<strong>Paper Title</strong><br>
Mixed Estimator of Kernel and Fourier Series in Nonparametric Regression<br>
<br>

<strong>Abstract</strong><br>
Lets paired the observation
 v
1 i
, v 2 i , ..., v pi , t 1 i , t 2 i , ..., t qi , y i  , i  1, 2,..., n , follow the additive nonparametric
  v i , t i    i , where   v
regression model y i 
p
, t i  
i
q
 g  v    h  t  ,
j
ji
s
j  1
si
s  1
v i   v 1 i , v 2 i ,..., v pi   , and t i   t 1 i , t 2 i ,..., t qi   . Random errors  i is a normal distribution with mean 0 and
variance
 2 . The aim of this study is obtain a mixed estimator   v , t  . In order to accomplish the aim, the regression
i
Φ    1 ,  2 ,  ,  p   and the regression curve component
 
curve g j v ji is approached by kernel with bandwidths
h s  t si  is fourier series where it is approached by T s  t si   b s t si 
p
p
The estimator
 g  v 
j
ji
is
1
2
M
a 0 s 
 a
ks
cos kt si with oscillation paremeter M.
k  1
p
 g ˆ  v 
j  j
ji
where
j  1
j  1
i
 g ˆ  v   V  Φ  y .
j  j
Based on Penalized Least Squares (PLS)
ji
j  1
method

q
Min R  a  
a
s
s
si
s  1
q
 h ˆ

  J  T  t  
with smoothing parameter
λ    1 ,  2 ,  ,  q   ,
q
the estimator
 h  t 
s
si
is
s  1
q
 s , M
 t  ,
si
and
s  1
 h ˆ
 s , M
 t   Xa ˆ  λ  . So that, the mixed estimator μ  v , t 
si
i
i
is
s  1
μ ˆ Φ , λ , M  v i , t i   Z  Φ , λ , M  y


  

where Z Φ , λ , M y  V Φ  S Φ , λ , M
  y .
 
Matrix V Φ

and S Φ , λ , M
parameter Φ , smoothing parameter λ and oscillation paremeter M. Optimal
Generalized Cross Validation (GCV).

are depended on bandwidths
Φ , λ and M can be obtained by using
Keywords- Mixed Nonparametric Regression, Kernel, Fourier Series