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Владикавказский математический журнал январь-март, 2007, Том 9, Выпуск 1
UDC 517.984
SOME ASYMPTOTIC PROPERTIES OF A KERNEL SPECTRUM ESTIMATE WITH DIFFERENT MULTITAPERS
Teamah A. A. M., Bakouch H. S.
Let X(t), t = 0, ±1,..., be a zero mean real-valued stationary time series with spectrum fxx (A), —n < A < n. Given the realization X(1), X(2),...,X(N), we construct L different multitapered periodograms iXx^3 (A), j = 1, 2,... , L, on non-overlapped and overlapped segments Xj)(t), 1 < t < N. Also, we give asymptotic expressions of the mean and variance of the average of these different multitapered periodograms. We obtain an estimate of fxx (A) via iXx''3 (A) and different kernels W/(j)(a), j = 1, 2,..., L; —n < a < n; 3 is a bandwidth. We find asymptotic expressions of the first and second-order moments of this estimate. Moreover, we propose a choice of the considered bandwidth. An asymptotic expression of the integrated relative mean squared error (IMSE) of the estimate is formulated.
Mathematics Subject Classification (2000): 62M10, 62M15.
Key words: Stationary time series; Non-overlapped and overlapped segments; Multitapering; Kernels; Bandwidth; Spectrum estimate.
1. Introduction
Multitapering method maintains the good bias properties that tapering provides and at the same time produces an estimate with less variability (see [7, 11, 12, 14]). Some asymptotic statistical properties of spectral estimates were studied by several authors (see [1, 3, 4]) using a tapered data. The authors of this paper argued in [9, 10] the asymptotic expressions of the first and second-order moments of some spectral estimates, on non-overlapped and overlapped segments via different tapers and different weight functions (kernels) for both continuous time and discrete time stationary processes.
In this paper we study the problem of estimating a spectral density function (spectrum) on non-overlapped and overlapped segments using different multitapers and different kernels with a bandwidth parameter, for a discrete parameter stationary time series. In section 2 we introduce an estimate of the spectral density function using different multitapers and different kernels. Moreover, we give asymptotic expressions of the mean and variance of the average of the constructed different multitapered periodograms. In section 3 we obtain the asymptotic expressions of the mean and variance for the suggested estimate, assuming that direct spectral estimates are uncorrelated. Also, we obtain an optimal choice of the bandwidth. Furthermore, we formulate an asymptotic expression of the integrated relative mean squared error of the estimate.
© 2007 A. A. M. Teamah, H. S. Bakouch
2. The model
Suppose that X(1),X(2),..., X(N) is a realization of N observations from a real-valued stationary and discrete parameter process X(t), t = 0, ±1,..., with a zero mean. The spectral density function of X (t) is
<x
fxx (A) = 2- E Cxx (t)e-iAT, -n < A < n; i = V—1, (1)
T = — <^
where Cxx (t) is the autocovariance function of X (t) and given by
Cxx (T) = j /xx (A)eiArdA, (2)
—n
ro
provided that ^ |Cxx(t)| < to.
T = — ro
If the process X(t) is invertible, then the inverse spectral density function is defined by
ro
/Xx(A) = ^ E dxx (t)e—iAT, (3)
T =—ro
where dxx (t) is the inverse autocovariance of X(t) (see [2]) and given by
dxx(t) = y /XX(A)eiAT dA,
—n
ro
such that ^ |dxx(t)| < to.
t=—ro
We construct L segments by dividing the given observations:
X(j)(t) = X[(j - 1)M + t], j = 1, 2,..., L; t = 1, 2,..., M + q; 0 < q < M, (4)
where X(j)(t) is the set of observations in the j — segment. If N = LM + q, 0 < q < M, then the number of overlapped segments L = (N — q)/M and each segment contains M + q observations. Also, if q = 0, then the number of non-overlapped segments L = N/M.
