The Confidence Interval for the Periodic Intensity Function in the Presence of Power Function Trend on the Nonhomogeneous Poisson Process


CAUCHY –Jurnal Matematika Murni dan Aplikasi 
Volume 7(1) (2021), Pages 73-83 
p-ISSN: 2086-0382; e-ISSN: 2477-3344 

Submitted: July 05, 2021 Reviewed: October 15, 2021 Accepted: Ocotober 30, 2021 
DOI: https://doi.org/10.18860/ca.v7i1.12848  

The Confidence Interval for the Periodic Intensity Function in the 
Presence of Power Function Trend on the Nonhomogeneous 

Poisson Process 

Ikhsan Maulidi1, Bonno Andri Wibowo2, Nina Valentika3, Muhammad Syazali4, 
Vina Apriliani5 

1Department of Mathematics, Syiah Kuala University, Banda Aceh, Indonesia 
2Department of Mathematics, IPB University, Bogor, Indonesia 

3Department of Mathematics, Pamulang University, Tangerang Selatan, Indonesia 
4Department of Mathematics, Universitas Pertahanan, Bogor, Indonesia 

5Department of Mathematics Education, UIN Ar-Raniry, Banda Aceh, Indonesia 

Email: ikhsanmaulidi@unsyiah.ac.id   bonno1818@gmail.com , 
dosen02339@unpam.ac.id, muhamad.syazali@idu.ac.id, vina.apriliani@ar-raniry.ac.id    

 ABSTRACT  

The nonhomogeneous Poisson process is one of the most widely applied stochastic processes. In 
this article, we provide a confidence interval of the intensity estimator in the presence of a 
periodic multiplied by trend power function. This estimator's confidence interval is an 
application of the formulation of the estimator asymptotic distribution that has been given in 
previous studies. By using the asymptotic theorem, the distribution was derived in the form of a 
confidence interval for the intensity function. In addition, constructive proof of the convergent in 
probability has been provided for all power functions. The results of this study contribute to the 
study of statistical analysis of the estimators that have been formulated previously. 

Keywords: asymptotic distribution; interval confidence; intensity function; Poisson process. 

INTRODUCTION 

There are many events in nature can be modeled by stochastic modeling processes. 
The stochastic process is a set of random variables that map the sample space to a state 
space. One of the stochastic processes is a counting process which states the number of 
events at a time interval. The counting process assuming the number of events has a 
Poisson distribution is called the Poisson process. Some basic theories related Poisson 
process can be seen in [1]–[3]. 

Due to the intensity function, the Poisson process is divided into two categories, 
namely the homogeneous Poisson process and the nonhomogeneous Poisson process. A 
homogeneous Poisson process has a constant intensity function (independent of time), 
while a nonhomogeneous Poisson process has a time-dependent intensity function. This 
nonhomogeneous Poisson process is widely applied to real phenomena, such as the 
phenomenon of earthquakes [4], traffic accidents [5], and radio burst rates [6]. 

On the other hand, the study of the nonhomogeneous Poisson process in the form a 
periodic intensity function also has been conducted in recent years. [7] studied the 

https://doi.org/10.18860/ca.v7i1.12848
mailto:ikhsanmaulidi@unsyiah.ac.id
mailto:bonno1818@gmail.com
mailto:dosen02339@unpam.ac.id
mailto:muhamad.syazali@idu.ac.id
mailto:vina.apriliani@ar-raniry.ac.id


The Confidence Interval for the Periodic Intensity Function in the Presence of Power Function 
Trend on the Nonhomogeneous Poisson Process 

Ikhsan Maulidi 74 

estimation of the intensity function in a nonhomogeneous Poisson process by including 
a trend component in the periodic intensity function. The trend component began with a 
trend in the form of an additive linear function. Then the study was continued with a 
trend in the form of a multiplicative linear function [8]. The estimation of the intensity 
function was carried out using the general kernel function approach, and [9] examined 
this Poisson process with a uniform kernel approach. Other related studies can be seen 
in [10]–[13]. 

[14] studied the estimation of the periodic Poisson process intensity function with 
the power function trend using a general kernel. Furthermore, the statistical properties 
of these estimators have also been proven. [15] has given strong consistency of these 
estimators. In addition, the asymptotic normality of the estimator has also been 
formulated and given a numerical simulation of the consistency of the estimator [16]. 
The results obtained in that study are the estimator of the periodic component which 
converges to the normal distribution by providing certain conditions.  

As an application of the asymptotic normality, it can be determined the confidence 
interval of the estimator for the periodic component. This study provides the theorems 
for the confidence interval for the intensity function parameters and their proofs. The 
contribution of this study is to provide the characteristics of the estimator, especially in 
terms of accuracy. With a certain number of samples (interval length), it can be 
determined how accurately the estimator predicts the value of the parameter in the 
form of a confidence interval.  

