Confidence Intervals for the Mean Function of a Compound Cyclic Poisson Process  in the Presence of Power Function Trend


CAUCHY –Jurnal Matematika Murni dan Aplikasi 
Volume 7(3) (2022), Pages 411-419 
p-ISSN: 2086-0382; e-ISSN: 2477-3344 
 

Submitted: May 07, 2022 Reviewed: May 15, 2022 Accepted: June 23, 2022 
DOI: http://dx.doi.org/10.18860/ca.v7i3.15989       

Confidence Intervals for the Mean Function of a Compound Cyclic 
Poisson Process  in the Presence of Power Function Trend 

Faisal Muhamad*, I Wayan Mangku, Bib Paruhum Silalahi 

Department of Mathematics, IPB University, Bogor, Indonesia 
 

Email: idipbfaisalmuhammad@apps.ipb.ac.id 

ABSTRACT 

Asymtotic normality of an estimator for the mean function of a compound cyclic Poisson process 
in the present of power function trend which introduced by Safitri in 2002. To provided 
information on parameters guarantees (mean function) covered in an interval, it is necessary to 
find a convidence interval for the mean function of a compound cyclic Poisson process in the 
presence of power function trend. The objectives of this paper are: (i) to construct confidence 
interval for the mean function of a compound cyclic Poisson process with significance level 0 <
𝛼 < 1, (ii) to prove that the probability that the mean function contained in the confidence 
interval converges to 1 − 𝛼, and (iii) to observe, using simulation study, that the probabilities of 
the mean function contained in the confidence intervals for bounded length of observation 
interval. This paper showed that a confidence interval for the mean function and a theorem 
about convergence of the probability that the mean function contained in confidence interval. 
The simulation study shows that the probability that the mean function contained in the 
confidence interval is in accordance with the theorem. The contribution of this study is to 
provide information for users regarding confidence interval for the mean function of a 
compound cyclic Poisson process in the presence of power function trend. 

Keywords: compound cyclic poisson process; power function tren; mean function; confidence 
interval; poisson process. 

INTRODUCTION 

There are many events in everyday life that are uncertain, such as the birth and 
death process [1] the queue process [2] and the estimation of total insurance claims [3], 
which can be modeled using a stochastic process. A stochastic process is process that 
describes series of random events at certain time intervals [4]. A special form of 
stochastic process is the compound Poisson process. A compound Poisson process is a 
process of adding sequencess random variables of independent and identically 
distributed (i.i.d) with certain distribution as many as Poisson random variables, and 
independent of the Poisson process. Based on the time aspect, stochastic process can be 
classified in two categories, namely discrete time stochastic process and continuous 
time stochastic process. A special form of continuous time stochastic process is the 
Poisson process. The Poisson process is a counting a process in which the number of 
events in a Poisson distribution time interval. 

http://dx.doi.org/10.18860/ca.v7i3.15989
mailto:idipbfaisalmuhammad@apps.ipb.ac.id


Confidence Intervals for the Mean Function of a Compound Cyclic Poisson Process  in the Presence 
of Power Function Trend 

Faisal Muhamad 412 

Based on the intensity aspect, the Poisson process can be classified in two 
categories, that is homogeneous Poisson process with constant intensity function (not 
dependent on time) and the non-homogeneous Poisson Process with intensity function 
depends on time. One type of non-homogeneous Poisson process is the cyclic or periodic 
Poisson process [5]. The period can be daily, weekly, yearly or in other forms [6]. This 
non-homogeneous Poisson process is widely applied to real phenomena, such as th 
phenomenon of earthquakes [7], healthcare [8], radio burst rates [9], and traffic 
accidents [10].  

The study of the compound periodic Poisson process is widely. This research 
begins with the estimation of the expected value function on the compound periodic 
Poisson process [11] [12], then it was developed with a power trend [13] [14]. The 
compound cyclic Poisson does not follow the usual distribution patern. One aspect 
which can be estimated is the mean value. Since this value depends on the time of 
observation, it is called the mean function. In [15], an estimator for the mean function of 
a compound cyclic Poisson process has been constructed and studied. The asymptotic 
normality of this estimator also has been proven. Furthemore, to give assurance 
information that the mean function is included in an interval, it is necessary to construct 
a confidence interval for mean function of the compound cyclic Poisson process in the 
presence of power function trend. 

