Bias control in the estimation of spectral functionals
We consider estimators for integrals of a spectrum and a bispectrum for random fields X(t), t belongs R^d, and present conditions guaranteeing the rate of convergence of bias to zero appropriate for dimensions d = 1, 2, 3.
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irk-123456789-44912009-11-20T12:00:33Z Bias control in the estimation of spectral functionals Sakhno, L. We consider estimators for integrals of a spectrum and a bispectrum for random fields X(t), t belongs R^d, and present conditions guaranteeing the rate of convergence of bias to zero appropriate for dimensions d = 1, 2, 3. 2007 Article Bias control in the estimation of spectral functionals / L. Sakhno // Theory of Stochastic Processes. — 2007. — Т. 13 (29), № 1-2. — С. 225-233. — Бібліогр.: 12 назв.— англ. 0321-3900 http://dspace.nbuv.gov.ua/handle/123456789/4491 en Інститут математики НАН України |
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We consider estimators for integrals of a spectrum and a bispectrum for random fields X(t), t belongs R^d, and present conditions guaranteeing the rate of convergence of bias to zero appropriate for dimensions d = 1, 2, 3. |
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Sakhno, L. Bias control in the estimation of spectral functionals |
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Sakhno, L. |
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Sakhno, L. |
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Bias control in the estimation of spectral functionals |
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Bias control in the estimation of spectral functionals |
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Bias control in the estimation of spectral functionals |
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Bias control in the estimation of spectral functionals |
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Bias control in the estimation of spectral functionals |
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bias control in the estimation of spectral functionals |
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Інститут математики НАН України |
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2007 |
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Bias control in the estimation of spectral functionals / L. Sakhno // Theory of Stochastic Processes. — 2007. — Т. 13 (29), № 1-2. — С. 225-233. — Бібліогр.: 12 назв.— англ. |
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AT sakhnol biascontrolintheestimationofspectralfunctionals |
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2025-07-02T07:43:23Z |
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Theory of Stochastic Processes
Vol.13 (29), no.1-2, 2007, pp.225-233
LUDMILA SAKHNO
BIAS CONTROL IN THE ESTIMATION OF
SPECTRAL FUNCTIONALS
We consider estimators for integrals of a spectrum and a bispectrum
for random fields X(t), t ∈ R
d, and present conditions guaranteeing
the rate of convergence of bias to zero appropriate for dimensions
d = 1, 2, 3.
1. Introduction
Let X(t), t ∈ R
d, be a real-valued measurable sixth-order weakly sta-
tionary random field with zero mean and spectral densities of second and
third orders f2 (λ), f3 (λ1, λ2), λ, λ1, λ2 ∈ R
d, that is the functions f2 (λ) ∈
L1
(
R
d
)
, f3 (λ1, λ2) ∈ L1
(
R
2d
)
such that for the cumulants of the second
and third orders we have:
c2 (t) =
∫
d
f2 (λ) ei(λ,t)dλ, c3 (t1, t2) =
∫
2d
f3 (λ1, λ2) ei(λ1,t1)+i(λ2,t2)dλ1dλ2.
We will consider the problem of estimation of the spectral functionals
J2(ϕ) =
∫
d
ϕ(λ)f2 (λ) dλ and J3(ψ) =
∫
2d
ψ(λ1, λ2)f3 (λ1, λ2) dλ1dλ2
for appropriate functions ϕ(λ) and ψ(λ1, λ2) (we suppose ϕ(λ)f2 (λ) ∈
L1
(
R
d
)
, ψ(λ1, λ2)f3 (λ1, λ2) ∈ L1
(
R
2d
)
) and based on the observations
X(t), t ∈ D
T
= [−T, T ]d.
The estimation of spectral functionals is relevant to many statistical
problems. These functionals can be used to represent some characteris-
tics of random fields in nonparametric setting. On the other hand, these
functionals appear in parametric estimation in spectral domain, e.g., when
so-called minimum contrast estimators are studied.
