Page 124 - Contributed Paper Session (CPS) - Volume 5

CPS1141 Mahdi Roozbeh

Similar to Proposition 2.1, we have the following result without proof.
Proposition 3.1. The partially generalized restricted and unrestricted ridge
estimators of have the following relation
1

Up to this point, we supposed that the null hypothesis Ho: =0 is true,
2
however, it must be tested that one can incorporate the PGURE in practice. For
this purpose, following Saleh (2006) and Yuzbashi and Ahmed (2015), we use
the following test statistic for testing the sparsity hypothesis Ho

Later, it will be shown that the test statistic £n has asymptotic chi-square
distribution with p2 degrees of freedom. The following result is a direct
conclusion of Theorem 2 of Knight and Fu (2000).
̂
̃
̃
̃
̃
T
T −1
-1
-1
Proposition 3.2. Let PG ( ) = ( + ) V . Then, under the

̂
regularity conditions (A1)-(A3), √(PG( ) − ) → (− , ).
−1
2 −1
0

,
According to Saleh (2006), the test statistic diverges as → ∞, under any fixed
alternatives : = . To overcome this difficulty, in sequel, we consider the
2

local alternatives

2
where = ( , . . . . . , ) ∈ ℝ is a fixed vector. For notational convenience,
1
2
let
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