Page 123 - Contributed Paper Session (CPS) - Volume 5

CPS1141 Mahdi Roozbeh
from the origin. In the following result, the relation between the submodel and
fullmodel estimators of will be obtained.
1

Proposition 2.1. Under the assumptions of this section,

3. Sparse Multicollinear Model
Under situations in which the matrix C(ωn) is ill-conditioned due to linear
̃
relationship among the covariates of matrix (multicollinearity problem) or
the number of independent variables (p) is larger than sample size (n), the
proposed estimators in previous section are not applicable, because, we
̃
always find a linear combination of the columns in which is exactly equal to
one other column. Mathematically, the design matrix has not full rank, rank
̃
() ≤ min (n, p) < p for p > n, and one may write = ( + ) for every in
̃
̃
̃
the null space of .
Therefore, without further assumptions, it is impossible to infer or estimate
from data. We note that this issue is closely related to the classical setting
̃
with p < n but rank () < p (due to linear dependence among covariables) or
ill-conditioned design leading to difficulties with respect to identifiability. We
̃
note, however, that for prediction or estimation of (that is the underlying
semiparametric regression surface), identifiability of the parameters is not
necessarily needed. From a practical point of view, high empirical correlations
among two or a few other covariables lead to unstable results for estimating
or for pursuing variable selection. To overcome this problem, we follow
Roozbeh (2015) and obtain the restricted ridge estimator by minimizing the
sum of squared partial residuals 3 with a spherical restriction and a linear
restriction = 0, i.e., the RSRM is transformed into an optimal problem with
2
two restrictions:

The resulting estimator is partially generalized restricted ridge estimator
(PGRRE), given by

where kn ≥ 0 is the ridge parameter as a function of sample size n. In a similar
fashion as in previous section, partially generalized unrestricted ridge
estimators (PGUREs) of and respectively have forms
1
2

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