Homework 1 STAT 547 solved

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1. Exercise 1.1 (b) & (c), Kokoszka and Riemherr (2017).
2. Exercise 1.4, Kokoszka and Riemherr (2017). The datasets in the book can be found
here: http://www.personal.psu.edu/mlr36/Documents/KRBook_DataSets.zip
3. For this and the next question, consider a multivariate random variable X ∈ R
d
, d ≥ 2.
Find the projection directions vk, k = 1, . . . , d for the principal component analysis
obtained in the following stepwise fashion:
v1 = arg max
kl1k=1
Var(l
|
1X)
vk = arg max
klkk=1
l
|
k
lj=0,
j=1,…,k−1
Var(l
|
kX), k = 2, . . . , d.
4. Let ξk = v
|
k
(X − µ), k = 1, . . . , d, where µ = E(X). Then
(a) E(ξk) = 0
(b) Var(ξk) = λk
(c) Cov(ξj
, ξk) = λkδjk, where δjk = 1 if j = k and 0 otherwise.
(d) Corr(Xj
, ξk) = √
λkvjk/
√σjj , where Xj and vjk are the jth entry of X and vj
,
respectively, and σjj is the jth diagonal entry of Σ = Cov(X).
5. Exercise 10.1, Kokoszka and Reimherr 2017
6. Exercise 10.3, Kokoszka and Reimherr 2017
7. Let X(t), t ∈ [0, 1] be a stochastic process for which the sample paths lie in L
2
([0, 1]).
Show that the solution to the following problem minimizing the residual variance coincides with the projection directions in the functional principal component analysis:
min EkX −
X
K
k=1
hX, ekiekk
2
.
The minimum is taken over orthonormal functions e1, . . . , eK, K ≥ 1.
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