Science Journal of Mathematics and Statistics
ISSN: 2276-6324
Published By
Science Journal Publication
http://www.sjpub.org
© Author(s) 2015. CC Attribution 3.0 License.
Research Article
International Open Access Publisher
Volume 2015 (2015), Article ID: sjms-109, 15 Pages, doi: 10.7237/sjms/109
Some Statistical Analysis for Continuous – Time Stationary
Processes with Missed Data
Eman A. Farag1, 2 and Mohamed A. Ghazal3
1Statistical
Department, Faculty of Science for Girls, King Abdulaziz University, P.O. Box 80200, Jeddah 21589, Saudi Arabia
2Mathematics
Department, Faculty of Science, Helwan University, Egypt
E-mail: eamali@kau.edu.sa & eman_farage@yahoo.com
3Mathematics
Department, Damietta Faculty of Science, Mansoura University, Egypt
Accepted on May 03, 2015
Abstract: In this paper, the statistical analysis for continuoustime stationary processes with missed data is presented. The
statistics of this process are constructed. The asymptotic
properties (first order, second order and k – th order cumulant)
for this process are investigated. The asymptotic distribution for
an estimate of the spectral density function is established
if X a t  observed
1

𝒅𝒂 (𝑡) = 
0

otherwise
(1.5)
Keywords: Time series, asymptotic distribution, missed data,
continuous time stationary process, spectral density function,
normal distribution.
Since 𝑃 {𝑑𝑎 (𝑡) = 1} = 𝑝𝑎 > 0, 𝑃 {𝑑𝑎 (𝑡) = 0} = 𝑞𝑎 , 𝑝𝑎 +
𝑞𝑎 = 1, 𝑎 = ̅̅̅̅̅
1 , 𝑟 and
(1.6)
Introduction
Several authors as e.g. Bloomfield (1970), Brillinger
(1970), Marshal (1980) studied the asymptotic time series
with missing observations. Ghazal (1999), Ghazal and
Farag (1998a) and (1998b) studied the asymptotic time
stationary processes with classical methods.
Let 𝑋 𝑟 (𝑡) = {𝑋𝑎 (𝑡) , 𝑎 = ̅̅̅̅̅
1 , 𝑟, 𝑡 𝜖 𝑹} be an r-dimensional
continuous time stationary process with mean 𝑚𝑎 , 𝑎 =
̅̅̅̅̅
1 , 𝑟 , 𝑅𝑎𝑎 (𝜏) , 𝜏 𝜖 𝑹 the continuous covariance function
which is defined by

𝑹𝑎𝑎 (𝜏) = 𝐸{𝑋𝑎 (𝑡 + 𝜏) 𝑋𝑎 (𝑡)}=
f a a   exp i    d ; t ,  R
(1.1)
R
The paper is organized as follows: Estimation of the
expectation and its asymptotic distribution is determined
in Section 2. Section 3 contains the asymptotic distribution
for continuous – time processes with missed data.
2. Estimation of the expectation and its asymptotic
distribution
and the spectral density function 𝑓𝑎𝑎 (𝜆), 𝜆 ∈ 𝑹, 𝒂 = ̅̅̅̅̅
1 ,𝑟
which is given by
We mention some results, which will be used to prove
some theorems.
𝑓𝑎𝑎 (𝜆) = (2 𝜋)− 1 ∫𝑹 𝑅𝑎𝑎 (𝜏) 𝑒𝑥𝑝(− 𝑖 𝜆 𝜏) 𝑑𝜏, 𝜆 ∈ 𝑹 . (1.2)
Theorem 2.1 For estimation𝑚
̂𝑎 , which is defined by
equation (1.3), then we has
Consider the statistic:
̂𝒂 =
𝒎
𝑇
∫ 𝑋 (𝑡)
𝑇 0 𝑎
1
𝑑𝑡,
(1.3)
Which is constructed by T, 𝑇 = [ 1 , ∞), 𝑋𝑎 (𝑡) is a
sequence of observations on 𝑋 𝑟 (𝑡). Let
𝑌𝑎 (𝑡) = 𝑋𝑎 (𝑡) 𝑑𝑎 (𝑡),
(1.4)
is irregularly data on the stationary process 𝑌
𝑎=
̅̅̅̅̅
1 , 𝑟 and 𝑑𝑎 (𝑡) is Bernoulli sequence of random variables,
which satisfies
𝑟 (𝑡),
𝐸𝑚
̂𝑎 = 𝑚𝑎
(2.1)
𝑇 𝑐𝑜𝑣{𝑚
̂𝑎 , 𝑚
̂𝑏 } = 2𝜋 ∫𝑹 𝑓𝑎𝑎 (𝑥) Φ 𝑇 (𝑥) 𝑑𝑥,
Where
Φ 𝑇 (𝑥) = (2 𝜋 𝑇)− 1
(2.3)
𝑠𝑖𝑛2 𝑥 𝑇 ⁄2
(𝑥⁄2)2
,
(2.2)
|Page2