Our main concern in the present paper is the bias evaluation for con-
sidered estimates and conditions which guarantee an appropriate rate of
2000 Mathematics Subject Classifications. 62G20, 62M15.
Key words and phrases. Spectral functionals, nonparametric estimation, peri-
odogram, bias, tapering
225
226 LUDMILA SAKHNO
convergence of the bias to zero, especially for the spatial data (d ≥ 2),
when the bias of estimates can be subject to so-called edge effects (see, e.g.,
[9], [10], [11]).
2. Bias control in the estimation of integrals of a spectrum
We will study here the estimator for the functional J2 (ϕ) based on the
tapered periodogram.
Consider the tapered data hT (t)X(t), t ∈ D
T
, where hT (t) = h
(
t1
T
, ..., td
T
)
=
∏d
i=1 h1
(
ti
T
)
, t = (t1, ..., td) ∈ R
d, is a taper (i.e., we suppose that the
taper factorizes), and h1 (t) , t ∈ R, is a positive measurable symmetric
(h1 (t) = h1 (−t)) function of bounded variation with bounded support:
h1(t) = 0 for |t| > 1.
Denote
Hk(λ) =
∫
h(t)ke−i(λ,t)dt, Hk,T (λ) =
∫
hT (t)ke−i(λ,t)dt = T dHk (Tλ) .
Let
dh
T
(λ) =
∫
hT (t) X(t)e−i(λ,t)dt, Ih
2,T (λ) =
1
(2πT )d H2(0)
dh
T
(λ)dh
T
(−λ)
be the finite Fourier transform and the periodogram of the second order
based on tapered data (it is supposed that H2(0) �= 0).
Consider the following estimator for the functional J2 (ϕ):
Ĵ2,T (ϕ) =
∫
d
ϕ(λ)Ih
2,T (λ)dλ.
The bias of Ĵ2,T (ϕ) can be represented as follows:
EĴ2,T (ϕ) − J2 (ϕ) =
=
∫ ∫
ϕ(λ) (f2 (λ + u)−f2(λ))Kh
2,T (u) dudλ (1)
=
∫ ∫
f2(λ) (ϕ (λ + u)−ϕ(λ)) Kh
2,T (u) dudλ (2)
=
∫
(g2(u) − g2(0))Kh
2,T (u) du, (3)
where Kh
2,T (u) = ((2πT )d H2(0))−1
∣∣H
1,T
(u)
∣∣2, and we have denoted g2(u) =∫
f2(λ)ϕ (λ + u) dλ.
Therefore, to evaluate the bias we need to analyze the asymptotic be-
havior of the expressions (1)-(3). Here we can apply standard arguments if
we impose conditions of regularity on the functions f2, ϕ, or, more generally,
ESTIMATION OF SPECTRAL FUNCTIONALS 227
on their convolution g2. We will also need some restriction on the kernels
Kh
2,T (u), more precisely, on the kernels Kh
2 (u) = ((2π)d H2(0))−1 |H1(u)|2
=
∏d
i=1 kh
2 (ui) , where kh
2 (u) =
(
2π
∫
h1 (t)2 dt
)−1 × ∣∣∫ h1 (t) e−itudt
∣∣2 .
We state some conditions which assure the desirable rate of convergence
of the bias of J2,T (ϕ) in the following theorem.
Theorem 1. Let the kernel kh
2 (u) satisfies the condition
(i)
∫ |u|lkh
2 (u) du < ∞, l = 1, 2,
and one of the following conditions holds:
(ii) f2 is twice boundedly differentiable and ϕ ∈ L1
(
R
d
)
;
(iii) ϕ is twice boundedly differentiable;
(iv) the convolution g2(u) is twice boundedly differentiable at zero.
