Page 157 - Special Topic Session (STS) - Volume 4

STS577 Mahdi Roozbeh
To be more specific and motivate our approach, one way of controlling β
not to deviate much from the origin, i.e., satisfying the null-hypothesis Ho : β
T
= 0, is simply not to let ||β|| 2 = β β get larger. In other words, one may think
2
of constrained hypothesis testing in which the penalty term ||β|| < λ, for some
λ > 0, is taken into account. This key element recalls the well-known ridge
regression approach, where the ridge estimator of β is given by

T
T

ˆ
β(k) = (X X + kI ) −1 X y, (2.6)
p

for some ridge parameter k > 0. Concentrating on the shrinkage factor (XTX +
−1
−1
kIp) , one may think of replacing (XTX) by (XTX + kIp) in (2.4) rather than
−1
eliminating any component, when XTX is not invertible. Hence, we propose a
rank-based test statistic for testing (2.3), when β0 = 0 by incorporating the
−1
shrinkage factor (XTX + kIp) as

−1 T
-2 T
−1
−1
Rn(k) = σa a (R(y))X(n X X + kIp) [X(k)] (n X X + kIp) X a(R(y)), (2.7)
-1 T
−1 T

-1 T
where X(k) =(n X X + kIp) -k(n X X + kIp) is an invertible matrix formulated
−2
-1 T
−1
based on XTX (see Eq. (4.2)), σa = 1 ∑ and k > 0 is a regularization
2
2
−1 =1
(ridge) parameter. It will be shown that Rn(k) has approximate χ2 distribution
with p d.f.. Our test statistic has some advantages compared to the previously
proposed results, which are listed below:
1. Our approach does not take any asymptotic assumption for

lim −1 = Σ, i.e., of small o(.) concept.
→∞
2. There is no need to have many knowledges about asymptotic theory in
high-dimension to derive the approximate distribution of the test
statistic.
3. The estimate of Σ can be easily achieved.
4. The shrinkage estimator based on Rn(k) performs much better than the
ridge estimator in the sense of having smaller risk values.
Theorem 1 Under the specified model (1.1), reject H o in favor of H A at
2
2
approximate level α iff R n(k)≥ χ (α), where χ (α) denotes the upper level α
2
critical value of χ distribution with p d.f.

3. Shrinkage Estimator
In this section, we define a Stein-type shrinkage estimator for β based on
the rank-statistic Rn(k). Further, we evaluate the unknown parameter of our
estimator making use of a generalized cross validation criterion. According to
the result of Hettmansperger and McKean (1998, Ch.3), under the assumptions
of Section 1, for p < n case, the rank-estimate of β is given by

146 | I S I W S C 2 0 1 9