𝑇 𝐷𝑚
̂𝑎 = 2𝜋 ∫𝑹 𝑓𝑎𝑎 (𝑥) Φ 𝑇 (𝑥) 𝑑𝑥 ,
(2.4)

g x  T x  dx  g 0 

 g x  g 0
Proof: Equation (2.1) comes directly by using formula
(1.3). From equation (1.1), we get
𝑇
= 𝑇
−2
0
0
Since
𝑇
∫0 𝑒𝑥𝑝(𝑖 𝑥 𝑡) 𝑑𝑡 =
sin 𝑥 𝑇⁄2
𝑥⁄2
𝑒𝑥𝑝(𝑖 𝑥 𝑇⁄2),
and by using formula (2.3), then equation (2.2) is
obtained. Equation (2.4) comes directly by putting 𝑎 = 𝑏
in equation (2.2). Now, the proof of the theorem is
completed.
Now, theorem (2.2) below shall study the properties of the
kernel Ψ 𝑇 (𝑥) on 𝑹, 𝑥 ∈ 𝑹 which is defined by:
Ψ 𝑇 (𝑥) = (2 𝜋 𝑇)− 𝟏 |
𝑇
∫0 𝑒𝑥𝑝(−
𝑖 𝑥 𝑡) 𝑑𝑡 | , 𝑇 = 1, 2, (2.5)
Theorem 2.2 For any 𝑥 ∈ 𝑹, the function Ψ 𝑇 (𝑥) is the
kernel on 𝑹 that satisfy the following properties
∞
(1)
∫− ∞ Ψ 𝑇 (𝑥) 𝑑𝑥 = 1, 𝑥 𝜖 𝑹.
(2)
−δ
lim ∫ ΨT (x)dx
𝑇→∞ −∞
∞
= lim ∫δ ΨT (x)dx = 0, for any
𝑇→∞
(2.7)
𝛿 >, 𝑥 ∈ 𝑹,
(3)
(2.6)
δ
lim ∫ ΨT (x)dx
𝑇→∞ −δ
A2  
=
A1  A2  A3 .
g x  at x  0 , we get

  x dx . Then, A
2
T


A2 is very small according any  is very small. Then
A2  0 . Suppose that g x  , x  R is bounded by
Now,
constant M, then according to formula (2.7),

A1  2 M
  x dx  0 .
T
 
T

Similarly,
A3 T

 0 .

 

Therefore,
 g x  x dx  g 0  0 ,
T
 
T
this

completes the proof of the theorem.
Theorem 2.3
If the spectral function f a b  x , x  R ,
a , b 1, r is bounded and continuous at a point x  0 ,
(1)
(2)
Proof: The proof is omitted.
Lemma 2.1
If the function 𝑔(𝑥), 𝑥 ∈ 𝑹 is bounded
and continuous at a point 𝑥 = 0 and the function Ψ 𝑇 (𝑥),
𝑥 ∈ 𝑹 satisfies the properties of Theorem 2.2, then
Proof: By using formula (2.6), we get
 T x  dx
then
= 1, for all 𝛿 > 0,𝑥 ∈ 𝑹.
ΨT (x)dx = g(0).
 g x  g 0
Now, from continuity of
(2.8)
∞
lim ∫ g(x)
𝑇→∞ −∞
T x  dx 

𝑇
∫ 𝑓𝑎𝑏 (𝑥) {∫ 𝑒𝑥𝑝(𝑖 𝑥 𝑡1 ) 𝑑𝑡1 ∫ 𝑒𝑥𝑝(− 𝑖 𝑥 𝑡2 ) 𝑑𝑡2 } 𝑑𝑥
𝑹
 T x  dx 


𝑐𝑜𝑣 (𝑚̂
̂𝑏 )
𝑎 ,𝑚
 g x  g 0


for all 𝑥 ∈ 𝑹 and 𝑎 , 𝑏 = ̅̅̅̅̅
1 , 𝑟.

