International Research Journal of Engineering and Technology (IRJET) e-ISSN: 2395 -0056
Volume: 04 Issue: 04 | Apr -2017 www.irjet.net p-ISSN: 2395-0072
ASYMPTOTIC PROPERTIES OF THE DISCRETE STABILITY TIME SERIES
WITH MISSED OBSERVATIONS BETWEEN TWO-VECTOR
VALUED STOCHASTIC PROCESS
M.A.Ghazal
1
, A.I.El-Deosokey
2
, M.A.Alargt
3
1
Department of Mathematics, Faculty of Science, University of Damietta, Egypt.
2
Lecture faculty of computer science and information system 6th of October University, Egypt.
3
Department of Mathematics, Faculty of Science, University of Damietta, Egypt.
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Abstract - In this paper, we defined the Expanded finite
Fourier transform of the strictly stability
) ( s r vector
valued time series where there are some randomly missed
observations, asymptotic moments are derived and the
application will be studied .
Key Words: Discrete time stability processes, Data tapers,
Finite Fourier transform, Missing values, Complex Normal
Distribution.
1.INTRODUCTION
Many authors, as e.g. Brillinger [1]; Dahlhaus[3]; Ghazal
and Farag [4] studied ''The estimation of the spectral
density, autocovariance function and spectral measure of
continuous time stationary processes''; E.A,El-Desokey[9]
studied ''Some properties of the discrete expanded finite
Fourier transform with missed observations''; M.A.Ghazal,
G.S. Mokaddis and A.El-Desokey[10],[11] are Studied ''The
Spectral Analysis of strictly stationary continuous time
series'' and ''Asymptotic Properties of spectral Estimates of
Second-Order with Missed Observations''. The paper is
organized as the following: Section1. Introduction, we
develop asymptotic properties of estimates the desired ,
) (u a In Section 2, the Asymptotic properties of Expanded
finite Fourier transform with missed observations was
discussed in section 3, section 4 we will apply our theoretical
study in two cases in climate and economy.
2. ASYMPTOTIC PROPERTIES OF ESTIMATES
THE DESIRED , ) (u a
Consider an ) ( s r vector-valued stability series
T
t Y t X t Z ) ( ) ( ) ( , (2.1)
,..... 2 , 1 , 0 t with ) (t X
-
r vector-valued and
) (t Y
s vector-valued.
We assume the series (2.1) is ) ( s r stability vector-valued
series with components
T
i j
t Y t X ) ( ) (
,
r j ,..., 2 , 1
s i ,....., 2 , 1 , all of
whose moments exist, we define the means as
y x
C t EY C t EX ) ( , ) ( (2.2)
The covariances
) ( ] ) ( ][ ) ( [ u C C t X C u t X E
xx
T
x x
,
) ( ] ) ( ][ ) ( [ u C C t Y C u t X E
xy
T
y x
, (2.3)
) ( ] ) ( ][ ) ( [ u C C t Y C u t Y E
yy
T
y y
,
and the second-order spectral densities
, ) (- ) ( ) 2 ( ) (
∑
1
u
xx xx
u i Exp u C f
, ) (- ) ( ) 2 ( ) (
∑
1
u
xy xy
u i Exp u C f (2.4)
∑
) (- ) ( ) 2 ( ) (
1
u
yy yy
u i Exp u C f
,
for .
In this section we consider the problem of determining an
s -vector , and an
r s filter )} ( { u a , so that
, ) ( ) (
u
u X u t a (2.5)
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