Page 159 - Special Topic Session (STS) - Volume 2

STS474 Yoshihiro Y.
(3) For = (which we assume as an integer), the kernel =
⁄

∑ ℎ exp() satisfies
=1

() sin ( 2)
⁄

(λ) = −1 [ sin( 2) ] ,

⁄
where () is a complex function, whose modulus is bounded and bounded

away from zero, with − 1 derivatives, all bounded in modulus as
increases for ∈ [−, ].
Then we define the discrete Fourier transform (DFT) ( , ) and the

2
1
periodogram ( , ) by

1
2
( , ) = 1

1
2
)),
∑ ℎ ℎ ( ) exp((
2 ∑ ℎ 2 ,=1 1 , 2 ,
=1
2
( , ) = | ( , )| ,

1
2
2
1
respectively where 1 2 is the bivariate Fourier frequency,
2 1 2 2 ( − 1)
= ( , ) = ( , ) , [ 2 ], ≤ , ≤ [ ],
1
2

2
1
2
and [] is the integer part of and {ℎ } is a sequence of data tapers of order

.
Next we define the normalized DFT by
( 1 , 2 ) = ( 1 , 2 ) ,

2
2
(( + ) −(+1) ) 1/2
2
1
( 1 , 2 ) = Re( (( 1 , 2 )),

( 1 , 2 ) = Im( (( 1 , 2 )),

where = , (0,0)/(8 ). H
2
0
Now we introduce the estimator. We denote by and the true
0
0
parameters, and by and any admissible values. Then define the closed
interval of admissible parameters of , ℋ = [∆ , ∆ ], where ∆ and ∆ are
2
1
2
0
1
numbers chosen such that 0 < ∆ < ∆ < 1. We can choose ∆ and ∆
1
2
1
2
arbitrarily close to 0 and 1 respectively. Next consider the objective function
(, ) = 1 ∑ {log (( 2 + 2 ) −−1 )
1 2
( 1, 2)∈
2
2
( 1 + 2 ) +1
+ ( 1 , 2 )},

where
= {( ) | , ≤ ( 1 ) + ( 2 ) ≤ , , 0 < , , ≤ 2 ≤ },
2
2
2
2

1
2
1, 2

< 1 < , 1

and is the cardinality of . plays an important role so that ( 1 , 2 )

and ( 1 , 2 ) are asymptotically independent if ( ) ≠ ( ). Then
1, 2
1, 2

the estimator exists.
̂ ̂
, = arg min (, ),

0<<∞,ℋ
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