210 Some Special Distributions(a)Find the total variation ofX.(b)Find the principal component vectorY.(c)Show that the first principal component accounts for 90% of the total varia-
tion.(d)Show that the first principal componentY 1 is essentially a rescaledX. Deter-
mine the variance of (1/2)Xand compare it to that ofY 1.Note that the R commandeigen(amat)obtains the spectral decomposition of the
matrixamat.
3.5.22.Readers may have encountered the multiple regression model in a previous
course in statistics. We can briefly write it as follows. Suppose we have a vector
ofnobservationsYwhich has the distributionNn(Xβ,σ^2 I), whereXis ann×p
matrix of known values, which has full column rankp,andβis ap×1 vector of
unknown parameters. The least squares estimator ofβis
̂β=(X′X)−^1 X′Y.(a)Determine the distribution of̂β.(b)LetŶ=Xβ̂. Determine the distribution ofŶ.(c)Let̂e=Y−Ŷ. Determine the distribution of̂e.(d)By writing the random vector (Ŷ′,̂e′)′as a linear function ofY, show that
the random vectorsŶand̂eare independent.(e)Show thatβ̂solves the least squares problem; that is,‖Y−X̂β‖^2 =min
b∈Rp
‖Y−Xb‖^2.3.6 t-andF-Distributions
It is the purpose of this section to define two additional distributions that are quite
useful in certain problems of statistical inference. These are called, respectively, the
(Student’s)t-distribution and theF-distribution.
3.6.1 Thet-distribution
LetW denote a random variable that isN(0,1); letV denote a random variable
that isχ^2 (r); and letW andV be independent. Then the joint pdf ofW andV,
sayh(w, v), is the product of the pdf ofWand that ofVor
h(w, v)={
√^1
2 πe−w^2 / 2 1
Γ(r/2)2r/^2 vr/ 2 − (^1) e−v/ (^2) −∞<w<∞, 0 <v<∞
0elsewhere.