Computation of Distribution Functions for Statistics Under Simple Order Restrictions

Abstract

To execute the multiple comparison methods introduced in Chaps. 1–4, we need the upper $100\alpha \%$ points for the distributions of the statistics. For cases with no order constraints on k means (see Chaps. 1, 2, and 4), we can use executable programs listed in Appendix B of Shiraishi (Multiple comparison procedures under continuous distributions. Kyoritsu Shuppan Co., Ltd., 2011) for the values of the upper $100\alpha \%$ points of the distributions. On the other hand, when there are order constraints on the location parameters introduced in Chap. 3, the multiple comparison methods require more complicated numerical calculations for the statistical distribution. The distributions in these cases are derived from the normal distribution. The key objective of the numerical calculation is the integral transformation of the Gaussian function. We set the function family $\mathbf {G}$ as a generalization of the Gaussian function and introduce Sinc approximation method suitable for the approximation and calculus of functions belonging to $\mathbf {G}$. As a specific application, in Sect. 7.2, we introduce calculation methods concerning the distribution function of the Hayter type statistics, which is the distribution of the max statistics in Chap. 3. Then, in Sect. 7.3, we introduce a numerical method to calculate the level probability for the multiple comparison methods that is based on the sum-of-squares statistics described in Sect. 3.3. The proofs for the theorems are not presented in this chapter. Please visit the following webpage for the proofs: http://www.st.nanzan-u.ac.jp/info/sugiurah/sincstatistics.