Asymptotic statistical properties of spectral estimates with different tapers for discrete time processes

doi:10.1016/S0096-3003(03)00299-6

Applied Mathematics and Computation

Volume 150, Issue 3, 17 March 2004, Pages 681-695

https://doi.org/10.1016/S0096-3003(03)00299-6 Get rights and content

Abstract

This paper concerns the problem of estimating the spectral density, spectral distribution and autocovariance functions on non-overlapped and overlapped intervals for a discrete parameter stationary time series with different tapers. Also, we obtain another estimate of spectral density function via different weight functions. Asymptotic statistical properties of these estimates and their limits will be considered.

Introduction

Many authors, as e.g. Brillinger [1]; Brillinger and Rosenblatt [2]; Dahlhaus [3]; Ghazal and Farag [5], have studied the asymptotic expressions of the first and second-order moments of spectral estimates via untapered data and tapered data. Ghazal [4] studied the statistical properties of the spectral density estimate on non-overlapped intervals. Ghazal and Farag [6] studied the statistical analysis of the spectral density estimate on overlapped intervals. Teamah and Bakouch [10] studied the statistical properties of the spectral estimates on overlapped and non-overlapped intervals via different tapers for the continuous time processes.

In this paper we study the problem of estimating the spectral density, spectral distribution and autocovariance function on overlapped and non-overlapped intervals with different tapers. We will use different weight function, spectral smoothing window, for each interval and combine tapering with smoothing, in the frequency domain, to obtain another estimate of spectral density. Statistical properties of these estimates will be studied. Furthermore, we study the limits of these properties.

Let X(t),t=0,±1,…, be a zero mean, r vector-valued, strictly stationary and discrete time series with spectral density f_xx(λ),−π⩽λ⩽π, spectral distribution F_xx(λ),−π⩽λ⩽π, and autocovariance function C_xx(k),k=0,±1,… The estimation problem depends on the values of X(t),t=1,2,…,N. Assume that X(t) has real-valued components X_a(t),a=1,2,…,r, all moments exist and the joint cumulant of X_a₁(t₁+τ),…,X_{a_n−1}(t_n−1+τ),X_{a_n}(τ) (see [2]) is given by $C_{a_{1},…,a_{n}} (t_{1},…,t_{n−1})= Cum {X_{a_{1}} (t_{1} +τ),…,X_{a_{n−1}} (t_{n−1} +τ),X_{a_{n}} (τ)}, a_{1},…,a_{n} =1,2,…,r; t_{1},…,t_{n−1},τ=0,±1,…; n=2,3,…,$ also one can write $C_{a_{1},…,a_{n}} (t_{1},…,t_{n−1})=∫_{−π}^{π} ⋯∫_{−π}^{π} f_{a_{1},…,a_{n}} (λ_{1},…,λ_{n−1})× exp i∑_{s=1}^{n−1} λ_{s} t_{s} d λ_{1},…, d λ_{n−1}, i= −1,$ where f_{a₁,…,a_n}(λ₁,…,λ_n−1) is the nth spectrum which is bounded and uniformly continuous.

Assumption 1.1

Let X(t) be a strictly stationary time series all of whose moments exist, for each s=1,2,…,n−1 and any n-tuple a₁,…,a_n (see [5]) we get $∑_{t_{1},…,t_{n−1}=−∞}^{∞} t_{s} C_{a_{1},…,a_{n}} (t_{1},…,t_{n−1}) <∞, n=2,3,…$

If X(t) satisfies Assumption 1.1, its cumulant spectral densities are $f_{a_{1},⋯,a_{n}} (λ_{1},…,λ_{n−1})= 1 (2π)^{n−1} ∑_{k_{1},…,k_{n−1}=−∞}^{∞} C_{a_{1},…,a_{n}} (k_{1},…,k_{n−1}) exp −i∑_{s=1}^{n−1} λ_{s} k_{s}, −π⩽λ⩽π; a_{1},…,a_{n} =1,2,…,r; n=2,3,…$

An estimate of C_xx(k) is given by $r ̂_{xx} (k)= 1 N ∑_{t=1}^{N−|k|} X(t)X(t+|k|), |k|⩽N−1,$ its Fourier transform is the periodogram, which is given by $I_{xx} (λ)= 1 2π ∑_{k=−(N−1)}^{N−1} r ̂_{xx} (k) e^{−iλk} = 1 2πN ∑_{t=1}^{N} X(t) e^{−iλt}^{2},$ that is, the periodogram can be described as squared Fourier transforms of data, or a Fourier transform of biased lagged product covariance estimates.