Then, as T → ∞,
EĴ2,T (ϕ) − J2 (ϕ) = O
(
T−2
)
. (4)
Proof. Consider the expression (1). We have
∫
(f2 (λ + u)−f2(λ)) Kh
2,T (u) du
=
T d
(2π)d H2(0)
∫
(f2 (λ + u)−f2(λ)) |H1(Tu)|2 du
=
1
(2π)d H2(0)
∫ (
f2
(
λ +
u
T
)
−f2(λ)
)
Kh
2 (u) du. (5)
If the condition (ii) holds, we can write in view of Taylor’s theorem:
f2
(
λ +
u
T
)
−f2(λ) =T−1
d∑
j=1
ujf
′
j (λ) + O
(
T−2
) |u|2
(uniformly in λ in O-term). Since the function kh
2 (u) is even and condition
(i) holds, the integral (5) can be evaluated as O (T−2) and since ϕ ∈ L1
(
R
d
)
we obtain that the expression (1) is also O (T−2) , as T → ∞. Analogously
we can deduce (4) from the expressions (2) or (3) applying the conditions
(iii) or (iv) respectively.
Remark. We can see that if the standard normalizing factor T d/2 is applied
(under the conditions of Theorem 1), then the bias will be of order T d/2−2,
that is we can handle dimensions d = 1, 2, 3 using the tapered periodogram
in the estimator for J2(ϕ).
We note that the representation for the bias in the form (1), where the
kernels Kh
2,T (u) are involved, and its further asymptotic analysis with the
appeal to the properties of these kernels (and smoothness properties of the
spectral density) is the approach the most commonly used in the literature
228 LUDMILA SAKHNO
for bias evaluation of estimators for the spectral density and spectral inte-
grals (see, [10], [11], among others, and also [4]–[7] for the case of untapered
periodogram). However, when estimating the integrals of spectral densities,
one has a possibility for a trade-off between the smoothness properties of a
spectral density f and that of weight function ϕ: as Theorem 1 shows, one
can relax conditions on f imposing at the same time stronger conditions on
ϕ. This allows to treat the case of processes with long-range dependence.
We can write down another representation which appears to be quite
helpful for the solution of the problem of bias control, namely, the following
one
EĴ2,T (ϕ) − J2 (ϕ) =
1
(2πT )d H2(0)
∫
ϕ (λ) e−i(λ,u)
∫
c2 (u) (6)
×
∫
(hT (t + u) − hT (t))hT (t) dtdudλ.
One can see that it is possible to achieve the desirable rate of convergence
to zero of the above expression by imposing appropriate restrictions on a
taper. We present here such possibilities for the case d = 1.
Theorem 2. Let d = 1 and the following conditions hold:
(i)
∫ |u| |c2 (u)| du < ∞;
(ii) there exists a finite K such that
∫ |h(t + u) − h(t)|dt < K|u| ∀u;
(iii) ϕ ∈ L1 (R) .
Then, as T → ∞, EĴ2,T (ϕ) − J2 (ϕ) = O (T−1) .
Proof follows immediately from the representation (6).
Note that the condition (ii) is a form of integrated Lipschitz condition.
It is satisfied, for example, by functions h(t) with uniformly bounded first
derivatives and by h(t) of bounded variation.
Theorem 2, which is a straightforward generalization of the result in [8]
(obtained in connection with the estimation of the spectral density itself),
relies on the very restrictive condition (i), which means that the process is
weakly dependent. However, dealing with the integrals of the periodogram,
we have the possibility to avoid the condition (i), replacing it with the
condition on the function ϕ, as the next theorem states.
Theorem 3. Let d = 1, the condition (ii) of Theorem 2 holds; ϕ ∈ L2 (R)
and its Fourier transform ϕ̃ satisfies the condition:
∫ |u| |ϕ̃ (u)| du < ∞.
Then, as T → ∞, EĴ2,T (ϕ) − J2 (ϕ) = O (T−1) .