METHODS 

The Estimator for Periodic Component of the Intensity Function 

Suppose that {𝑁(𝑡), 𝑡 ≥ 0} is a nonhomogeneous Poisson process with intensity 
function 𝜆 which locally integrable and unknown. Suppose also that 𝜆 is a periodic 
function with the trend of the power function, then λ which depends on the time 
variable 𝑠 can be expressed as 

𝜆(𝑠) = 𝜆𝑐
∗ (𝑠). 𝑎𝑠𝑏 .                                                              (1) 

The values of the 𝑎 and 𝑏 constants are assumed to be known, so that what is not 
known is the function of the periodic component of the intensity function, namely 𝜆𝑐

∗ . 
Equation (1) can also be stated as follows 

𝜆(𝑠) = 𝜆𝑐 (𝑠). 𝑠
𝑏 ,                                                               (2) 

with 𝜆𝑐 (𝑠) = 𝑎𝜆𝑐
∗ (𝑠). [14] has been given the kernel type estimator for 𝜆𝑐 (𝑠) by using  

general kernel functions. The estimator for periodic component of the intensity is 

�̂�𝑐,𝑛,𝑘 (𝑠) =
𝜏

𝑛
∑

1

ℎ𝑛 (𝑠 + 𝑘𝜏)
𝑏

∞

𝑘=0

∫ 𝐾 (
𝑥 − (𝑠 + 𝑘𝜏)

ℎ𝑛
) 𝑁(𝑑𝑥).                        (3)

𝑛

0

 

On Equation (3), the constant  𝜏 is a period of the intensity function which 
satisfies  

𝜆𝑐 (𝑠 + 𝑘𝜏) = 𝜆𝑐 (𝑠), for 𝑘 ∈ 𝑍. 
With 𝑛 is the length of the time interval used. In this case, since the Poisson process is a 
discrete stochastic process, it is clear that 𝑛 is a natural number. The function 𝐾 called a 
kernel function if it satisfies the following properties: (K1) 𝐾 is a probability density 
function, (K2) 𝐾 is bounded, and (K3) 𝐾 is defined in [-1,1] [17]. 
  



The Confidence Interval for the Periodic Intensity Function in the Presence of Power Function 
Trend on the Nonhomogeneous Poisson Process 

Ikhsan Maulidi 75 

The Asymptotic Normality of the Estimator 

Theorem 1 (The Asymptotic Normal Distribution for �̂�𝒄,𝒏,𝒌(𝒔), 𝟎 < 𝒃 < 𝟏) 

Suppose that the intensity 𝜆 satisfies (1) and locally integrable. The kernel function 
𝐾 satisfies (K1), (K2), (K3), 𝜆𝑐  has a bounded second derivative around of 𝑠, 0 < 𝑏 < 1, 
𝑛1−𝑏 ℎ𝑛 → 0, 𝑛

𝑏+1ℎ𝑛 → ∞, ℎ𝑛 ↓ 0 as 𝑛 → ∞, 

a) If (𝑛1+𝑏 ℎ𝑛
5 )

1

2 → 0, then 

(𝑛1+𝑏 ℎ𝑛 )
1

2(�̂�𝑐,𝑛,𝐾 (𝑠) − 𝜆𝑐 (𝑠))  
𝑑
→  Normal(0, 𝜎2)                                   (4) 

as 𝑛 → ∞, with 𝜎2 =
𝜏𝜆𝑐(𝑠)

(1−𝑏)
∫ 𝐾2(𝑧)𝑑𝑧.

1

−1
 

b) If (𝑛1+𝑏 ℎ𝑛
5 )

1

2 → 1, then 

(𝑛1+𝑏 ℎ𝑛 )
1

2(�̂�𝑐,𝑛,𝐾 (𝑠) − 𝜆𝑐 (𝑠))  
𝑑
→  Normal(𝜇, 𝜎2)                                      (5) 

as 𝑛 → ∞, with 𝜇 =
𝜆𝑐

′′(𝑠)

2
∫ 𝑧2𝐾(𝑧)𝑑𝑧

1

−1
 and 𝜎2 =

𝜏𝜆𝑐(𝑠)

(1−𝑏)
∫ 𝐾2(𝑧)𝑑𝑧.

