Spectral Density Estimation of Continuous Time Series

Ahmed Elhassanein^{1,2}

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1. Introduction

Although spectral analysis is one of the oldest tools for time series analysis, it is still one of the most widely used analysis techniques in many branches of sciences, [1] - [6] . For zero mean r-vector valued strictly stationary time series, the spectral estimation has been studied, [7] - [17] . Time series with missing observations frequantly appear in paractice. If a block of observations is periodically unobtainable, Jones [18] provides a development for spectral estimation of a stationary time series. The theory of amplitude-modulated stationary processes has been developed by Parzen [19] and applied to periodic missing observations problems [20] . The case where an observation is made or not according to the out come of a Bernoulli trial has been discussed by Scheinok [21] . Bloomfield [22] considered the case where a more general random mechanism is involved. Broersen et al. [23] and [24] developed models for time series with missing observation and discussed their use for spectral estimation. Unbiased spectral estimators have been formulated assuming wavelet models of stationary time series by [25] . Their asymptotic properties have been also investigated.

In this paper, we will discuss the spectral analysis of a strictly stationary r-vector valued continuous time series with randomly missing observations in joint segments of observations. The paper is organized as follows. Section 2 introduces the basic definitions and assumptions. The modified series is defined in Section 3. Section 4 considers the expanded finite Fourier transform and its properties. The modified periodogram, the spectral density estimator and its properties are given in Section 5.

2. Observed Series

Let be a zero mean r-vector valued strictly stationary time series with

(2.1)

and

(2.2)

where denotes the matrix of absolute values, the bar denotes the complex conjugate and '' denotes the matrix transpose. We may then define the matrix of second order spectral densities by

(2.3)

Using the assumed stationary, we then set down

Assumption I. is a strictly stationary continuous series all of whose moments exist. For each and any k-tuple we have

(2.4)

where

(2.5)

(;).

Because cumulants are measures of the joint dependence of random variables, (2.4) is seen to be a form of mixing or asymptotic independence requirement for values of well separated in time. If satisfies Assumption I we may define its cumulant spectral densities by

(2.6)

(). If the cross-spectra are collected together in the matrix of (2.3).

Assumption II. Let is bounded, is of bounded variation

and vanishes for all t outside the interval, that is called data window.

3. Modified Series

Let be a process independent of such that, for every t

note that

The success of recording an observation not depend on the fail of another and so it is independent. We may then define the modified series

with components,

where

4. Expanded Finite Fourier Transform in L-Joint Segments of Observations

In the case when there are some randomly missing observations, Elhassanein [17] constructed the expanded finite Fourier transform on disjoint segments of observations. In this section the expanded finite Fourier transform is constructed in L-joint segments of observations for a strictly stationary r-vector valued time series. Expression for its mean, variance and cumulant will be derived. The results introduced here may be regarded as a generalization to [13] and [17] . Let be an observed stretch of data with some randomly missing observations. Let, where L is the number of joint segments and N is the length of each segment and M is the length of joint parts, , where we get the results in [17] . The expanded finite Fourier transform of a given stretch of data, is defined by

(4.1)

where and is the data window satisfies Assump- tion II.

Theorem 4.1. Let be a strictly stationary r-vector valued time series with mean zero, and satisfy Assumption I. Let be defined as (3.1), and satisfy Assumption II, for then

(4.2)

(4.3)

where

and

where

for then

(4.4)

(4.5)

where is uniform in as, and

Proof. We will prove (4.5), by (4.1) we get

let and since

for some constants and, we get

where

since satisfy Assumption II for then

which implies to, using (2.6) the proof is completed. ,

5. Estimation

Using expanded finite Fourier transform (4.1), we construct the modified periodogram as

(5.1)

such that

where the bar denotes the complex conjugate. The smoothed spectral density estimate is constructed as

(5.2)

Theorem 5.1. Let be a strictly stationary r-vector valued continuous time series with mean zero, and satisfy Assumption I. Let be given by (3.6), and satisfy Assumption II for then

(5.3)

(5.4)

(5.5)

where the summation extends over all partitions

into pairs of the quantities

excluding the case with for some, where is uniform in.

Proof. By (5.1), we have

then by (4.3) the proof of (5.3) is completed. From (5.1), and by Theorem (2.3.2) in [10] p. 21, we have

By Theorem (4.1) the proof of (5.4) is completed. From (5.1), we have

By Theorem (2.3.2) in [10] p. 21, we get

where the summation extends over all indecomposable partitions of the transformed table

Then, by Theorem (4.1), we get the proof of (5.5). ,

Theorem 5.2. Let be a strictly stationary r-vector valued time series

with mean zero, and satisfy Assumption I. Let be

given by (3.6), for and satisfy Assumption II for Then are asymptotically independent variates. Also if. then is asymptotically independent of the previous variates. Where, denotes an symmetric matrix-valued Wishart variate with covariance matrix and degree of freedom and denotes an Hermitian matrix-valued complex Wishart variate with covariance matrix and degree of freedom.

Proof. The proof comes directly from Theorem (4.2), for more details about Wishart distribution see [26] . ,

Theorem 5.3. Let be a strictly stationary r-vector valued time series with mean zero, and satisfy Assumption I. Let be given by (3.7), then

(5.6)

(5.7)

Proof. By (5.2), we have

then by (5.3) the proof of (5.6) is completed. From (5.2), we get

which completes the proof of (5.7). ,

Theorem 5.4. Let be a strictly stationary r-vector valued time series with mean zero, and satisfy Assumption I. Let be given by (5.2),

, for, Then

are asymptotically independent variates. Also if. then is asymptotically indepen- dent of the previous variates.

Proof. The proof comes directly by Theorem (5.3) and Theorem (7.3.2) in [26] p. 162. ,

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