R. Fisher,et al.  Limiting forms of the frequency distribution of the largest or smallest member of a sample , 1928, Mathematical Proceedings of the Cambridge Philosophical Society.

[2]  Calyampudi R. Rao.  Large sample tests of statistical hypotheses concerning several parameters with applications to problems of estimation , 1948, Mathematical Proceedings of the Cambridge Philosophical Society.

[3]  A. Jenkinson.  The frequency distribution of the annual maximum (or minimum) values of meteorological elements , 1955 .

[4]  D. Lawley.  A GENERAL METHOD FOR APPROXIMATING TO THE DISTRIBUTION OF LIKELIHOOD RATIO CRITERIA , 1956 .

[5]  S. D. Silvey,et al.  The Lagrangian Multiplier Test , 1959 .

[6]  J. D. T. Oliveira,et al.  Decision results for the parameters of the extreme value (Gumbel) distribution based on the mean and the standard deviation , 1963 .

[7]  Calyampudi Radhakrishna Rao,et al.  Linear Statistical Inference and its Applications , 1967 .

[8]  Prem S. Puri,et al.  On Optimal Asymptotic Tests of Composite Statistical Hypotheses , 1967 .

[9]  W. E. Bardsley.  A test for distinguishing between extreme value distributions , 1977 .

[10]  Michael A. Stephens,et al.  Goodness of fit for the extreme value distribution , 1977 .

[11]  M.A.J. van Montfort,et al.  On testing a shape parameter in the presence of a location and a scale parameter , 1978 .

[12]  A. Otten,et al.  The power of two tests on the type of distributions of extremes , 1978 .

[13]  J. Tiago de Oliveira,et al.  Statistical Choice of Univariate Extreme Models , 1981 .

[14]  P. Prescott,et al.  Maximum likeiihood estimation of the parameters of the three-parameter generalized extreme-value distribution from censored samples , 1983 .

[15]  D. Carter,et al.  Monthly extreme air temperatures at Rothamsted , 1983 .