As an application of the asymptotic normality, it can be determined the 
confidence interval of the estimator for the periodic component. In [16] studied the  
confidence intervals for the mean and variance functions of compound Poisson process 
with power function intensity have been studied, while in [17] confidence intervals for 
the mean and variance functions of compound Poisson process with exponential of 
linear function intensity have been studied.  Specifically, this research was conducted to 
(i) to construct confidence interval for the mean function of a compound cyclic Poisson 
process in the presence of power function trend, (ii) to prove convergence to 1 − 𝛼 of 
probability that the mean function included in the confidence interval, and (iii) to check 
using simulation study that the probabilities of the mean function contained in the 
confidence intervals for bounded length of observation interval. The contribution of this 
study is to provide information for users regarding confidence interval for the mean 
function of a compound cyclic Poisson process in the presence of power function trend.  

 

METHODS  

The Estimator for the Mean Function  

 Suppose that {𝑁(𝑡), 𝑡 ≥ 0} is a nonhomogeneous Poisson process with 
(unknown) intensity function 𝜆 which is assumed to be locally integrable. Suppose that 𝜆 
has two components, namely a cyclic component (𝜆𝑐 ) with (known) period 𝜏 >  0 and a 
power function trend component (𝑎𝑠𝑏 ). In other words, for all 𝑎 ≥ 0 and 𝑠 ≥ 0, the 
intensity function 𝜆(𝑠) can be expressed as  

𝜆(𝑠) = 𝜆𝑐 (𝑠) + 𝑎𝑠
𝑏 . (1) 

 

 

The value of b  is assumed to be known real number and 0 < 𝑏 <
1

2
. We do not  

assumed any parametric form for the cyclic component 
c
 , except that it is cyclic or 

periodic, which satisfies  



Confidence Intervals for the Mean Function of a Compound Cyclic Poisson Process  in the Presence 
of Power Function Trend 

Faisal Muhamad 413 

𝜆𝑐 (𝑠) = 𝜆𝑐 (𝑠 + 𝑘𝜏) (2) 

for all 𝑠 ≥ 0  and all k∈ ℕ.  
 Suppose that {𝑌(𝑡), 𝑡 ≥ 0} is a process where 

𝑌(𝑡) = ∑ 𝑋𝑖
𝑁(𝑡)

𝑖=1
 

(3) 

with {𝑋𝑖 , 𝑖 ≥ 1} is a sequence of independent and identically distributed random 
variables which having mean 𝜇 < ∞ and variance 𝜎2 < ∞, and also independent of 
{𝑁(𝑡), 𝑡 ≥ 0}. The process {𝑌(𝑡), 𝑡 ≥ 0} is called a compound cyclic Poisson process with 
power function trend [7]. 
 Suppose that 𝜓(𝑡) is notation of the mean function of 𝑌(𝑡), that is 

𝜓(𝑡) = 𝐸(𝑌(𝑡)) = 𝐸[𝑁(𝑡)]𝐸[𝑋1] = 𝛬(𝑡)μ  (4) 

with 𝜇 = 𝐸(𝑋𝑖) and 

𝛬(𝑡) = ∫ 𝜆(𝑠) 𝑑𝑠
𝑡

0
. (5) 

Let 𝑡𝑟 = 𝑡 − ⌊
𝑡

𝜏
⌋ 𝜏, where ⌊

𝑡

𝜏
⌋ represents the largest integer  less than or equal to 

𝑡

𝜏
,  

𝑡

𝜏
∈ ℝ 

and 𝑘𝑡,𝜏 = ⌊
𝑡

𝜏
⌋. Then, for any real number 𝑡 ≥ 0, 𝑡 can be expressed as 𝑡 = 𝑘𝑡,𝜏 𝜏 + 𝑡𝑟  with 

0 ≤ 𝑡𝑟 ≤ 𝜏.  

