Robert_V._Hogg,_Joseph_W._McKean,_Allen_T._Craig

9.8. The Distributions of Certain Quadratic Forms 555

(a)Using the baseball data, determine the estimate and the confidence interval for

the correlation coefficient between height and weight for professional baseball

players.

(b)Separate the pitchers and hitters and for each obtain the estimate and confi-

dence for the correlation coefficient between height and weight. Do they differ

significantly?

(c)Argue that the difference in the estimates of the correlation coefficients is the

mle ofρ 1 −ρ 2 for two independent samples, as in Part (b).

9.7.8.Two experiments gave the following results:

n x ysx sy r

100 10 20 5 8 0.70

200 12 22 6 10 0.80

Calculaterfor the combined sample.

9.8 The Distributions of Certain Quadratic Forms

Remark 9.8.1.It is essential that the reader have the background of the multi-

variate normal distribution as given in Section 3.5 to understand Sections 9.8 and

9.9.

Remark 9.8.2.We make use of thetraceof a square matrix. IfA=[aij]isan
n×nmatrix, then we define the trace ofA, (trA), to be the sum of its diagonal
entries; i.e.,

trA=

∑n

i=1

aii. (9.8.1)

The trace of a matrix has several interesting properties. One is that it is a linear
operator; that is,

tr (aA+bB)=atrA+btrB. (9.8.2)

A second useful property is: IfAis ann×mmatrix,Bis anm×kmatrix, andC

is ak×nmatrix, then

tr (ABC)=tr(BCA)=tr(CAB). (9.8.3)

The reader is asked to prove these facts in Exercise 9.8.7. Finally, a simple but
useful property is that tra=a, for any scalara.

We begin this section with a more formal but equivalent definition of a quadratic
form. LetX=(X 1 ,...,Xn)beann-dimensional random vector and letAbe a
realn×nsymmetric matrix. Then the random variableQ=X′AXis called a