(2.9)
lim T cov  m̂a , m̂b   2  f a b 0, a ,b  1 , r
T
 
lim T D m̂a  2  f a a 0, a  1 , r
T
 
(2.10)
(2.11)
Proof: The proof can be easily obtained by using lemma
2.1.
Theorem 2.4
Let X
r
t    X a t  ,t  R , a  1 , r ,
be r – dimensional continuous stationary process with
mean zero. Then the statistic
 a T  
1
T
T
 X t  dt ,
a
(2.12)
0
is asymptotically normal distribution with mean zero and
dispersion given by
How to Cite this Article: Eman A. Farag and Mohamed A. Ghazal, "Some Statistical Analysis for Continuous – Time Stationary Processes with Missed Data",
Science Journal Of Mathematics and Statistics, Volume 2015 (2015), Article ID: sjms-109, 4 Pages, doi: 10.7237/sjms
|Page3
lim D a T   2  f a a 0 , a  1 , r .
Then
T
 
Proof: We begin by noting that
to the condition that E
E  T    0 according
X a t   0 , a  1 , r . Next we note
from formula (1.2) that
cov  a T  , b T   
=
Then

1
T
T T


f a b x  
 expi x  t
1
0 0
T

1
2   f a b x   2  T   exp  i x t1  dt1

0

T

exp  i x t 2
0

f a b x  T x  dx .
lim D  a T    2  f a a 0 , a  1 , r .

0
a1

ak
1
k
T
2

   ca1 ak  t1  t k , ,t k 1  t k dt1 dtk
T
T
0
0



T
T

0 T

 T
k
2
  ca1  ak  u1 , ,u k 1  du1 duk 1 dt .

T


process
equations
(1.5)
Ra a   ,   R
and
is
the
d a t  ,t  R , a  1 , r satisfies
(1.6), ma is the mean,
covariance


function
and
.
(1)
Ya t  , t  R , a  1 , r , then
ma  ma pa ,
(2)
(3)
 p a Raa 0  ma2 p a q a

Ra a    
 p 2 R  
 a aa
f aa    p f a a     2 
2
a

1
if   0
if   0
(3.2)

pa qa ma2  Ra a 0
(3.3)
T
Suppose


the
(1.4) where a sequence
Putting t1  t k  u1 , ,t k 1  t k  u k 1 , then
cum  a1 T  ,, ak T 
If
(3.1)
cum  a1 T  , , ak T  =
k
2
on the processes
   cum  X t  ,  , X t  dt  dt
0
3. Asymptotic distribution of stationary process with

 dt missed
dx observations
f aa   ,   R , a  1 , r is the spectral density function
cum  a1 T  , , ak T  =
2
asymptotically
X a ( t ) and Ya t  , a  1 , r corresponding by equation
T
 
T
is
normal distribution with mean zero and dispersion given
Lemma 3.1
Taking the limits on both sides and then by using lemma
2.1 and then putting a  b , we get
T
a
0
equation (1.4).


T
 X t  dt
T

cov  a T  , b T    2
k
 a T  
T
1
Now, we will study the moments and the asymptotic
distribution for Ya t , t  R which is defined by
.
By using equation (2.5), then


 t 2   dt1 dt2 dx
by 2  f a a 0 , which completes the proof.
=

Finally,

1 k
cum  a1 T  , , ak T   O  T 2  T

 0,



for all k  2 .
Proof: Formula (3.1) can be deduced from equation (1.4)
and properties of d a t  , a
 1 , r . To prove formula (3.2),
we get
ca1 ak  u1 , ,u k 1  du1 duk 1  , k  2 ,3 ,
Ra a    E X a  t   X a ( t ) E d a  t   d a t  
E X a  t   E X a ( t ) E d a  t   E d a t  .
Since
E X a  t 
 X b ( t )  Ra b    ma2 ,
(3.4)
How to Cite this Article: Eman A. Farag and Mohamed A. Ghazal, "Some Statistical Analysis for Continuous – Time Stationary Processes with Missed Data",
Science Journal Of Mathematics and Statistics, Volume 2015 (2015), Article ID: sjms-109, 4 Pages, doi: 10.7237/sjms

