GOODNESS-OF-FIT TESTS BASED ON ESTIMATED EXPECTATIONS OF PROBABILITY INTEGRAL TRANSFORMED ORDER STATISTICS

JAN W. H. SWANEPOEL AND FRANCOIS C. VAN GRAAN

Department of Statistics, Potchefstroom University for CHE, Potchefstroom 2520, South Africa

(Received February 5, 1999; revised February 13, 2001)

Abstract. New goodness-of-fit tests, based on bootstrap estimated expectations of probability integral transformed order statistics, are derived for the location-scale model. The resulting test statistics are location and scale invariant, and are sensitive to discrepancies at the tails of the hypothesized distribution. The limiting null distributions of the test statistics are derived in terms of functionals of a certain Gaussian process, and the tests are shown to be consistent against a broad family of alternatives. Critical points for all sample sizes are provided for tests of normality. A simulation study shows that the proposed tests are more powerful than established tests such as Shapiro-Wilk, Cramér-von Mises and Anderson-Darling, for a wide range of alternative distributions.

Key words and phrases: Bootstrap, consistency, Gaussian process, Monte Carlo simulation, tests for normality, Shapiro-Wilk test.