ESTIMATOR DERET FOURIER UNTUK ESTIMASI KURVA REGRESI NONPARAMETRIK BIRESPON

Tripena, Agustini

MAGISTRA
Vol 25, No 84 (2013): Magistra Edisi Juni

In the last decade Fourier series estimator in nonparametric regression (one response) a lot
of attention from researchers because of its flexibility. In this paper will be developed in Fourier series estimator
in nonparametric regression of two responses (Birespon). Data are given in pairs (t1j, y1j), j = 1, 2, ... n1 and (t2j,
y2j), j = 1, 2, ... n2,. The relationship between t1j, t2j, y1j and y2j following nonparametric regression model
Birespon: y1j = f1 (t1j) +  1j and y2j = f2 (t2j) +  2j
Form of regression curves f1 and f2 are unknown and assumed to be contained within the space of continuous
functions (0,  ). Random error  1j mutually independent with mean zero and variance 2
1 , and  1j also
mutually independent with mean zero and variance 2
2 . Random error  1j and  2j are correlated with the
Cor(  1j,  2j) = r. Regression curve f1(t) and f2(t) respectively were approached by a continuous and differentiable
function:
1 1( )jd t = 1 1 01 1 1
1
1 cos
2
K
j k j
k
t kt  

   , And 2 2( )jd t = 2 2 02 2 2
1
1 cos
2
K
j k j
k
t kt  

   ,
 1  ,  2  , 01  , 02  , 1k  , k = 1, 2, ..., n1, 2k  , k = 1, 2, ..., n2 are parameters that are
unknown in the Fourier series model.
Estimated nonparametric regression curve Birespon f1(t) and f2(t) obtained from the complete optimization
Penalized Weighted Least Square (PLST):







  
 

0 0
2
2
2221
2
111
2
2
2
1
1 dttfdttfywy ]([ ]([ ))(,,()(
n (n
""
21
Completion of this form of optimization PLST Fourier series estimator that can be presented in the form:
1
2
11
1 2
22
ˆ ( )ˆ ( ) ( , )
ˆ ( )
yf t
f t B
yf t



 
   
    
    



. Fourier series estimator in nonparametric regression Birespon is biased
to nonparametric regression curve
1 1
2 2
( )
( )
( )
f t
f t
f t
 
  
 
Although biased, but this estimator is a linear estimator, which is very supportive in building statistical
inference for nonparametric regression curve Birespon...

Key words: Fourier series estimator, Nonparametric Regression Birespon, Penalized Weighted Least Square