Abstract
We consider multiple comparisons tests for the differences among mean responses in k normal populations．Hayter (1990) proposed single-step procedure as all-pairwise comparison test between ordered treatments. Lee and Spurrier (1995) also discussed single-step procedure as successive comparison test. Shiraishi (2014) proposed closed testing procedures which are superior to Hayter (1990) and Lee and Spurrier (1995). To execute the closed testing procedures, Shiraishi (2014) utilized the upper 100α★th percentiles of the distribution of maxi<i′ (−tii′), where α★=1−(1−α)ℓ/M (2 ≦ ℓ ≦ M ≦ k) and tii''s denote the two sample t test statistics. We investigate the properties of the density functions which appear in the distribution of maxi<i′ (−tii′). Based on it, we show these density functions are approximated efficiently with using sinc method described in Lund and Bowers (1992) and Stenger (1993). Finally, we describe the algorithm to calculate upper 100α★%th percentiles of the closed testing procedure superior to Hayter (1990). Numerical examples are given to demonstrate the performance of our scheme.