  Let 𝜃 =  
1

𝜏
∫ λ𝑐 (𝑠) 𝑑𝑠

𝜏

0
 denotes the global intensity of the periodic component in 

the process {N(t ), t ≥ 0} and it is assumed that 𝜃 > 0. This 𝜃 can be written as 𝛬𝑐 (𝑡𝑟) +
𝛬𝑐

𝑐 (𝑡𝑟 ) with 

𝛬𝑐 (𝑡𝑟 ) = ∫ λ𝑐 (𝑠) 𝑑𝑠
𝑡𝑟

0

 (6) 

and  

𝛬𝑐
𝑐 (𝑡𝑟) = ∫ λ𝑐 (𝑠) 𝑑𝑠.

𝜏

𝑡𝑟

 (7) 

By using (6) and (7) and substituting (1) into (5), then for any  t ≥ 0, 𝛬(𝑡) can be written 
as 

𝛬(𝑡) = (𝑘𝑡,𝜏 + 1)𝛬𝑐 (𝑡𝑟)+𝑘𝑡,𝜏 𝛬𝑐
𝑐 (𝑡𝑟 ) +  

𝑎

𝑏+1
𝑡𝑏+1. (8) 

By substituting (8) into (4), we have 

𝜓(𝑡) = ((𝑘𝑡,𝜏 + 1)𝛬𝑐 (𝑡𝑟)+𝑘𝑡,𝜏 𝛬𝑐
𝑐 (𝑡𝑟 ) +  

𝑎

𝑏+1
𝑡𝑏+1) 𝜇. (9) 

Estimation of Mean Function  

In [11] an estimator of the mean function 𝜓(𝑡) has been formulated as follows 

�̂�𝑛,𝑏 (𝑡) = ((𝑘𝑡,𝜏 + 1)�̂�𝑐,𝑛,𝑏 (𝑡𝑟 ) + 𝑘𝑡,𝜏 �̂�𝑐,𝑛,𝑏
𝑐
(𝑡𝑟 ) +

�̂�𝑚,𝑏
𝑏 + 1

𝑡𝑏+1) �̂�𝑛 (10) 

where 

�̂�𝑐,𝑛,𝑏 (𝑡𝑟 ) =
(1 − 𝑏)𝜏1−𝑏

𝑛1−𝑏
∑

1

𝑘𝑏

𝑘𝑛,𝜏

𝑘=1
𝑁([𝑘𝜏, 𝑘𝜏 + 𝑡𝑟 ]) − �̂�𝑚,𝑏 (1 − 𝑏)𝑛

𝑏 𝑡𝑟 , (11) 

�̂�𝑐,𝑛,𝑏
𝑐
(𝑡𝑟) =

(1 − 𝑏)𝜏1−𝑏

𝑛1−𝑏
∑

1

𝑘𝑏

𝑘𝑛,𝜏

𝑘=1
𝑁([𝑘𝜏 + 𝑡𝑟 , 𝑘𝜏 + 𝜏]) + �̂�𝑚,𝑏(1 − 𝑏)𝑛

𝑏 (𝑡𝑟 − 𝜏), (12) 



Confidence Intervals for the Mean Function of a Compound Cyclic Poisson Process  in the Presence 
of Power Function Trend 

Faisal Muhamad 414 

�̂�𝑚,𝑏 =
(1 + 𝑏)𝑁([0, 𝑚])

𝑚(1+𝑏)
−

(1 + 𝑏)

𝑚𝑏
�̃�𝑛 , (13) 

�̃�𝑛 =
(1 − 𝑏)

𝑛1−𝑏 𝜏𝑏𝑏2
∑

1

𝑘𝑏

𝑘𝑛,𝜏

𝑘=1
𝑁([𝑘𝜏, 𝑘𝜏 + 𝜏]) −

(1 + 𝑏)(1 − 𝑏)𝑛𝑏 𝑁([0, 𝑛])

𝑛(1+𝑏)𝑏2
, (14) 

�̂�𝑛 =
1

𝑁[0, 𝑛]
∑ 𝑋𝑖

𝑁([0,𝑛])

𝑖=1
. (15) 

with �̂�𝑛 = 0  when 𝑁([0, 𝑛]) = 0. 

