By Theodore W. Anderson
Perfected over 3 variants and greater than 40 years, this box- and classroom-tested reference:* makes use of the tactic of extreme probability to a wide volume to make sure moderate, and every now and then optimum procedures.* Treats the entire simple and critical subject matters in multivariate statistics.* provides new chapters, in addition to a couple of new sections.* offers the main methodical, up to date details on MV records to be had.
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Extra info for An Introduction to Multivariate Statistical Analysis (Wiley Series in Probability and Statistics)
Now. L and covariance matrix ~ of rank r, it can be written as (34) (except for 0 probabilities), where X has an arbitrary distribution, and Y of r (5. p) components has a suitable distributiun. If 1 is of rank r, there is a p X P nonsingular matrix B such that (36) B~B'=(~ ~), where the identity is of order r. =v= V(I») ( V(2) , say. Since the variances uf the elements of probability 1. Now partition (39) V(2) are zero, V(2) = V(2) with B- 1 = (C D), where C consists of r columns. Then (37) is equivalent to (40) Thus with probability 1 (41) which is of the form of (34) with C as A, V(1)as Y, and Dv(2) as A.
L,)/U"j. The mean squared difference between the two standardized variables is (53) The smaller (53) is (that is, the larger p is), the more similar Yl and Yz are. If p> 0, XI and X 2 tend to be positively related, and if p < 0, they tend to be negatively related. If p = 0, the density (52) is the product 0: the marginal densities of XI and X 2 ; hence XI and X z are independent. It will be noticed that the density function (45) is constant on ellipsoids (54) for every positive value of c in a p-dimensional Euclidean space.
Transformation of Variables Let the density of XI' ... ' Xp be f(x l , ... , x p). Consider the p real-valued,' functions ~ (33) i= 1, ... ,p. We assume that the transformation from the x-space to the y-space is one-to-one;t the inverse transformation is (34) i = 1, ... ,p. 'More precisely. (XI' ... 'X p ) is positive. , ... , y" be defined by (35) i = 1, . ,p. Y; = Yi( XI'···' Xp), Then the density of YI , ... , ... , Yp) is the Jacobian (37) J(Yl' ... 'Yp) = aX I aYI ax! ayz aX I ayp ax z ax z ayz axz ayp axp ay!
An Introduction to Multivariate Statistical Analysis (Wiley Series in Probability and Statistics) by Theodore W. Anderson