1

−1
 

Theorem 2 (The Asymptotic Normal Distribution for �̂�𝒄,𝒏,𝒌(𝒔), 𝒃 = 𝟏) 

Suppose that the intensity 𝜆 satisfies (1) and locally integrable. The kernel function 
𝐾 satisfies (K1), (K2), (K3), 𝜆𝑐  has a bounded second derivative around of 𝑠,  𝑏 = 1, 

ln (𝑛)ℎ𝑛 → 0, 
𝑛2ℎ𝑛

𝑙𝑛(𝑛)
→ ∞, ℎ𝑛 ↓ 0 as 𝑛 → ∞, 

a) If (
𝑛2ℎ𝑛

5

ln (𝑛)
)

1

2
→ 0, then 

(
𝑛2ℎ𝑛
ln (𝑛)

)

1

2

(�̂�𝑐,𝑛,𝐾 (𝑠) − 𝜆𝑐 (𝑠))  
𝑑
→  Normal(0, 𝜎2)                                    (6) 

as 𝑛 → ∞, with 𝜎2 = 𝜏𝜆𝑐 (𝑠) ∫ 𝐾
2(𝑧)𝑑𝑧

1

−1
. 

b) If (
𝑛2ℎ𝑛

5

ln (𝑛)
)

1

2
→ 1, then 

(
𝑛2ℎ𝑛
ln (𝑛)

)

1

2

(�̂�𝑐,𝑛,𝐾 (𝑠) − 𝜆𝑐 (𝑠))  
𝑑
→  Normal(𝜇, 𝜎2)                                    (7) 

as 𝑛 → ∞, with 𝜇 =
𝜆𝑐

′′(𝑠)

2
∫ 𝑧2𝐾(𝑧)𝑑𝑧

1

−1
 and 𝜎2 = 𝜏𝜆𝑐 (𝑠) ∫ 𝐾

2(𝑧)𝑑𝑧.
1

−1
 

Theorem 3  (The Asymptotic Normal Distribution for �̂�𝒄,𝒏,𝒌(𝒔),  𝒃 > 𝟏) 

Suppose that the intensity 𝜆 satisfies (1) and locally integrable. The kernel function 
𝐾 satisfies (K1), (K2), (K3), and 𝜆𝑐  has a bounded second derivative around of 𝑠, 𝑏 > 1, 
𝑛2ℎ𝑛 → ∞, ℎ𝑛 ↓ 0 as 𝑛 → ∞, 

a) If (𝑛2ℎ𝑛
5 )

1

2 → 0, then 

(𝑛2ℎ𝑛)
1

2
 
(�̂�𝑐,𝑛,𝐾 (𝑠) − 𝜆𝑐 (𝑠))  

𝑑
→  Normal(0, 𝜎2)                                    (8) 

as 𝑛 → ∞, with 𝜎2 = 𝜏2−𝑏 𝜆𝑐 (𝑠)𝜁(𝑏) ∫ 𝐾
2(𝑧)𝑑𝑧

1

−1
, 

and 𝜁(𝑏) = lim
𝑛→∞

(∑
1

𝑘𝑏
∞
𝑘=1 𝐼(𝑦 + 𝑠 + 𝑘𝜏 ∈ [0, 𝑛])) . 



The Confidence Interval for the Periodic Intensity Function in the Presence of Power Function 
Trend on the Nonhomogeneous Poisson Process 

Ikhsan Maulidi 76 

b) If (𝑛2ℎ𝑛
5 )

1

2 → 1, then 

(𝑛2ℎ𝑛 )
1

2(�̂�𝑐,𝑛,𝐾 (𝑠) − 𝜆𝑐 (𝑠))  
𝑑
→  Normal(𝜇, 𝜎2)                                    (9) 

as 𝑛 → ∞, with 𝜇 =
𝜆𝑐

′′(𝑠)

2
∫ 𝑧2𝐾(𝑧)𝑑𝑧

1

−1
, 𝜎2 = 𝜏2−𝑏 𝜆𝑐(𝑠)𝜁(𝑏) ∫ 𝐾

2(𝑧)𝑑𝑧
1

−1
, and 𝜁(𝑏) =

lim
𝑛→∞

(∑
1

𝑘𝑏
∞
𝑘=1 𝐼(𝑦 + 𝑠 + 𝑘𝜏 ∈ [0, 𝑛])) . 

 
The proofs of Theorem 1, Theorem 2, and Theorem 3 above can be proved through 

a rough analysis, [18].  It is recommended to study the basic theory to proof these 
theorems in [19]–[21]. 

RESULTS AND DISCUSSION  

Suppose that ф denotes the standard normal distribution with ф−1 is the inverse. 
Based on Theorem 1, Theorem 2, and Theorem 3 above, it can be given some confidence 
interval for 𝜆𝑐  with significant level 1 − 𝛼 as follows: 

Corollary 1 (The confidence interval for 𝝀𝒄 for 𝟎 < 𝒃 < 𝟏) 

Suppose that all conditions on Theorem 1 are satisfied, the for a significant level α 
where 0 < 𝛼 < 1, the confidence interval for 𝜆𝑐  for 0 < 𝑏 < 1 has been given in the 
following conditions: 

a) If (𝑛1+𝑏 ℎ𝑛
5 )

1

2 → 0 then 

𝐼𝜆𝑐 = (�̂�𝑐,𝑛,𝑘 (𝑠) −
𝜎ф−1 (1 −

𝛼

2
)

√𝑛1+𝑏ℎ𝑛
, �̂�𝑐,𝑛,𝑘 (𝑠) +

𝜎ф−1 (1 −
𝛼

2
)

√𝑛1+𝑏ℎ𝑛
 ), 

where 𝜎2 =
𝜏𝜆𝑐(𝑠)

(1−𝑏)
∫ 𝐾2(𝑧)𝑑𝑧.

