WEIGHTED QUANTILE CORRELATION TESTS FOR GUMBEL,
WEIBULL AND PARETO FAMILIES
Sándor Csörgõ
Tamás Szabó
Abstract: Weighted quantile correlation tests are worked out for the Gumbel location and
location-scale families. Our theoretical emphasis is on the determination of computable
forms of the asymptotic distributions under the null hypotheses, which forms are
based on the solution of an associated eigenvalue-eigenfunction problem. Suitable
transformations then yield corresponding composite goodness-of-fit tests for the Weibull
family with unknown shape and scale parameters and for the Pareto family with an
unknown shape parameter. Simulations demonstrate slow convergence under the
null hypotheses, and hence the inadequacy of the asymptotic critical points. Other
rounds of extensive simulations illustrate the power of all three tests: Gumbel against
the other extreme-value distributions, Weibull against gamma distributions, and
Pareto against generalized Pareto distributions with logarithmic slow variation.
2000 AMS Mathematics Subject Classification: Primary: 62F05, 62E20; Secondary:
60F05.
Keywords and phrases: Composite goodness of fit; Gumbel, Weibull and Pareto
families; weighted quantile correlation tests; asymptotic distributions; speed of convergence;
power.