Proof. The left hand side of (6) can be written in the following form:
1
2πTH2(0)
∫ ∫
f2 (γ) eiγudγ
∫
ϕ (λ) e−iλu
×
∫
(hT (t + u) − hT (t)) hT (t) dtdudλ = I,
ESTIMATION OF SPECTRAL FUNCTIONALS 229
and taking into account (ii), we have |I| ≤ 1
2πTH2(0)
∫
f2 (γ) dγ
∫ |u| |ϕ̃ (u)| du,
which implies the statement of the theorem.
The next theorem shows that the better rate of convergence for the bias
can be achieved with another set of conditions.
Theorem 4. Let d = 1, the taper h is twice boundedly differentiable and
suppose that ϕ ∈ L2 (R) is an even function and its Fourier transform ϕ̃
satisfies the condition:
∫ |u|l|ϕ̃ (u) c2 (u) |du < ∞, l = 1, 2.
Then, as T → ∞, EĴ2,T (ϕ) − J2 (ϕ) = O (T−2) .
Proof is analogous to the proof of Theorem 1 and based again on Taylor’s
theorem which is applied now to hT (t + u) − hT (t) in the left hand side of
(6).
Note that with Theorem 4, as well as with Theorem 3 one can treat the
processes with long range dependence.
We will not present here the asymptotics for the variance of Ĵ2,T (ϕ);
this standard expression, where the second-order and fourth-order spectral
densities are involved, can be found, for example, in [2], [10] (under different
sets of conditions)
3. Bias control in the estimation of integrals of a
bispectrum
We study two estimates for the functional J3 (ψ) . Firstly, we consider
the estimate
Ĵh
3,T (ψ) =
∫
R2d
ψ (λ1, λ2) Ih
3,T (λ1, λ2) dλ1dλ2, (7)
based on the tapered periodogram of the third order
Ih
3,T (λ1, λ2) =
1
(2π)2dT dH3(0)
dh
T
(λ1)d
h
T
(λ2)d
h
T
(−λ1 − λ2),
and we suppose H3(0) �= 0.
Then we can write analogously to the second-order case the following
representation for the bias of Ĵh
3,T (ψ) :
EĴ3,T (ψ) − J3 (ψ)
=
∫
R4d
ψ (λ1, λ2) (f3 (λ1 + u1, λ2 + u2)−f3 (λ1, λ2)) (8)
×Φh
3,T (u1, u2) du1du2dλ1dλ2
=
∫
R4d
f3 (λ1, λ2) (ψ (λ1 + u1, λ2 + u2)−ψ (λ1, λ2)) (9)
×Φh
3,T (u1, u2) du1du2dλ1dλ2
=
∫
R2d
(g3 (u1, u2) − g3(0, 0))Φh
3,T (u1, u2) du1du2, (10)
230 LUDMILA SAKHNO
where Φh
3,T (u1, u2) =
(
(2π)2dT dH3(0)
)−1
H
1,T
(u1)H1,T
(u2)H1,T
(−u1 − u2)
and g3 (u1, u2) =
∫
R2d ψ (λ1, λ2) f3 (λ1 + u1, λ2 + u2) dλ1dλ2. Analogously
to the second-order case the asymptotic analysis of the above expression
leads to the next theorem. We will denote kh
3 (u) =
(
(2π)2
∫
h1 (t)3 dt
)−1 ×∫
h1 (t) e−itudt
∫
h2
1 (t) e−itudt.
Theorem 5. Let kh
3 (u) satisfies the condition
(i)
∫ |u|lkh
3 (u) du < ∞, l = 1, 2,
and one of the following conditions holds:
(ii) f3 is twice boundedly differentiable and ψ∈ L1
(
R
2d
)
;
(iii) ϕ is twice boundedly differentiable;
(iv) the convolution g3 (u1, u2) is twice boundedly differentiable at zero.