1

−1
 

b) If (𝑛1+𝑏 ℎ𝑛
5 )

1

2 → 1 then 

𝐼𝜆𝑐 = (�̂�𝑐,𝑛,𝑘 (𝑠) −
𝜎ф−1 (1 −

𝛼

2
) + 𝜇

√𝑛1+𝑏 ℎ𝑛
, �̂�𝑐,𝑛,𝑘 (𝑠) +

𝜎ф−1 (1 −
𝛼

2
) − 𝜇

√𝑛1+𝑏 ℎ𝑛
 ), 

where 𝜇 =
𝜆𝑐

′′(𝑠)

2
∫ 𝑧2𝐾(𝑧)𝑑𝑧

1

−1
 and 𝜎2 =

𝜏𝜆𝑐(𝑠)

(1−𝑏)
∫ 𝐾2(𝑧)𝑑𝑧.

1

−1
 

Corollary 2 (The confidence interval for 𝝀𝒄 for 𝒃 = 𝟏) 

Suppose that all conditions on Theorem 2 are satisfied, the for a significant level α 
where 0 < 𝛼 < 1, the confidence interval for 𝜆𝑐  for 𝑏 = 1 has been given in the following 
conditions: 

a) If (
𝑛2ℎ𝑛

5

ln (𝑛)
)

1

2
→ 0 then 

𝐼𝜆𝑐 = (�̂�𝑐,𝑛,𝑘 (𝑠) − 𝜎√
ln(𝑛)

𝑛2ℎ𝑛
ф−1 (1 −

𝛼

2
) , �̂�𝑐,𝑛,𝑘 (𝑠) + 𝜎√

ln(𝑛)

𝑛2ℎ𝑛
ф−1 (1 −

𝛼

2
) ), 

where 𝜎2 = 𝜏𝜆𝑐 (𝑠) ∫ 𝐾
2(𝑧)𝑑𝑧.

1

−1
 



The Confidence Interval for the Periodic Intensity Function in the Presence of Power Function 
Trend on the Nonhomogeneous Poisson Process 

Ikhsan Maulidi 77 

b) If (
𝑛2ℎ𝑛

5

ln (𝑛)
)

1

2
→ 1 then 

𝐼𝜆𝑐 = (�̂�𝑐,𝑛,𝑘 (𝑠) − (𝜎ф
−1 (1 −

𝛼

2
) + 𝜇)√

ln(𝑛)

𝑛2ℎ𝑛
, �̂�𝑐,𝑛,𝑘 (𝑠) + (𝜎ф

−1 (1 −
𝛼

2
) − 𝜇)√

ln(𝑛)

𝑛2ℎ𝑛
 ),  

where  𝜇 =
𝜆𝑐

′′(𝑠)

2
∫ 𝑧2𝐾(𝑧)𝑑𝑧

1

−1
 and 𝜎2 = 𝜏𝜆𝑐 (𝑠) ∫ 𝐾

2(𝑧)𝑑𝑧.
1

−1
 

Corollary 3 (The confidence interval for 𝝀𝒄 for 𝒃 > 𝟏) 

Suppose that all conditions on Theorem 3 are satisfied, the for a significant level α 
where 0 < 𝛼 < 1, the confidence interval for 𝜆𝑐  for 𝑏 > 1 has been given in the following 
conditions: 

a) If (𝑛2ℎ𝑛
5 )

1

2 → 0 then 

𝐼𝜆𝑐 = (�̂�𝑐,𝑛,𝑘 (𝑠) −
𝜎ф−1 (1 −

𝛼

2
)

√𝑛2ℎ𝑛
, �̂�𝑐,𝑛,𝑘 (𝑠) +

𝜎ф−1 (1 −
𝛼

2
)

√𝑛2ℎ𝑛
 ), 

where 𝜎2 = 𝜏2−𝑏 𝜆𝑐 (𝑠)𝜁(𝑏) ∫ 𝐾
2(𝑧)𝑑𝑧

1

−1
 

b) If (𝑛1+𝑏 ℎ𝑛
5 )

