This corollary could have been inferred from Hannan (1960) p.32, and a stronger result is available in Hannan (1974) The explicit representations of the

In document Some aspects of time series frequency estimation (Page 96-100)

5 .3 .2 P r o p e r t ie s o f t h e A R p a r a m e te r e s t im a t e s a*

T h e autoregressive p a ra m e te rs dk are e stim a te d by s u b stitu tin g th e estim ated a u to covariances r(j ) in to th e Y ule-W alker eq u atio n s (5.2.6), so th a t

R( k - l ) d fc = -ffc .

A rg u m en ts th a t follow are b ased extensively on m odifications of those used in B erk (1974) w hich w ere also used in C h a p te r 4.

W e observe

( a k - a k) = a k4- R( k1) 1f k

= R ( k - l ) ~ 1{ R ( k - l )<** + ?*}

= R( k - a k]T (5.3.1)

w here R ( k ) m is th e k x (k + 1) m a trix form ed from th e (k + 1) x ( k + 1) m atrix

R( k ) by deleting its first row. From th e au g m en ted Y ule-W alker eq u atio n s (5.2.7)

we h av e R(fc)*[l a]t]T = 0, w hich we can s u b s titu te in to (5.4.9), giving

( a k - &k ) = R(k- - H (fc),}[l a k}T

= { R ( k - I ) " 1 -

R(k- l ) - 1}{Ä(fc). - Ä(Jfc).}[l

J k]T

+ R ( k - l ) - 1{ R ( k ) . - R ( k ) , } [ l J k]T . (5.3.2)

Frequency e stim ate s are b ased on lin ear co m b in atio n s of th e (d* — ajt). T he re su lts a b o u t th e ä k th a t will be used in th e sequel are su m m arised in th e following.

T h e o r e m 5 .3 .3 . Under th e conditions o f Theorem 5.2.1, for c j0 n ot too close to

0 or 7T, as N —+ oo

(i) a* —► Ofc in probability.

that k 2 / N —* 0

(ii) I N i q ( k y ( & k- a k) - N%q(k)*R(k- 1 )"> {Ä(fc)„ - Ä (fc).} [l « I ] T

—+ 0 in probability .

(Hi) N 2q(k)(6ik — afc) has asym ptotic distribution with m ean zero and variance

$*;= q (k)* R (k — 1 )~ 1V R ( k — 1 )~ 1q(k), where V is defined

V = /fccr4 + p2<j2[2 cos mw0 cos /u?o] /, m = 1, 2 , .. . , k .

W e recall from (5.2.11) th a t we have R (k — l ) -1 = cr~2I k + 0 ( h - 1 ), so th a t for large k we can express th e variance te rm V by

j y __ 4 t ^ <j2 _ T V = a 1 ---T r kr k Pz Rrr2

R(k - l y ' V R ^ k -

l ) " 1 = Ä(fc - l ) - 1 ^

- l )“ 1* 4 - — OckOiTk

— h - - ^ r ^ k o c l + o i k - 1 ) so th a t

? * = k « l 2 - ^ l« ( * ) * < * tl2 •

(5.3.3)

pz

In th e sequel, th e following re p re se n ta tio n will be used for

M s

- 1

,k =

q(k)*R(k- l ) " 1 { £ (* ). -

Ä(fc)„}[l:<xT]T .

(5.3.4)

Define

A = q(k)*R(k - 1 )_1 , B = R ( k ) . - R( (5.3.5)

w here A is 1 x k, B i s f c x ( f c + 1), C i s 1 x (fc -b 1), a n d use th ese in th e re la tio n vec ( A B C t ) = (C ® A ) vec (B ) . (5.3.6)

H ere 0 d en o tes th e K ronecker p ro d u c t, a n d th e v ec(-) o p e ra to r ap p lied to an

m x n m a trix B form s th e ( m n x 1) colum n vector co n tain in g th e colum ns of B below one a n o th er. H ere n o te th a t w ith th ese definitions A B C r is a scalar, so th a t ( A B C r ) = v e c ( A B C T ), w hence (5.3.4) can be expressed

M N,k = ([1 • a if] 0 q ( k ) * R ( k — l ) - 1 ) vec (R ( k)* — R( k ) * ) . (5.3.7) T h e o re m 5.3.3 proved th a t, for q(k) a v ecto r w ith uniform ly b o u n d ed L 2

n o rm , \ N 2q(k)*(ock — a * ) — N * A i N , k \ = ° p (l)- T h e n ex t resu lt n eed ed is for th e

sp ecial case q(k)* = {—z je ,Ja,° / k 3/ 2}, j = 1, T h e o r e m 5 .3 .4 .

N- 1

N * M N - i , k = N ~ i (C ® A) { De ( s ) - 60,m<r2} (5.3.8)

3 — 0

is a zero -m ea n m a rtingale array w here D is th e k (k + 1) x 1 vecto r defined b y D( i ) = e(s — m ) 4- 2pcos m uo cos(so;o 4- <j>)

where, i f i = n k 4- h, n — 0 , . . . , k and h = 1 , . . . , k th en

m = I n k 4- h — n ( k 4 - 1)| (5.3.9) = \hn\ .

T h is co m p licated n o ta tio n is necessary to rep ro d u ce th e T oep litz s tru c tu re of

( R ( k ) m — R( k) +) u n d e r th e v ec(-) o p e ra tio n . T h e scalar [(C 0 A )D ] depends on s

a n d k.

T h is re su lt will have several useful ap p licatio n s, th e first of w hich is th e fol­ low ing

T h e o r e m 5 .3 .5 . I f uo E (0,7r), N —► oo and k —»■ oo s.t. k 3/ N —+ 0, then u n ifo rm ly in 0 E ( 0 , 7r),

(a n d as o b serv ed below (5.2.1), f k i ß ) 9(ß) a.s. as k —♦ oo, w here g(9) is the “noise” sp e c tru m , in th is case g(9) = cr2/2w).

5 .4 B ia s o f t h e F r e q u e n c y E s t im a t e s 5 .4 .1 B ia s d u e t o fin ite a u t o r e g r e s s io n

T h e re are tw o ty p es of bias th a t a p p e a r in frequency e stim ate s p ro d u ced by th e au to reg ressiv e p rocedures, e ith e r A lg o rith m 5.1 or A lg o rith m 5.2. T h e first is sim ply a re su lt of th e obvious fact th a t th e finite autoregressive tra n sfe r function does n o t h av e its m in im u m m odulus ex actly a t 9 = u>o, nor does it have a ro o t e x ac tly o n th e u n it circle w ith arg u m en t 9 = u;0.

S to ica et al. (1986) show th a t for a frequency e stim a te fo rm ed following Al­ g o rith m 5.2, th e a rg u m en t w* of th e zero of Qfc(z) closesTto th e u n it circle satisfies

In document Some aspects of time series frequency estimation (Page 96-100)