MANOVA Test Statistics

The main purpose of this chapter is to give a method for obtaining error bounds for asymptotic expansions of the null distributions of Hotelling’s $$T^2$$ (or Lawley–Hotelling criterion, $$T_{LH}$$ ), the likelihood-ratio criterion $$T_{LR}$$ and the Bartlett–Nanda–Pillai criterion $$T_{BNP}$$ in the MANOVA model when the sample size is large. The results for $$T_{LH}$$ and $$T_{LR}$$ are obtained by expressing these statistics in terms of a multivariate scale mixture, and using error bounds evaluated in $$L_1$$ -norm. The error bound is given for the limiting distribution of $$T_{BNP}$$ by using a relationship between $$T_{BNP}$$ and $$T_{LH}$$ . Further, we give error bounds for these criteria when the sample size and the dimension are large.