1

2 → 1 then 

𝐼𝜆𝑐 = (�̂�𝑐,𝑛,𝑘 (𝑠) −
𝜎ф−1 (1 −

𝛼

2
) + 𝜇

√𝑛2ℎ𝑛
, �̂�𝑐,𝑛,𝑘 (𝑠) +

𝜎ф−1 (1 −
𝛼

2
) − 𝜇

√𝑛2ℎ𝑛
), 

where 𝜇 =
𝜆𝑐

′′(𝑠)

2
∫ 𝑧2𝐾(𝑧)𝑑𝑧

1

−1
, 𝜎2 = 𝜏2−𝑏 𝜆𝑐 (𝑠)𝜁(𝑏) ∫ 𝐾

2(𝑧)𝑑𝑧
1

−1
, 

and 𝜁(𝑏) = lim
𝑛→∞

(∑
1

𝑘𝑏
∞
𝑘=1 𝐼(𝑦 + 𝑠 + 𝑘𝜏 ∈ [0, 𝑛])) . 

 

To strengthen the reasons for the above confidence intervals, it is given the 

probability convergence theorems for these confidence interval. 

 

Theorem 4. Convergence in Probability of the Confidence Interval for 𝝀𝒄 and 𝟎 <

𝒃 < 𝟏 

If λ̂c,n,k is the estimator for periodic component of the intensity function that is 

given in equation (3). Also,  Iλc,n  is a confidence interval that is given in Corollary 1, then 

for the value 0 < b < 1 satisfies 

P(λc,n(s)ϵIλc,n ) → 1 − α + o(1), 

provided n → ∞. 

The proof of Theorem 4: 

Case (a) Assumption  (𝒏𝟏+𝒃𝒉𝒏
𝟓 )

𝟏

𝟐 → 𝟎 

P(λc(s)ϵIλc ) = 𝑃 (λ̂c,n,k −
𝜎ф−1 (1 −

𝛼

2
)

√𝑛1+𝑏 ℎ𝑛
≤  λc,n(s) ≤ λ̂c,n,k +

𝜎ф−1 (1 −
𝛼

2
)

√𝑛1+𝑏 ℎ𝑛
) 



The Confidence Interval for the Periodic Intensity Function in the Presence of Power Function 
Trend on the Nonhomogeneous Poisson Process 

Ikhsan Maulidi 78 

P(λc,n(s)ϵIλc,n ) = P (−
𝜎ф−1 (1 −

𝛼

2
)

√𝑛1+𝑏 ℎ𝑛
≤  λc,n(s) − λ̂c,n,k ≤

𝜎ф−1 (1 −
𝛼

2
)

√𝑛1+𝑏ℎ𝑛
) 

P(λc,n(s)ϵIλc,n ) = 𝑃 (−
𝜎ф−1 (1 −

𝛼

2
)

√𝑛1+𝑏 ℎ𝑛
≤  λ̂c,n,k − λc,n(s) ≤

𝜎ф−1 (1 −
𝛼

2
)

√𝑛1+𝑏 ℎ𝑛
) 

P(λc,n(s)ϵIλc,n ) = 𝑃 (−𝜎ф
−1 (1 −

𝛼

2
) ≤  √𝑛1+𝑏 ℎ𝑛 (λ̂c,n,k − λc,n(s)) ≤ 𝜎ф

−1 (1 −
𝛼

2
)), 

Let 𝑌 = √𝑛1+𝑏ℎ𝑛 (λ̂c,n,k − λc(s)),  then based on Theorem 1 𝑌~Normal(0, 𝜎
2),  By using 

Central Limit Theorem 

𝑍 =
𝑌

𝜎
~Normal(0,1). 

Therefore 

P(λc(s)ϵIλc ) = 𝑃 (−ф
−1 (1 −

𝛼

2
) ≤  𝑍 ≤ ф−1 (1 −

𝛼

2
)) 

P(λc,n(s)ϵIλc,n ) = P (𝑍 ≤ ф
−1 (1 −

𝛼

2
)) − 𝑃 (𝑍 < −ф−1 (1 −

𝛼

2
)). 

Since the normal distribution has a symmetricity property,  

𝑃 (𝑍 < −ф−1 (1 −
𝛼

2
)) = 𝑃 (𝑍 ≥ −ф−1 (1 −

𝛼

2
)), 

then 

P(λc(s)ϵIλc ) = P (𝑍 ≤ ф
−1 (1 −

𝛼

2
)) − (𝑍 ≥ −ф−1 (1 −

𝛼

2
)) 

P(λc,n(s)ϵIλc,n ) = P (𝑍 ≤ ф
−1 (1 −

𝛼

2
)) − 1 + P (𝑍 ≤ ф−1 (1 −

𝛼

2
)) 

P(λc,n(s)ϵIλc,n ) = ф (ф
−1 (1 −

𝛼

2
)) − 1 + ф (ф−1 (1 −

𝛼

2
)) 

P(λc,n(s)ϵIλc,n ) = 1 −
𝛼

2
− 1 + 1 −

𝛼

2
= 1 − 𝛼, provided 𝑛 → ∞. 