Data tapering is a smoothing technique for raw data in the time domain, where the data taper h(t) gives less weight on the boundary point. Given a sample of N observations from X(t), t=1,2,…,N, to reduce the large fluctuations of the periodogram, we divide the available sample of N observations into L intervals: $X^{(j)} (t)=X[(j−1)M+t], t=1,2,…,M+K; 0⩽K<M; j=1,2,…,L,$ where X^(j)(t) is the jth interval. If N=LM+K,0<K<M, then a number of overlapped intervals L=(N−K)/M and each interval contains M+K observations, that is, the number of overlapping observations between successive intervals is K. Also, if K=0, then N=LM, where L is a number of non-overlapped intervals of M observations each. The case when the intersection of crossed intervals contains K observations is studied (see [3], [6]) and also, the case when K=0 (see [4]) for a taper in all intervals.

The interval X^(j)(t) has the following tapered periodogram $I_{xx}^{(j)} (λ)= 1 2π∑_{t=1}^{M+K} h^{(j)} (t) ∑_{t=1}^{M+K} h^{(j)} (t)X^{(j)} (t) e^{−iλt}^{2}, j=1,2,…,L,$ where h^(j)(t), j=1,2,…,L, is the taper corresponding to the jth interval, which equals zero outside the interval [1,M+K]. The taper h^(j)(t),t=1,2,…,M+K, gives less weight to the data at the ends of each interval, hence making the consecutive interval sequences less correlated to another. This decorrelation will be a more effective reduction of variance by averaging the periodograms in (1.8). The expanded finite Fourier transform for the jth interval is defined by $d_{a}^{(j)} (λ)= 1 2π∑_{t=1}^{M+K} h^{(j)} (t) ∑_{t=1}^{M+K} h^{(j)} (t)X_{a}^{(j)} (t) e^{−iλt}, j=1,2,…,L; −π⩽λ⩽π,$ where X_a^(j)(t)=X_a[(j−1)M+t], a=1,2,…,r; 0⩽K<M; t=1,2,…,M+K; j=1,2,…,L.

From , , one can write $I_{ab}^{(j)} (λ)=d_{a}^{(j)} (λ)d_{b}^{∗(j)} (λ), j=1,2,…,L; a,b=1,2,…,r,$ where the asterisk denotes complex conjugate.

The averaged tapered periodoram, Welch’s spectrum estimate, (see [8], [9]) is given by $Φ ̂_{ab} (λ)= 1 L ∑_{j=1}^{L} I_{ab}^{(j)} (λ),−π⩽λ⩽π.$

If X(t) satisfies Assumption 1.1, then for each interval (see [7]), one gets $Cum {d_{a_{1}}^{(j)} (λ_{1}),…,d_{a_{n}}^{(j)} (λ_{n})}= (2π)^{n−2} (M+K)^{n−1} [H_{a_{1},…,a_{n}}^{(j)} (0)]^{−1/2} H_{a_{1},…,a_{n}}^{(j)} ∑_{s=1}^{n} λ_{s} ×f_{a_{1},…,a_{n}}^{(j)} (λ_{1},…,λ_{n−1})+O{(M+K)^{−n/2}}, a_{1},…,a_{n} =1,2,…,r; n=2,3,…,$ where d_a^(j)(λ),a=1,2,…,r, is given by (1.9), the error term O{(M+K)^−n/2} is uniform in λ₁,…,λ_n and $H_{a_{1},…,a_{n}}^{(j)} ∑_{s=1}^{n} λ_{s} =∑_{t=1}^{M+K} h_{a_{1}}^{(j)} (t),…,h_{a_{n}}^{(j)} (t) exp −i∑_{s=1}^{n} λ_{s} t, j=1,2,…,L.$

Section snippets

Statistical properties of the averaged tapered periodogram

In this section we obtain some statistical properties of the averaged tapered periodogram and determine their behaviour as M→∞:

Statistical properties of the spectral distribution estimate

Let X(t),t=0,±1,…, be a time series with a spectral density f_xx(λ) with respect to the spectral distribution F_xx(λ), that is $F_{xx} (λ)=∫_{0}^{λ} f_{xx} (α) d α,0⩽λ⩽π$ then, the spectral distribution corresponding to interval j is given by $F^{(j)} (λ)=∫_{0}^{λ} f^{(j)} (α) d α,0⩽λ⩽π, j=1,2,…,L,$ F^(j)(λ) can be estimated using the tapered periodogram according to $F ̂^{(j)} (λ)=∫_{0}^{λ} I^{(j)} (α) d α, 0⩽λ⩽π.$

