Asymptotic statistical properties of spectral estimates with different tapers for discrete time processes
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 fxx(λ),−π⩽λ⩽π, spectral distribution Fxx(λ),−π⩽λ⩽π, and autocovariance function Cxx(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 Xa(t),a=1,2,…,r, all moments exist and the joint cumulant of Xa1(t1+τ),…,Xan−1(tn−1+τ),Xan(τ) (see [2]) is given byalso one can writewhere fa1,…,an(λ1,…,λn−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,…,an (see [5]) we get
If X(t) satisfies Assumption 1.1, its cumulant spectral densities are
An estimate of Cxx(k) is given byits Fourier transform is the periodogram, which is given bythat 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: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 periodogramwhere 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 bywhere Xa(j)(t)=Xa[(j−1)M+t], a=1,2,…,r; 0⩽K<M; t=1,2,…,M+K; j=1,2,…,L.
From , , one can writewhere the asterisk denotes complex conjugate.
The averaged tapered periodoram, Welch’s spectrum estimate, (see [8], [9]) is given by
If X(t) satisfies Assumption 1.1, then for each interval (see [7]), one getswhere da(j)(λ),a=1,2,…,r, is given by (1.9), the error term O{(M+K)−n/2} is uniform in λ1,…,λn and
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 fxx(λ) with respect to the spectral distribution Fxx(λ), that isthen, the spectral distribution corresponding to interval j is given byF(j)(λ) can be estimated using the tapered periodogram according to
Hence
From formula (1.11), one getswhere
Statistical properties of the autocovariance estimate
Let X(t),t=0,±1,…, be an r vector-valued stationary time series with autocovariance functionwhere E[X(t)]=0, t=0,±1,…, its spectral density function is given byfor the intervals, an estimate of C(j)(k) (see [5]) isAlsothe Fourier transform of Eq. (4.4) is
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:
As an estimate of the spectral density fab(j)(λ) isthenwherewhich is an estimate of fab(λ).
Now, we give some statistical properties for .
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