Case (b). Assumption (𝒏𝟏+𝒃𝒉𝒏
𝟓 )

𝟏

𝟐 → 𝟏 

P(λc(s)ϵIλc ) 

= 𝑃 (λ̂c,n,k −
𝜎ф−1 (1 −

𝛼

2
) + 𝜇

√𝑛1+𝑏 ℎ𝑛
≤  λc,n(s) ≤ λ̂c,n,k +

𝜎ф−1 (1 −
𝛼

2
) − 𝜇

√𝑛1+𝑏 ℎ𝑛
) 

= P (−
𝜎ф−1 (1 −

𝛼

2
) + 𝜇

√𝑛1+𝑏 ℎ𝑛
≤  λc,n(s) − λ̂c,n,k ≤

𝜎ф−1 (1 −
𝛼

2
) − 𝜇

√𝑛1+𝑏 ℎ𝑛
) 

= 𝑃 (−
𝜎ф−1 (1 −

𝛼

2
) − 𝜇

√𝑛1+𝑏ℎ𝑛
≤  λ̂c,n,k − λc,n(s) ≤

𝜎ф−1 (1 −
𝛼

2
) + 𝜇

√𝑛1+𝑏 ℎ𝑛
) 



The Confidence Interval for the Periodic Intensity Function in the Presence of Power Function 
Trend on the Nonhomogeneous Poisson Process 

Ikhsan Maulidi 79 

= 𝑃 (− (𝜎ф−1 (1 −
𝛼

2
) − 𝜇) ≤  √𝑛1+𝑏 ℎ𝑛 (λ̂c,n,k − λc,n(s)) ≤ (𝜎ф

−1 (1 −
𝛼

2
) + 𝜇)) 

Suppose that 𝑌 = √𝑛1+𝑏 ℎ𝑛 (λ̂c,n,k − λc(s)) then based on Theorem 1b 𝑌~Normal(𝜇, 𝜎
2),  

and 

𝑍 =
𝑌 − 𝜇

𝜎
~Normal(0,1). 

Therefore 

P(λc(s)ϵIλc ) = 𝑃 (−ф
−1 (1 −

𝛼

2
) ≤  𝑍 ≤ ф−1 (1 −

𝛼

2
)) 

P(λc,n(s)ϵIλc,n ) = P (𝑍 ≤ ф
−1 (1 −

𝛼

2
)) − 𝑃 (𝑍 < −ф−1 (1 −

𝛼

2
)). 

Since the normal distribution has a symmetricity property  

𝑃 (𝑍 < −ф−1 (1 −
𝛼

2
)) = 𝑃 (𝑍 ≥ −ф−1 (1 −

𝛼

2
)), 

so 

P(λc(s)ϵIλc ) = P (𝑍 ≤ ф
−1 (1 −

𝛼

2
)) − (𝑍 ≥ −ф−1 (1 −

𝛼

2
)) 

P(λc,n(s)ϵIλc,n ) = P (𝑍 ≤ ф
−1 (1 −

𝛼

2
)) − 1 + P (𝑍 ≤ ф−1 (1 −

𝛼

2
)) 

P(λc,n(s)ϵIλc,n ) = ф (ф
−1 (1 −

𝛼

2
)) − 1 + ф (ф−1 (1 −

𝛼

2
)) 

P(λc,n(s)ϵIλc,n ) = 1 −
𝛼

2
− 1 + 1 −

𝛼

2
= 1 − 𝛼, provided 𝑛 → ∞. 

Theorem 5. Convergence in Probability of the Confidence Interval for 𝝀𝒄 and 𝒃 = 𝟏 

If λ̂c,n,k is the estimator for periodic component of the intensity function that is 

given in equation (3). Also,  Iλc,  is an confidence interval that is given in Corollary 2, then 

for the value b = 1 satisfies 

P(λc(s)ϵIλc ) → 1 − α + o(1), 

provided n → ∞.  