Then, as T → ∞,
EĴh
3,T (ϕ) − J2 (ϕ) = O
(
T−2
)
. (11)
Proof. Analogously to Theorem 1, the proof is based on Taylor’s formula
and properties of the kernels
Φh
3,T (u1, u2) =
1
(2π)2dT dH3(0)
H
1,T
(u1)H1,T
(u2)H1,T
(−u1 − u2)
=
1
(2π)2dH3(0)
T 2d H1(Tu1)H1(Tu2)H1(T (−u1 − u2))
=
1
(2π)2d
(∫
h1 (t)3 dt
)d
T 2d
d∏
i=1
k1(Tu
(i)
1 )k1(Tu
(i)
2 )k1(T (−u
(i)
1 − u
(i)
2 )),
where k1(u) =
∫
h1 (t) e−itudt. Taking into account the above representation
for Φh
3,T (u1, u2), consider the expression (8). Changing the variables: Tu1 =
v1, Tu2 = v2 in (8) and developing f3 in Taylor’s series up to the second
order term, we have the possibility to integrate over v1, v2 the products∏d
i=1 k1(v
(i)
1 )k1(v
(i)
2 )k1(−v
(i)
1 − v
(i)
2 ), so that we obtain the sum of the terms
of the form
const ×
∫
R2d
ψ (λ1, λ2)
∫
v
(i)
l kh
3 (v
(i)
l )dv
(i)
l dλ1dλ2, l = 1, 2, i = 1, ..., d,
which equal to zero (as kh
3 is even), and the sum of the terms of the form
∫
R2d
ψ (λ1, λ2)
∫ (
v
(i)
l
)2
kh
3 (v
(i)
l )dv
(i)
l dλ1dλ2, l = 1, 2, i = 1, ..., d, (12)
supplied by the multiplier of order O (T−2) , in view of (i) and (ii) the
integrals (12) are bounded, therefore, the result (11) follows. Analogously,
the derivation of (11) can be based on (iii) or (iv).
ESTIMATION OF SPECTRAL FUNCTIONALS 231
Next, we consider the estimator for J3 (ψ) of the form
Ĵw
3,T (ψ) =
∫
R2d
ψ (λ1, λ2) f̂w
3,T (λ1, λ2) dλ1dλ2, (13)
where the estimator for the bispectrum f̂w
3,T (λ1, λ2) we propose to construct
in the following way:
f̂w
3,T (λ1, λ2) =
1
|D
T
|
∫
D
T
∫
D
T
∫
D
T
wT (u)wT (v) e−i(u,λ1)−i(v,λ2)
×X (t) X (t + u)X (t + v) dtdudv, (14)
it is supposed here (for convenience in notations) that we are given the
observations
{
X (t) , t ∈ D
2T
= [−2T, 2T ]d
}
; the averaging weights are de-
fined as follows: wT (u) = w
(
t
T
)
=
∏d
i=1 w1
(
ti
T
)
=
∏d
i=1 w1,T (ti) , t ∈ D
T
,
where the function w1 (t) has bounded support, w1 (t) = 0 for |t| > 1,
and its Fourier transform (supposed to exist) W1 (u) =
∫
w1 (t) e−itλdt
is an even function such that
∫
R
W1 (u) du = 1. We define W1,T (u) =∫
w1,T (t) e−itλdt = TW1 (Tu) , note that W1,T (u) is even and integrates to
1:
∫
R
W1,T (u) du = 1. Denote WT (u) =
∏d
i=1 W1,T (ui) .
