ABSTRACT
N/2] when H0 is true. The expansion converges extremely rapidly, and therefore the level of significance derived from it will be quite adequate even for moderate values of N. For large N, the Box result is equivalent to the large sample result of Wilks (1938); that is, under H0, - 21og ? is
distributed as central with pr degrees of freedom as N? 8 . The Box approximation (with p=r) is, under H0,
For the likelihood ratio criterion, exact tables are available only for p=4. The Lawley-Hotelling test criterion cannot be used for small samples sizes and appropriate p, since only a result asymptotic in sample size is available (see Anderson, 1958, p. 224; Pillai, 1954). Morrow (1948) has shown that, under H0, N
tr(AB-1) has central when N? 8 . The largest and the smallest root criteria of Roy can be used in the general case, although percentage point tables are available only for the restricted values of the parameters. Appropriate tables are given by Foster and Rees (1957), Foster (1957, 1958), Heck (1960), and Pillai (1960). Different criteria for this problem have been compared on the basis of their power functions, in some detail, by Smith et al. (1962) and Gabriel (1969).