The proof of Theorem 5  

Case (a) Assumption (
𝑛2ℎ𝑛

5

ln (𝑛)
)

1

2
→ 𝟎 

P(λc(s)ϵIλc ) 

= 𝑃 (λ̂c,n,k − 𝜎√
ln(𝑛)

𝑛2ℎ𝑛
ф−1 (1 −

𝛼

2
) ≤  λc,n(s) ≤ λ̂c,n,k + 𝜎√

ln(𝑛)

𝑛2ℎ𝑛
ф−1 (1 −

𝛼

2
)) 

= P (−𝜎√
ln(𝑛)

𝑛2ℎ𝑛
ф−1 (1 −

𝛼

2
) ≤  λc,n(s) − λ̂c,n,k ≤ 𝜎√

ln(𝑛)

𝑛2ℎ𝑛
ф−1 (1 −

𝛼

2
)) 



The Confidence Interval for the Periodic Intensity Function in the Presence of Power Function 
Trend on the Nonhomogeneous Poisson Process 

Ikhsan Maulidi 80 

= 𝑃 (−𝜎√
ln(𝑛)

𝑛2ℎ𝑛
ф−1 (1 −

𝛼

2
) ≤  λ̂c,n,k − λc,n(s) ≤ 𝜎√

ln(𝑛)

𝑛2ℎ𝑛
ф−1 (1 −

𝛼

2
)) 

= 𝑃 (−𝜎ф−1 (1 −
𝛼

2
) ≤  √

𝑛2ℎ𝑛
ln(𝑛)

 (λ̂c,n,k − λc,n(s)) ≤ 𝜎ф
−1 (1 −

𝛼

2
)). 

Let = √
𝑛2ℎ𝑛

ln(𝑛)
 (λ̂c,n,k − λc(s)),  then based on Theorem 2 𝑌~Normal(0, 𝜎

2),  By using 

Central Limit Theorem 

𝑍 =
𝑌

𝜎
~Normal(0,1). 

Therefore  

P(λc(s)ϵIλc ) = 𝑃 (−ф
−1 (1 −

𝛼

2
) ≤  𝑍 ≤ ф−1 (1 −

𝛼

2
)) 

By using the same arguments before, it is obtained  

P(λc(s) ϵ Iλc ) = 1 − 𝛼, provided 𝑛 → ∞. 

Case (b). Assumption  (
𝑛2ℎ𝑛

5

ln (𝑛)
)

1

2
→ 𝟏 

P(λc(s)ϵIλc ) 

= 𝑃 (λ̂c,n,k − (𝜎ф
−1 (1 −

𝛼

2
) + 𝜇)√

ln(𝑛)

𝑛2ℎ𝑛
≤  λc(s) ≤ λ̂c,n,k + (𝜎ф

−1 (1 −
𝛼

2
) − 𝜇)√

ln(𝑛)

𝑛2ℎ𝑛
)  

= P (−𝜎ф−1 (1 −
𝛼

2
) + 𝜇)√

ln(𝑛)

𝑛2ℎ𝑛
≤  λc(s) − λ̂c,n,k ≤ 𝜎ф

−1 (1 −
𝛼

2
) − 𝜇)√

ln(𝑛)

𝑛2ℎ𝑛
) 

= 𝑃 (−ф−1 (1 −
𝛼

2
) − 𝜇)√

ln(𝑛)

𝑛2ℎ𝑛
≤  λ̂c,n,k − λc(s) ≤ 𝜎ф

−1 (1 −
𝛼

2
) + 𝜇)√

ln(𝑛)

𝑛2ℎ𝑛
) 

= 𝑃 (− (𝜎ф−1 (1 −
𝛼

2
) − 𝜇) ≤  √

𝑛2ℎ𝑛
ln(𝑛)

 (λ̂c,n,k − λc(s)) ≤ (𝜎ф
−1 (1 −

𝛼

2
) + 𝜇)), 

Suppose that 𝑌 = √
𝑛2ℎ𝑛

ln(𝑛)
 (λ̂c,n,k − λc(s)),  then according to Theorem 1b 

𝑌~Normal(𝜇, 𝜎2) and 

𝑍 =
𝑌 − 𝜇

𝜎
~Normal(0,1). 

Therefore 

P(λc,n(s) ϵ Iλc,n ) = 𝑃 (−ф
−1 (1 −

𝛼

2
) ≤  𝑍 ≤ ф−1 (1 −

𝛼

2
)). 

The same arguments  gave us  

P(λc(s) ϵ Iλc ) = 1 − 𝛼, provided 𝑛 → ∞. 



The Confidence Interval for the Periodic Intensity Function in the Presence of Power Function 
Trend on the Nonhomogeneous Poisson Process 

Ikhsan Maulidi 81 

 
Theorem 6. Convergence in Probability of the Confidence Interval for 𝝀𝒄,𝒏  

and 𝒃 > 𝟏 

If λ̂c,n,k is the estimator for periodic component of the intensity function that is 

given in equation (3). Also,  Iλc,n  is a confidence interval that is given in Corollary 3, then 

for the value b > 1 satisfies 

P(λc(s)ϵIλc ) → 1 − α + o(1), 

provided n → ∞.  

