UDC 517.984 SOME ASYMPTOTIC PROPERTIES OF A KERNEL SPECTRUM ESTIMATE WITH DIFFERENT MULTITAPERS

Assume X1,X2,…,XN are realizations of N observations from a real-valued discrete parameter third-order stationary process Xt,t=0±1,±2,…, with bispectrum fXXX(λ1,λ2) where “−π≤λ1,λ2≤π”. Based on the previous assumption, L different multitapered biperiodograms IXXX(mt)j(λ1,λ2);j=1,2,…,L on overlapped segments (Xt(j);1≤t<N) can be constructed. Further, the mean and variance of the average of these different multitapered biperiodograms can be expressed as asymptotic expressions. According to different bispectral windows/kernels (Wβ(j)(α1,α2), where “−π⩽α1,α2⩽π” andβ is the bandwidth) and IXXX(mt)j(λ1,λ2), the bispectrum fXXX(λ1,λ2) can be estimated. The asymptotic expressions of the first- and second-ordered moments as well as the integrated relative mean squared error (IMSE) of this estimate are derived. Finally, some estimation results based on numerically generated data from the selected process “DCGINAR(1)” are presented and discussed in detail.


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.

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 where C XX (τ ) is the autocovariance function of X(t) and given by provided that If the process X(t) is invertible, then the inverse spectral density function is defined by where d XX (τ ) is the inverse autocovariance of X(t) (see [2]) and given by We construct L segments by dividing the given observations: where X (j) (t) is the set of observations in the j th 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 f XX (λ): where is the multitapered periodogram of X (j) (t) and given by XX (λ) and equals zero outside the interval [1, M + q] and K is the number of components of multitaper in each segment.

XX (λ)
In this section we obtain the asymptotic expressions of expectation, variance and integrated relative mean squared error of the smoothed (kernel) spectrum estimate f (mt)sp XX (λ): 3.1.Expected value.Taking expectation of equation ( 7), we get Making use of the transformation µ = λ + βα, with small β; λ ∈ (−π, π], then equation ( 9) becomes from a Taylor expansion for f XX (λ + βα) about λ, equation ( 10) has the form: where f XX (λ) is the second derivative of the spectrum f XX (λ).Therefore It is clear that the bias of
From equations ( 12) and ( 13) the mean squared error (MSE)of f (mt)sp 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 β 2 L −1 and [βKL 2 (M + q)] −1 .Then it follows that the variance and the squared bias terms of f (mt)sp XX (λ) are balanced for [βKL 2 (M +q)] −1 ≈ β 4 L −2 .This implies an optimal choice of bandwidth equals to β ≈ [K(M + q)] −1/5 .Hence, β → 0 as M → ∞.Using equations ( 12), (13) and the optimal choice of β, we get that is

XX
(λ) becomes less variability as at least K or L increases.