Now, we define the average of different multitapered periodograms as an estimate of /xx (A):
1 L
/??(A) = LE lXmX)j (A), (5)
j=i
where lXX)j (A) is the multitapered periodogram of X(j)(t) and given by
1 K
(A) = ¿E (A) k=1
with
' M+q
I<&k (A) = 2nE jk (t) t=i
-1 M+q 2
E j(t)Xj)(t)e"
, , , .......e—¿At
t=1
hj,fe(t) is called the data taper for the (j, k)— direct spectral estimate Ij^j' k (A) and equals zero outside the interval [1, M + q] and K is the number of components of multitaper in each segment.
Let hjk (t) be a set of orthonormal tapers, that is
M+q
Y1 hj'r (t)hj,k (t) t=1
1, if r = k; 0, otherwise,
(see [12]). From the orthonormality we deduce that direct spectral estimates I^o^(A), j = 1, 2,..., L; k = 1, 2,..., K, are uncorrelated (see [13]), and then
I (d)j,k
Ixx
1
M+q
(A) = ^ E hj,k(t)X(j)(t)e t=i
-¿At
In fact, the direct spectral estimate iX^jj'^(A) has the asymptotic properties (see [8, 13])
E
(A) « fxx(A), Var iXr (A) " fXx(A).
(d)j,k.
(6)
E
Equations (5), (6) and the uncorrelation of direct spectral estimate /^^'^(A) imply /??(A)] - fxx (A) and Var [/^(A)] - fix (A)/KL. Obviously, Var [/^(A) becomes less variability as at least K or L increases. The case when L = 1 was investigateo in [13].
Smoothing the multitapered periodograms (A), j = 1, 2,..., L, in equation (5) by
the different kernels (weight functions) W^j)(a), —n < a ^ n, and taking their average, we get
f
(mt)s
(a) = T^e/W(j)
№ j=1y V ß
J 1 — n
(7)
which is a smoothed estimate (kernel estimate) of fxx (A) with W^ (a) = a), such
that W(j)(a)da = 1; W(j) (—a) = W(j)(a) . ^ is called the bandwidth. Also, we can deduce that
/*(mt)sp
(A) = Ll f
j=1
(mt)j
L^-xx (A),
(8)
where
Äx)j (A) = 1/ Wj) ß
is the j — smoothed multitapered periodogram of X(j)(t).
3. Statistical properties of fx?x:)sp (A)
In this section we obtain the asymptotic expressions of expectation, variance and integrated relative mean squared error of the smoothed (kernel) spectrum estimate fx^^ (A): 3.1. Expected value. Taking expectation of equation (7), we get
E
f(x)sp (A)
J- £ fw j)( Mfxx (^ d^.
j = 1—n
(9)
2
7T
7T
Making use of the transformation ^ = A + with small A G (—then equation (9) becomes
L
E
f Fxp (A) J « ZE J W (j)(«)/xx (A + £a) da, from a Taylor expansion for fxx(A + £a) about A, equation (10) has the form:
(10)
E
fS)sp (A) = y£ I W(j) (a)
1
L
j=1-r
2^,2
£2 a
fxx (A) + £afXX (A) + /XX (A) + O(£2)
fxx (A) + fL/xX (A)E J a2W(j) (a) da + O(£2), (11)
da =
j=i-
where fXXX (A) is the second derivative of the spectrum fXX (A). Therefore
Bias
fS)sp (A)'
It is clear that the bias of f (A) is of the order £2L 1.
2L
L °°
fXX (A) E / a2W(j) (a) da.