The proof of Theorem 6  

Case a. Assumption (𝑛2ℎ𝑛
5 )

1

2 → 0 

P(λc(s)ϵIλc ) 

= 𝑃 (λ̂c,n,k −
𝜎ф−1 (1 −

𝛼

2
)

√𝑛2ℎ𝑛
≤  λc(s) ≤ λ̂c,n,k +

𝜎ф−1 (1 −
𝛼

2
)

√𝑛2ℎ𝑛
) 

= P (−
𝜎ф−1 (1 −

𝛼

2
)

√𝑛2ℎ𝑛
≤  λc(s) − λ̂c,n,k ≤

𝜎ф−1 (1 −
𝛼

2
)

√𝑛2ℎ𝑛
) 

= 𝑃 (−𝜎
𝜎ф−1 (1 −

𝛼

2
)

√𝑛2ℎ𝑛
≤  λ̂c,n,k − λc(s) ≤ 𝜎

𝜎ф−1 (1 −
𝛼

2
)

√𝑛2ℎ𝑛
) 

= 𝑃 (−𝜎ф−1 (1 −
𝛼

2
) ≤  √𝑛2ℎ𝑛  (λ̂c,n,k − λc(s)) ≤ 𝜎ф

−1 (1 −
𝛼

2
)). 

Let 𝑌 = √𝑛2ℎ𝑛  (λ̂c,n,k − λc(s), then according to Theorem 3a 𝑌~Normal(0, 𝜎
2) and 

𝑍 =
𝑌

𝜎
~Normal(0,1). 

Therefore 

P(λc(s)ϵIλc ) = 𝑃 (−ф
−1 (1 −

𝛼

2
) ≤  𝑍 ≤ ф−1 (1 −

𝛼

2
)). 

The same arguments gave us  

P(λc(s) ϵ Iλc ) = 1 − 𝛼, provided 𝑛 → ∞. 

Case b. Assumption  (𝑛2ℎ𝑛
5 )

1

2 → 𝟏 

P(λc(s)ϵ Iλc )
 

= 𝑃 (λ̂c,n,k −
𝜎ф−1 (1 −

𝛼

2
) + 𝜇

√𝑛2ℎ𝑛
≤  λc(s) ≤ λ̂c,n,k +

𝜎ф−1 (1 −
𝛼

2
) − 𝜇

√𝑛2ℎ𝑛
) 

= P (−
𝜎ф−1 (1 −

𝛼

2
) + 𝜇

√𝑛2ℎ𝑛
≤  λc(s) − λ̂c,n,k ≤ 𝜎

𝜎ф−1 (1 −
𝛼

2
) − 𝜇

√𝑛2ℎ𝑛
) 



The Confidence Interval for the Periodic Intensity Function in the Presence of Power Function 
Trend on the Nonhomogeneous Poisson Process 

Ikhsan Maulidi 82 

= 𝑃 (−
𝜎ф−1 (1 −

𝛼

2
) − 𝜇

√𝑛2ℎ𝑛
≤  λ̂c,n,k − λc(s) ≤

𝜎ф−1 (1 −
𝛼

2
) + 𝜇

√𝑛2ℎ𝑛
) 

= 𝑃 (− (𝜎ф−1 (1 −
𝛼

2
) − 𝜇) ≤  √𝑛2ℎ𝑛  (λ̂c,n,k − λc(s)) ≤ (𝜎ф

−1 (1 −
𝛼

2
) + 𝜇)). 

Suppose that 𝑌 = √𝑛2ℎ𝑛  (λ̂c,n,k − λc(s),  then based on Theorem 3b 𝑌~Normal(𝜇, 𝜎
2) 

and 

𝑍 =
𝑌 − 𝜇

𝜎
~Normal(0,1). 

Therefore 

P(λc(s) ϵ Iλc ) = 𝑃 (−ф
−1 (1 −

𝛼

2
) ≤  𝑍 ≤ ф−1 (1 −

𝛼

2
)). 

By using the same arguments, it is obtained that  

P(λc(s) ϵ Iλc ) = 1 − 𝛼, provided 𝑛 → ∞. 

CONCLUSIONS 

From the results that have been studied, the formula to determine the confidence 
interval for parameter of the periodic component of the nonhomogeneous Poisson 
process with the intensity in the form of periodic function has been obtained. These 
confidence intervals have been given for each case of the values of b, this is because the 
results of previous studies show that the variance of the estimator is given in a different 
function for each case of the values of b. These confidence intervals have been proved to 
converge in probability 1 − 𝛼. 

The recommendation for further research that can be done is providing numerical 
simulations for each confidence interval case, there are 6 cases. The simulation can be 
started by determining the bandwidth function ℎ𝑛  which satisfies all the conditions in 
the given case and determining the probability of the estimator being in the confidence 
interval. 

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Trend on the Nonhomogeneous Poisson Process 

Ikhsan Maulidi 83 

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