Asymptotic Normally of the Estimator for the Mean Function 

Theorem 1 (The Asymptotic Normally of the Estimator for the Mean Function) 

Suppose that the intensity 𝜆 statisfies (1) and locally integrable.  If 𝑌(𝑡) statisfies 
(2), then 

 

√𝑛1−𝑏 (�̂�𝑛,𝑏(𝑡) − 𝜓(𝑡)) 

𝑑
→ Normal (0, (𝑘𝑡,𝜏 + 1)

2
𝑎(1 − 𝑏)𝜏𝑡𝑟 𝜇

2 + 𝑘𝑡,𝜏
2

𝑎(1 − 𝑏)𝜏(𝜏 − 𝑡𝑟 )𝜇
2) 

(16) 

as 𝑛 → ∞.  
 The proofs of Theorem 1 can be proved through a rough analysis [11].   

RESULTS AND DISCUSSION  

Our main results are a confidence interval for the mean function 𝝍(𝒕) and a 
theorem about convergence of the probability that 𝝍(𝒕) contained in the confidence 
interval.  

Corollary 1 (The confidence interval for 𝝍(𝒕)) 

For given a significant level 𝛼, where 0 < 𝛼 < 1, the confidence interval for 𝜓(𝑡)  in the 

case 0 < 𝑏 <
1

2
 is given by 

 

𝐼𝜓,𝑛 = [�̂�𝑛,𝑏 (𝑡) − 
−1 (1 −

𝛼

2
) √𝑉𝑛,𝑏   ,  �̂�𝑛,𝑏(𝑡) + 

−1 (1 −
𝛼

2
) √𝑉𝑛,𝑏 ] 

with  

𝑉𝑛,𝑏 =
(𝑘𝑡,𝜏 + 1)

2
�̂�𝑚,𝑏 (1 − 𝑏)𝜏𝑡𝑟 �̂�𝑛

2
+ 𝑘𝑡,𝜏

2
�̂�𝑚,𝑏 (1 − 𝑏)𝜏(𝜏 − 𝑡𝑟 )�̂�𝑛

2

𝑛1−𝑏
, 

where  denotes the standard normal distribution and 𝑉𝑛,𝑏 denotes the studenize 

version of (16). 

Theorem 2 (Convergence of Probability that 𝝍(𝒕) ∈ 𝑰𝝍,𝒏) 

For confidence interval 𝐼𝜓,𝑛 of 𝜓(t) given in Corollary 1, we have that   

𝑃(𝜓(𝑡) ∈ 𝐼𝜓,𝑛) → 1 − 𝛼 

as 𝑛 → ∞. 



Confidence Intervals for the Mean Function of a Compound Cyclic Poisson Process  in the Presence 
of Power Function Trend 

Faisal Muhamad 415 

Proof of Theorem 2: 

The probability that 𝜓(𝑡) contained in the confidence interval 𝐼𝜓,𝑛 can be computed as 

follows. 
. 

𝑃 (�̂�𝑛,𝑏 (𝑡) − 
−1 (1 −

𝛼

2
) √𝑉𝑛,𝑏 ≤ 𝜓(t) ≤ �̂�𝑛,𝑏 (𝑡) + 

−1 (1 −
𝛼

2
) √𝑉𝑛,𝑏 )  

 

= 𝑃 (−−1 (1 −
𝛼

2
) √𝑉𝑛,𝑏 ≤ −�̂�𝑛,𝑏 (𝑡) + 𝜓(t) ≤ 

−1 (1 −
𝛼

2
) √𝑉𝑛,𝑏 )   

 

= 𝑃 (−1 (1 −
𝛼

2
) √𝑉𝑛,𝑏 ≥ �̂�𝑛,𝑏(𝑡) − 𝜓(t) ≥ −

−1 (1 −
𝛼

2
) √𝑉𝑛,𝑏 )   

 

= 𝑃 (−−1 (1 −
𝛼

2
) √𝑉𝑛,𝑏 ≤ �̂�𝑛,𝑏 (𝑡) − 𝜓(t) ≤ 

−1 (1 −
𝛼

2
) √𝑉𝑛,𝑏 )  

 

= 𝑃 (−−1 (1 −
𝛼

2
) ≤

�̂�𝑛,𝑏(𝑡)−𝜓(t)

√𝑉𝑛,𝑏
≤ −1 (1 −

𝛼

2
)).  

