Page 176 - Applied Probability

t
The matrix H k = M k M is the household indicator matrix described
k
k
k
in the text. When X has the representation σ k M k W , one should
k
k
replace X in the complete data by σ k W . With this change, show
2
that the EM update for σ is
k
1 8. The Polygenic Model 161
t
2
σ n+1,k = [tr(Υ nk )+ ν ν nk ],
nk
s
where ν nk and Υ nk are
t
2
= σ M Ω −1 (y − Aµ)
ν k
k k
4
t
2
= σ I − σ M Ω −1
Υ k M k
k k k
evaluated at the current parameter vector γ n .
8. Demonstrate the following facts about the Kronecker product of two
matrices:
(a) c(A ⊗ B)=(cA) ⊗ B = A ⊗ (cB) for any scalar c.
t
t
t
(b) (A ⊗ B) = A ⊗ B .
(c) (A + B) ⊗ C = A ⊗ C + B ⊗ C.
(d) A ⊗ (B + C)= A ⊗ B + A ⊗ C.
(e) (A ⊗ B) ⊗ C = A ⊗ (B ⊗ C).
(f) (A ⊗ B)(C ⊗ D)= (AC) ⊗ (BD).
(g) If A and B are invertible square matrices, then
(A ⊗ B) −1 = A −1 ⊗ B −1 .
(h) If λ is an eigenvalue of the square matrix A with algebraic mul-
tiplicity r and µ is an eigenvalue of the square matrix B with
algebraic multiplicity s, then λµ is an eigenvalue of A ⊗ B with
algebraic multiplicity rs.
(i) If A and B are square matrices, tr(A ⊗ B) = tr(A) tr(B).
(j) If A is an m × m matrix, and B is an n × n matrix, then
m
n
det(A ⊗ B) = det(A) det(B) .
All asserted operations involve matrices of compatible dimensions.
(Hint: For part (h), let A = USU −1 and B = VTV −1 be the Jordan
canonical forms of A and B. Check that S ⊗ T is upper triangular.)
9. In some variance component models, several pedigrees share the same
theoretical mean vector µ and variance matrix Ω. Maximum likeli-
hood computations can be accelerated by taking advantage of this re-
dundancy. In concrete terms, we would like to replace a random sam-
ple y 1 ,...,y k from a multivariate normal distribution with a smaller