On some problems associated with D2- statistics and p-statistics [by] P.K. Bose and S.B. Chaudhuri

his monograph deals with the recursion formulae and tables for the percentage points of classical D2, studentized D2 and for p = 2 and 3 the maximum value of ^-statistics (characteristic roots of S^S^S^'1, S? and S2 are independent Wishart and p the order of matrix ?$). The main part of the book (namely, Chapter II (except Section 2.5), Chapter III (except Section 3.3) and Chapter IV) is a collection of their published papers in Sankhy?( 1947 and 1954) and Cal. Statist. Assoc. Bull. (1949 and 1956). The first chapter (pp.1-3) deals with a general method of reduction of the probability integral into basic elementary and auxiliary functions. These basic elementary functions have to be calculated numerically and the latter by suitable recursion chain. It should be mentioned here that 5% values of the transformed classical D2-statistic upto four decimal figures for p = 1(1)7 and ? = 0(.2)5.0 are contained in a paper by Fisher, dealing with the limiting distribution of the square of the multiple correlation coefficient. This work which appeared in the Proceedings of the Royal Society, Series A, 1928, Vol. 121, pp. 654-673 has not been noticed by the authors though the method employed by the authors is practically similar. In Section 2.5, a method indicates when to replace a distribution of a standardised random variable by a normal distribution. This is applied to the classical D2-distribution and the Table 1.7 gives the maximum error involved in the normal approxi mation to the classical ?^-distribution. From this, it is suggested that the normal approxi mation can be used with a maximum error of 1% if ? ^ 20. Patnaik (Biometrika, 1949, Vol. 36, 202-232) gives a central ^-approximation for the non-central ^-distribution and this formula works well for the smaller values of ?. Noting that the distribution of the trans formed classical D2-statistic is a non-central #2, a reference to Patnaik's work in this volume would have been highly appropriate. Chapter V (pp. 33 to 42) deals with a numerical example only to show the use of these statistics and the tables given in Chapters II, III and IV, though a discussion of computa tional aspects would have made the volume attractive. The book does not give a thorough understanding of D2-statistie, studentized D2 or ^-statistics for a beginner in these topics. It is not written in the style of a text book. There are a few printing mistakes, e.g., on p. 5, in the third line from bottom, ?'2, should read as - A2 instead of 2A2.