|Page4
Then
 E X a t  X a t  E d a t  d a t   E X a t  E X a t 

Ra a    
 E X  t   X t  E d  t   d t   m 2 p 2
a
a
a
a
a
a

By using equation (3.4) and properties of d a t  , a
we have


 1,r ,

2
a

1 
 0

1
D ma  2
T
p f a a     2 
1
 t 2  dt1 dt2 
T pa
Now, we will study the limiting distribution for
the statistic defined by (3.5).

Let
ma be defined by equation (3.5). If
f a a   is continuous at a point
  0 and bounded on R
then
lim T D ma  2  f a a 0 


pa2 Ra a 0  2  pa2 f aTa 
 

1
pa q a Ra a 0   2 

1
2
a
m pa qa
.
Then equation (3.3) is obtained and then the proof is
completed.
Theorem 3.1
1
T pa
1
pa
 2
 ma q a 




f
y
dy

 aa



(3.8)
(3.7) as T
 Y t  dt ,

  .
References
1.
P. Bloomfield, Spectral analysis with random missed
observations, J. Roy. Statist. Soc. B 32 (1970) 369 – 380.
2.
D.R. Brillinger, Statistical inference for irregularly observed
processes, J. Roy. Statist. Soc. B 32 (1970) 93 – 114.
3.
M.A. Ghazal, On spectral density estimate on non-crossed
intervals observation, Int. J. Appl. Math. I (8) (1999) 875 – 882.
4.
M.A. Ghazal and E.A. Farag, Estimation the spectral density;
autocovariance function and spectral measure of continuous –
time stationary, The Annual Conference ISSR, Cairo University,
vol. 33, Part I,1998a, pp. 1-20.
5.
M.A. Ghazal and E.A. Farag, Asymptotic distribution of spectral
density estimate of continuous – time series on crossed
intervals observation, The Egyptian Statistical Journal of ISSR,
Cairo University, vol. 42, No. 2, 1998b, pp. 197 – 214.
Let
T
a
0
(3.5)
Then
E ma  ma ,
(3.6)

 2
1 
2
D ma 
  f a a  y  dy  ma q a  
T p a   
T

where  T
aa
0 0
Ra a 0  ma2 qa
Proof: The proof can be obtained directly from equation
=
2
a
,
  R t
the spectral density
0






R

exp

i


d

=
  a a


 2  1  pa Ra a 0  ma2 pa qa  
T T
By using equation (1.1), then formula (3.7) is obtained and
the proof is completed.
Corollary 3.1

ma 
0 0
.
p a Ra a 0  ma2 p a q a  p a2 Ra a   exp   i  
p a2
+
   E Ya t1  Ya t 2   E Ya t1  E Ya t 2   dt1
 0
2
a
Then formula (3.2) is obtained. Finally, from the definition
of spectral density and equation (3.2), we get
f aa     2 
if   0
T T
1
D ma  2 2
T pa
By using equations (1.4) and (3.2), we get
 Ra a 0  m p a  m p ,

Ra a    
 R    m 2 p 2  m 2 p 2 ,
a
a
a
a
 aa
2
a
Proof: Formula (3.5) comes directly from equation (1.4).
To
formula (3.6), then by
E dprove
if using
 0 equation (3.5) and
a t  E d a t  ,
the definition of the dispersion to get
(3.7)



f a a  y   T  y  dy
6.
R.J. Marshal, Autocorrelation estimation of time series with
randomly missing observation, Biometrica 67 (1980) 557 –
570.
 y  is defined by equation (2.3).
How to Cite this Article: Eman A. Farag and Mohamed A. Ghazal, "Some Statistical Analysis for Continuous – Time Stationary Processes with Missed Data",
Science Journal Of Mathematics and Statistics, Volume 2015 (2015), Article ID: sjms-109, 4 Pages, doi: 10.7237/sjms
d