By the Studentize version of (16), we have that  
�̂�𝑛,𝑏(𝑡)−𝜓(t)

√𝑉𝑛,𝑏

𝑑
→ Normal(0,1),  

as 𝑛 → ∞. Therefore 𝑃(𝜓(𝑡) ∈ 𝐼𝜓,𝑛) converges to 

 𝑃 (−−1 (1 −
𝛼

2
) ≤ 𝑍 ≤ −1 (1 −

𝛼

2
))  

as 𝑛 → ∞, where 𝑍 is the standard normal random variable.  
Further we can simplify the above probability as follows. 

𝑃 (−−1 (1 −
𝛼

2
) ≤ 𝑍 ≤ −1 (1 −

𝛼

2
))  

=  𝑃 (𝑍 ≤ −1 (1 −
𝛼

2
)) − 𝑃 (𝑍 ≤  −1 (1 −

𝛼

2
)) 

= 𝑃 (𝑍 ≤ −1 (1 −
𝛼

2
)) − 𝑃 (𝑍 ≥ −1 (1 −

𝛼

2
)) 

= 𝑃 (𝑍 ≤ −1 (1 −
𝛼

2
)) − (1 − 𝑃 (𝑍 ≤ −1 (1 −

𝛼

2
))) 

=  (−1 (1 −
𝛼

2
)) − (1 −  (−1 (1 −

𝛼

2
))) 

= (1 −
𝛼

2
) − (1 − (1 −

𝛼

2
)) 

= 1 − 𝛼. 
This completes the proof of Theorem 2. 

Simulation of the Confidence Interval for the Mean Function 

The purpose of this simulation is to check the probability that the mean function 
𝜓(𝑡) is contained in the confidence intervals for some different significant levels, period, 
and length of observation interval, using generated data. This simulation was carried out 
with the help of R software and Scilab software for illustration the results. 



Confidence Intervals for the Mean Function of a Compound Cyclic Poisson Process  in the Presence 
of Power Function Trend 

Faisal Muhamad 416 

 The programing stage is carried out by generating the realization of compound 
periodic Poisson process with a power function trend with the formulation of the 
intensity function: 

𝜆(𝑠) = sin
2𝜋𝑠

𝜏
+ 1 + 𝑎𝑠𝑏 . 

In this simulation, we choose significant levels 𝛼 = 1%, 5% and 10%,  𝜏 = 1, 𝑠 = 2.5, 
𝑎 = 0.1, 𝑏 = 0.4, 𝑛 = 20, 50 and 100 with 1000 repetitions.  

Table 1. Simulation results of confidence interval for the mean function 𝜓(𝑡) 

            𝛼 𝑛 A B C D E 

1% 
20 985 15 98.5% 1.5% 0.5% 
50 988 12 98.8% 1.2% 0.2% 

100 990 10 99.0% 1.0% 0.0% 

5% 
20 941 59 94.1% 5.9%% 0.9% 
50 950 50 95.0% 5.0% 0.0% 

100 952 48 95.2% 4.8% 0.2% 

10% 
20 891 109 89,1% 10.9% 0.9% 
50 907 93 90.7% 9.3% 0.7% 

100 917 83 91.7% 8.3% 1.7% 
(A= the number confidence interval containing the parameter, B= the number confidence 
interval that  do not contain the parameter, C= percentage of confidence interval containing 
the parameter, D= percentage of  confidence interval that does not contain the parameter, E= 
absolute error between 𝛼 and percentage of confidence interval that does not contain the 
parameters) 

 Based on simulation results,   percentage of confidence interval that does not 
contain parameter at 𝑠 = 2.5 and 𝜏 = 1 with  𝛼 = 1%, 5% and 10% fir observation 
interval [0, 𝑛] with 𝑛 = 20, 50 and 100 respectively from 0.0% − 0.5%, 0.0% − 0.9% and 
from 0.7% − 1.7%. The error that obtained between 𝛼 and percentage of confidence 
interval that does not contain parameters also tend to be small, between 0% and 1.7%. 
This shows that the result of the simulation of the confidence interval for the mean 
function 𝜓(𝑡) for the compound Poisson process with different significant levels is in 
accordance with the theory obtained. The simulation results based on the first 200 
estimators can be seen in Figure 1. 



