Page 158 - Special Topic Session (STS) - Volume 4

STS577 Mahdi Roozbeh
T
T
= (X X) −1 X yˆ , (3.1)

̂
ψ

where yˆ ψ is the minimizer of Dψ (η) = ||y –η||ψ over η ∈ ΩF and ||v||ψ =
ˆ
∑ (( )) . Thus, β ψ is the solution to the rank-normal equations

=1

T
X a(R(y−Xβ)) = 0. In the same fashion as in formulating the ridge estimator,
we define a robust ridge estimator as
-1 T −1 T
(k) = (n X X + kI ) X yˆ , (3.2)
̂
ψ

p

where k > 0 is the ridge parameter.
One way of improving upon the robust ridge estimator, is to incorporate
the information consists in β = 0. Preliminary test estimator, emanating from
ˆ
testing the null-hypothesis Ho, which leads to select one of the extremes 0 or β
(k) depending on the output of test, is one way of improvement. However, this
ψ
estimator is heavily dependent on the size of the test and has discrete nature.
Hence, we consider its continuous version, and shrink the ridge estimator
toward the origin by proposing the following Stein-type shrinkage estimator
(SSE)

ˆ

(k, d) = (1- ) β (k) (3.3)
̂

ψ
()
−1
ˆ
= (k) − dR (k) β (k), d > 0.
̂

ψ

n
The SSE depends on ridge parameter k and shrinkage parameter d that
must be evaluated in practice. To this end and finding the optimal values, we
use the generalized cross validation (GCV) criterion for selecting the optimal
values of both parameters, simultaneously. The GCV has been applied for
obtaining the optimal ridge parameter in a ridge regression model (Golub et
al., 1979) and for obtaining the optimal ridge parameter and bandwidth of the
kernel smoother in semiparametric regression model (Amini and Roozbeh,
2015) as well as partial linear models (Speckman, 1988). Our proposed GCV
criterion creates a balance between the precision of the estimators and the
biasedness caused by the ridge and shrinkage parameters. The GCV function is
then defined as

1 ||(−(,)|| 2
(, ) = 2 (3.4)
1
(1− ((,)))

-1 T −1 T

where L(k, d) = (1- )X(n X X + kI ) X .

p
()

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