Page 155 - Special Topic Session (STS) - Volume 4

STS577 Mahdi Roozbeh

=
y = x  +  , i 1 2 , ,..., n (1.1)
i
i
i

where  = ( 1 , 2 ,..., p ) is the vector of regression coefficients and  is the
i
i th error component, having a continuous cumulative distribution function
(c.d.f.), f(.) And finite fisher information, ( ),
 ( ' f ) x  2 dF (x ) d 2 F (x )

( I ) f =     f (x )dx  , f (x ) = , ( ' f ) x = (1.2)

R  f (x )  dx dx 2

Now, consider a regression model in the presence of multicollinearity. The
existence of multicollinearity may lead to wide confidence intervals for the
individual parameters or linear combination of the parameters and may
produce estimates with wrong signs. For our purpose we employ the ridge
regression concept due to Hoerl and Kennard (1970), to combat
multicollinearity. There are a lot of works adopting ridge regression
methodology to overcome the multicollinearity problem. To mention a few
recent researches, see Hassanzadeh Bashtian et al. (2011), Amini and Roozbeh
(2015), Arashi et al. (2015), Arashi and Valizadeh (2015) and Roozbeh (2016).
When < , a classical method to deal with this problem is the famous
F-test statistic. However, it is shown that the power of F-test is adversely
impacted by an increased dimension. Moreover, the F-test statistics is
undefined when the dimension of data is greater than the within sample
degrees of freedom since the pooled sample covariance matrices are not
positive definite. In order to overcome this issue, we propose a robust test
based on rank regression in case > .
We organize this article as follows: In Section 2, we propose a robust ridge
test statistic based on rank regression. In Section 3, some regularity conditions
are given, while approximate distribution of the test statistic is given in Section
4. Definition of a Stein-type shrinkage estimator and evaluating the ridge and
shrinkage parameters using GCV criterion, are the content of Section 5.
Numerical studies are the context of Section 6. We conclude our results in
section 7.

2. Robust Test Statistics
The problem of our study is to find a rank-based test statistic for testing
the following set of hypotheses:

H : β = β vs H : β≠β . (2.3)
0
A
o
0
Testing hypotheses similar to (2.3), has been considered by many authors.
To be more specific about their finding and predispose our result, we first
consider the following rank-based score test (see Hettmansperger and

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