(12)
3.2. Variance. Since the tapered periodogram ordinates Ixx'k (^i), 1 ^ i < (M + q)/2,
), then equat
A - A T(d)j„
are asymptotically independent (see [6, 8]), then equation (8) can be put in the form:
k ^
fx?3 (A) = ^ EE / W(j) ( ^ ) IXX * M d^
k=1
£ 2n
K
£K (M + q)
EEw (j)
fe=1
Using equations (6) and (8), we get
4n2
Var
fs)sp (A)
£2L2K(M + q)2 j=1
EE
W(j)
A -
£
2n
^LK(M + q) j=1
J = 1-n
W(j)
f2 x
A — ^
£
f2X M
Putting ^ = A + £a Var
fS)sp (A)
2n
£K (M + 2n
— ^y |_W(j)(a)J fix (A + £a) da
j=1-
« £K(M + q)L2f-(A) E/ [W(j)(a) which is of the order [£KL2(M + q)] 1 .
da,
2
2
2
From equations (12) and (13) the mean squared error (MSE)of fx^c"* (A) is
MSE
/S)sp (A)'
= I Bias
/(mt) sp
xx
(A)
+ Var
S)sp (A)
The MSE of an estimate can be small only if both bias term and variance term are small. We show that the two terms are of the orders P2L-1 and [PKL2(M + q)]-1. Then it follows that the variance and the squared bias terms of (A) are balanced for [PKL2 (M+q)]-1 ~
P4L-2. This implies an optimal choice of bandwidth equals to P ~ [K(M + q)]-1/5. Hence, P — 0 as M —► to. Using equations (12), (13) and the optimal choice of P, we get
Bias
/K)sp (A)
Var
/^tU (A)
— 0, M —► to,
(14)
that is (A) is a consistent estimate of /xx(A) as M — to. Also, Var (A)
becomes less variability as at least K or L increases.
3.3. Integrated relative mean squared error. We take IMSE as a measure for the goodness of fit of a spectral estimate. IMSE of (A) is defined by (see [5]):
IMSE
/S)sp (A)
= E
/S)sp (A) -/xx(A)"1 2
/xx(A)
dA =
E
/(mt)sp (A)
(A)
1
dA + / Var
/
(mt) sp ( A )
(A)
(15)
(A)
dA.
Hence,
IMSE
/(mt)sp(A)] =2n + f [/-^(A)]2 (E [/(mt)sp(A)])*dA-
- 2 I /-1 (A)E /Xmt)sp (A) dA + / (A)]z Var /XXsp (A)
p (mi) s
dA.
From equations (3), (11) and (13), then formula (15) has the form:
IMSE
+
Zimi)sp(A) « 2n + (2n)—2 ^ £ dxx(r) dxx(r):
—n
O
T =—O r=—o
L :
J /xx(A) + g/Xx(A)£ J a2W(j)(a)da
e—iA(T+r) dA-
— n
+ q) dxx(t)dxx(r) y y /Xx(a) £ W^f
—i
—n
e dxx(t)J /xx(A) + 2L/Xx(A)£ J a2W(j)(a)da
e—iAT dA+
T =—O r=—o
—O —n
j=1
2
7T
2
7T
7T
7T
7T
7T
—n
—n
2
X
7T
OO n
If we consider the optimal choice of fl, fl « [K(M + q)] 1/5, then equation (16) can be put in the form:
IMSE
/S)sp (A) « 2n + (2n)-2 ^ £ dxx(T)dxx(r)
T = -<X r = -rx
— n
1 £ dxx (t)
f /xx(A)e-iXTdA
— n
<x <x
fx x (A)e-iA(T+r)dA
—n
<x
2n - n-1 Y dxx(T)Cxx(-T)+
+ (2n)-2 Y^ Y dxx(t)dxx(r)
T = — <x r = — <x
fx x (A)e-iA(T+r)dA
M (17)
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10.
n
n
T =
n
Received 12 March, 2006 Teamah A. A. M.
Department of Mathematics, Faculty of Science, Tanta University, Tanta, EGYPT E-mail: abdelmoneim2002uk@yahoo.co.uk
Hassan Salah Bakouch
Department of Mathematics, Faculty of Science, Tanta University, Tanta, EGYPT E-mail: hbakouch@yahoo.com
