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 chance to a wide quantity to make sure moderate, and occasionally optimum procedures.* Treats the entire uncomplicated and demanding issues in multivariate statistics.* provides new chapters, besides a few new sections.* offers the main methodical, updated info on MV information to be had.
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Extra info for An Introduction to Multivariate Statistical Analysis (Wiley Series in Probability and Statistics)
1. The Characteristic Function The characteristic function of a multivariate normal distribution has a form similar to the density function. From the characteristic function, moments and cumulants can be found easily. 1. The characteristic function of a random vector X is 4>(t) (1) = $el/'x defined for every real vector t. , To make this definition meaningful we need to define the expected value of a complex-valued function of a random vector. 2. Let the complex-valued function g(x) be written as g(x) glCe) + igix), where gl(X) andg 2(x) are real-valued.
LII) + ~(x(2) - 1-1-(2») is called the regression function. Let u i ",+1. ,' be the i,jth element of l:11'2' We call these partial cuuarial/ces; if,i'" + I. "" is a partial variance. 2 (8) Pij·q + I .... p yu U i j-q+I, .... Iu.. P i,j=l, ... ,q, ' is the partial correlation between Xi and Xj holding Xq + 1, ... , Xp fixed. 5 CONDITIONAL DISTRIBUTIONS; MULTIPLE CORRELATION 35 The numbering of the components of X is arbitrary and q is arbitrary. Hence, the above serves to define the conditional distribution of any q components of X given any other p - q components.
Suppose u'Y is normally distributcd for every u. Then the characteristic function of such linear combination is (19) Now set t = 1. 3 below). 2. If every linear combination of the components of a vector Y is normally distributed, then Y is normally distributed. 2 to hold. For instance, if Y = (Yl , Y 2 )' and Yl and Y 2 are not independent, then Yl and Y2 can each have a marginal normal distribution. An example is most easily given geometrically. Let Xl' X 2 have a joint normal distribution with means O.