OPTIMAL IMS FOR VARIABLE SELECTION 191
D
−1/2
MD
−1/2
. Note, in particular, that θ
j
= 0 if and only if β
j
= 0, so the vari-
able selection problem has not been changed; furthermore, the matrix L has all ones
on the diagonal.
10.5.2 Variable selection assertions
Recall the dimension-reduced association
ˆ
θ = θ +
ˆ
σU in (10.7), where θ =
(θ
1
,.. . ,θ
p
)
>
and U is a p-vector distributed as t
p
(0,L,n − p −1), and L is a ma-
trix with ones on the diagonal. The goal is identify which of θ
1
,.. . ,θ
p
are non-zero.
Consider the collection of complex assertions A
j
= {θ : θ
j
6= 0}, j = 1,. . ., p.
Consider first a particular A
j
. This can be written as A
j
= A
j1
∪A
j2
, where A
j1
=
{θ : θ
j
< 0} and A
j2
= {θ : θ
j
> 0}. We claim that these sub-assertions are both
simple in the sense of