Page 122 - Contributed Paper Session (CPS) - Volume 5

CPS1141 Mahdi Roozbeh
2
symmetric, positive definite known matrix and σ is an unknown parameter. To
estimate the parameters of the model (2.1), we first remove the non-
parametric effect, apparently. Assuming to be known, a natural
̂
nonparametric estimator of f(.) is (t)= ()( − ), with k(t) = (Kωn (t,t1), . .
. , Kωn (t,tn)), where Kωn (.) is a kernel function of order m with bandwidth
̂
parameter ωn. For the existence of (, ) at the optimal convergence rate
n −4/5 , in semiparametric regression models with probability one, we need some
conditions on kernel function. See Muller (2000) for more details. Replacing
̂
f(t) by () in (2.1), the model is simplified to

̃
̃
= + , (2.2)

̃
̃
Where = ( − ). = ( − ) K is the smoother matrix with

i,jth component ( ). We can estimate the linear parameter in (2.1)
,
under the assumption cov () = , by minimizing the generalized sum of
2
squared errors
̃ ̃ −1 ̃ ̃
( = ( − ) ( − ). (2.3)
,
The unique minimizer of (2.3) is the partially generalized least squares
estimator (PGLSE) given by

Motivated by Fallahpour et al. (2012), we partition the regression
parameter as = ( , ) , where the subvector has dimension pi, i = 1,

2

1
2 and p1 + p2 = p. Thus the underlying model has form

̃
̃
̃
̃
where is partitioned according to (1, 2) in such a way that i is a n × pi
submatrix, i = 1, 2. With respect to this partitioning, the PGLSEs of and
1
2
are respectively given by

The sparse model is defined when Ho: = 0 is true. In this paper, we refer
2
restricted semiparametric regression model (RSRM) to the sparse model. For
the RSRM, the partially generalized restricted least squares estimator (PGRLSE)
has form

According to Saleh (2006), the PGRLSE performs better than PGLSE when
model is sparse, however, the former estimator performs poorly as deviates
2

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