The bias of the estimator Ĵw
3,T (ψ) can be represented in the form:
EĴw
3,T (ψ) − J3 (ψ)
=
∫
R4d
ψ (λ1, λ2) (f3 (λ1 + u1, λ2 + u2)−f3 (λ1, λ2))
×WT (u1)WT (u2) du1du2dλ1dλ2
=
∫
R4d
f3 (λ1, λ2) (ψ (λ1 + u1, λ2 + u2)−ψ (λ1, λ2))
×WT (u1)WT (u2) du1du2dλ1dλ2
=
∫
R2d
(g3 (u1, u2) − g3(0, 0))WT (u1) WT (u2) du1du2,
where g3 (u1, u2) =
∫
f3 (λ1, λ2) ψ (λ1 + u1, λ2 + u2) dλ1dλ2. Comparing the
above formula with (8)-(10), we note that the kernel under the integral sign
factorizes, that is instead of Φh
3,T (u1, u2) we have a product WT (u1) WT (u2) ,
which makes the analysis simpler and completely the same as in the second-
order case. Asymptotic analysis of the above representation leads to our
next result.
Theorem 6. Let the function W1 (u) be such that
(i)
∫ |u|lW1 (u) du < ∞, l = 1, 2,
and one of the following conditions holds:
232 LUDMILA SAKHNO
(ii) f3 is twice boundedly differentiable and ψ∈ L1
(
R
2d
)
;
(iii) ψ is twice boundedly differentiable;
(iv) the convolution g3 (u1, u2) is twice boundedly differentiable at zero.
Then, as T → ∞, EĴh
3,T (ϕ) − J2 (ϕ) = O (T−2) .
Proof is analogous to the proof of Theorem 1.
Remark. Note that our estimator for the bispectrum (14) was tailored
in such a specific way with the purpose to obtain the expression for the
bias of Ĵw
3,T (ψ) , which is analogous to the expressions (1)-(3) for the bias
of Ĵ2,T (ϕ), and therefore, allows us to handle the bias exactly as in the
second-order case. The estimate (14) is of the form of weighted third-
order sample moments, however with a truncated region of integration.
Due to this truncation and specific weights we obtain the desirable rate of
convergence of the bias for d = 1, 2, 3. The idea to use a truncated versions of
a periodogram of the second order can be found in [12] for the discrete-time
case (there the restriction on the region of summation of sample covariances
is defined by means of some monotonically increasing function). Here we
“simplified” the truncation in comparison with the mentioned authors, in
our scheme the observation from D
2T
\D
T
are treated as “less significant” as
they involved in the expression (14) to less extent (for the case of discrete
time stochastic processes the estimate for a spectrum and a bispectrum
of the form (14) can be found in [1]). Also, the second-order truncated
periodogram is not guaranteed to be non-negative, but dealing with the
third-order spectra we do not need to worry about this. We note also that
other schemes of truncation (e.g., like that used in [12]) can be of use here
and can be combined with the procedure of averaging of estimates built
on overlapping d-dimensional intervals covering the region of observation,
which can lead to better estimates.
We will not present here the asymptotic expressions for the variance of
estimates (7) and (13). For (7) this expression, which includes the spectral
densities of second-, third-, fourth- and sixth-order, can be found in [2] (see
also [3] for more detailed expression for the case of untapered periodogram
which can be easily rewritten for tapered case as only the factor depending
on the taper should be supplied), and for (13) the analogous expression can
be written (however, this expression will not depend on weight functions,
that is appear in the form, similar to the untapered case). Note that for the
case of estimate (13) conditions guaranteeing the corresponding asymptotic
representation for the variance become of simpler form than those for the
estimate (7) due to the fact that all kernels, which appear under the integral
sign in the corresponding derivations, factorize.
ESTIMATION OF SPECTRAL FUNCTIONALS 233
4. Conclusions
The biases of the estimators for integrals of a spectrum based on the
tapered periodogram and estimators for integrals of a bispectrum based on
the tapered third-order periodogram and on weighted sample third-order
moments have been considered. For dimensions d = 1, 2, 3 conditions have
been obtained which guarantee the appropriate rate of convergence of a bias
to zero.
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Department of Probability Theory and Mathematical Statistics,
Mechanics and Mathematics Faculty,
Kyiv National Taras Shevchenko University, Kyiv, Ukraine
E-mail address: lms@univ.kiev.ua
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