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. ,