Hence $F ̂_{a_{1}b_{1}} (λ_{1})=∫_{0}^{λ_{1}} 1 L ∑_{j=1}^{L} I_{a_{1}b_{1}}^{(j)} (α) d α.$

From formula (1.11), one gets $F ̂_{a_{1}b_{1}} (λ_{1})=∫_{0}^{λ_{1}} Φ ̂_{a_{1}b_{1}} (α) d α, 0⩽λ_{1} ⩽π,$ where $F ̂_{a_{1}b_{1}} (λ_{1})= 1 L ∑_{j=1}^{L} F ̂_{a_{1}b_{1}}^{(j)} (λ_{1}), a_{1},b_{1}$

Statistical properties of the autocovariance estimate

Let X(t),t=0,±1,…, be an r vector-valued stationary time series with autocovariance function $C_{xx} (k)=E{X(t)X(t+k)}=∫_{−π}^{π} f_{xx} (λ) e^{iλk} d λ, k,t=0,±1,…; −π⩽λ⩽π,$ where E[X(t)]=0, t=0,±1,…, its spectral density function is given by $f_{xx} (λ)= 1 2π ∑_{k=−∞}^{∞} C_{xx} (k) e^{−iλk}, −π⩽λ⩽π,$ for the intervals, an estimate of C^(j)(k) (see [5]) is $r ̂_{xx}^{(j)} (k)= (M+K)∑_{t=1}^{M+K} (h^{(j)} (t))^{2}^{−1/2} ∑_{t=1}^{M+K−|k|} h^{(j)} (t+|k|)h^{(j)} (t)X^{(j)} (t+|k|)X^{(j)} (t), |k|⩽M+K−1.$ Also $I_{ab}^{(j)} (λ)= 1 2π ∑_{k=−(M+K−1)}^{M+K−1} r ̂_{ab}^{(j)} (k) e^{−iλk}, −π⩽λ⩽π,$ the Fourier transform of Eq. (4.4) is $r ̂_{ab}^{(j)}$

Statistical properties of the smoothed spectral density estimate

Assumption 5.1

Let ψ(α),−π<α⩽π, be a weight function, it is a smoothing technique for the periodogram in the frequency domain, which is bounded, non-negative and symmetric about zero: $∫_{−π}^{π} ψ(α) d α=1 (see [1]).$

As an estimate of the spectral density f_ab^(j)(λ) is $f ̂_{ab}^{(j)} (λ)=∫_{−π}^{π} ψ_{ab}^{(j)} (λ−α)I_{ab}^{(j)} (α) d α, j=1,2,…,L,$ then $f ̂_{ab} (λ)= 1 L ∫_{−π}^{π} ∑_{j=1}^{L} ψ_{ab}^{(j)} (λ−α)I_{ab}^{(j)} (α) d α,$ where $f ̂_{ab} (λ)= 1 L ∑_{j=1}^{L} f ̂_{ab}^{(j)} (λ)$ which is an estimate of f_ab(λ).

Now, we give some statistical properties for $f ̂_{ab} (λ)$ .

References (10)

D.R. Brillinger
Asymptotic properties of spectral estimates of second order
Biometrika
(1969)
D.R. Brillinger et al.
Asymptotic theory of estimates of k-th order spectra
R. Dahlhaus
On a spectral density estimate obtained by averaging periodograms
J. Appl. Probab.
(1985)
M.A. Ghazal
On a spectral density estimate on non-crossed intervals observation
Int. J. Appl. Math.
(1999)
M.A. Ghazal, E.A. Farag, Estimating the spectral density, autocovariance function and spectral measure of...

There are more references available in the full text version of this article.

Cited by (3)

Some Asymptotic Properties of a Kernel Bispectum Estimate with Different Multitapers
2022, Mathematics
Bispectral density estimation of continuous time series with missed observations
2021, Journal of Statistics Applications and Probability
A bivariate beta-gamma autoregressive process (BVBGAR(1))
2009, Communications in Statistics - Theory and Methods

View full text

Asymptotic statistical properties of spectral estimates with different tapers for discrete time processes

Abstract

Introduction

Section snippets

Statistical properties of the averaged tapered periodogram

Statistical properties of the spectral distribution estimate

Statistical properties of the autocovariance estimate

Statistical properties of the smoothed spectral density estimate

Asymptotic properties of spectral estimates of second order

Biometrika

Asymptotic theory of estimates of k-th order spectra

On a spectral density estimate obtained by averaging periodograms

J. Appl. Probab.

On a spectral density estimate on non-crossed intervals observation

Int. J. Appl. Math.