Confidence Intervals for the Mean Function of a Compound Cyclic Poisson Process  in the Presence 
of Power Function Trend 

Faisal Muhamad 417 

 
Figure 1. Confidence interval for the mean function 𝜓(𝑡) based on the first 200 estimators with 

𝑠 = 2.5, 𝜏 = 1, 𝛼 = 1% and 𝑛 = 100  

 
 The illustration in Figure 1 shows some of the results of the confidence interval 
simulation for the mean function 𝜓(𝑡) at 𝑠 = 2.5 and 𝜏 = 1 based on the first 200 
estimators with significance level of 𝛼 = 1% and 𝑛 = 100. It can be seen in the figure 
that the horizontal line is the true value of the mean function 𝜓(𝑡)  and vertical lines are 
the confidence intervals of the mean function 𝜓(𝑡). If the horizontal and vertical lines do 
not intersect each other, this indicates that the value of the mean function 𝜓(𝑡) is not in 
that interval. In Figure 1, there are three non intersecting lines which indicates there are 
three confidence intervals based on the first 200 estimators do not contain the value of 
the mean function 𝜓(𝑡). Since in Table 1 there are 10 confidence intervals that do not 
contain the mean function 𝜓(𝑡), this shows that there are seven  confidence intervals 
based on the 201-st to 1000-th estimators do not contain the value of the mean function  
𝜓(𝑡). 
 The illustration results in Figure 1 show that the probability of the mean function 
𝜓(𝑡) is contained in the confidence interval already close to 1 − 𝛼 for 𝜏 = 1 and 5, 𝛼 =
1%, 5% and 10% for bounded time interval observation. 

CONCLUSIONS 

According to the main results, it can be concluded that confidence interval for the 
mean function 𝜓(𝑡) of compound cyclic Poisson process in the presence of power 
function trend is  

𝐼𝜓,𝑛 = [�̂�𝑛,𝑏 (𝑡) − 
−1 (1 −

𝛼

2
) √𝑉𝑛,𝑏  ,  �̂�𝑛,𝑏 (𝑡) + 

−1 (1 −
𝛼

2
) √𝑉𝑛,𝑏 ], 

where  denotes the standard normal distribution and  

𝑉𝑛,𝑏 =
(𝑘𝑡,𝜏 + 1)

2
�̂�𝑚,𝑏 (1 − 𝑏)𝜏𝑡𝑟 �̂�𝑛

2
+ 𝑘𝑡,𝜏

2
�̂�𝑚,𝑏 (1 − 𝑏)𝜏(𝜏 − 𝑡𝑟 )�̂�𝑛

2

𝑛1−𝑏
. 

 Convergence of the probability that the mean function 𝜓(𝑡) contained in the 
confidence interval is 



Confidence Intervals for the Mean Function of a Compound Cyclic Poisson Process  in the Presence 
of Power Function Trend 

Faisal Muhamad 418 

 
𝑃(𝜓(𝑡) ∈ 𝐼𝜓,𝑛) → 1 − 𝛼, as 𝑛 → ∞.   

 The simulation results show that the probability of the mean function 𝜓(𝑡) 
included in the confidence interval already close to 1 − 𝛼 for a finite length observation 
interval. 

A recommendation for futher research can be to use different intensity functiom 
and different observation function from this study at thr simulation stage, so that they 
can show more diverse simulation results. 
 

REFRENCES 

[1] E. Roflin, "Analysis of Time Series with Calendar Effects," Management Science, vol. 
26, pp. 106-112, 2000. 

[2] S. Udayabaskaran and V. T. Dora Pravina, "Transient Analysis of an M/M/1 
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Confidence Intervals for the Mean Function of a Compound Cyclic Poisson Process  in the Presence 
of Power Function Trend 

Faisal Muhamad 419 

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