Confidence Interval for Variance Function of a Compound Periodic Poisson Process with a Power Function Trend

ABSTRACT


A. INTRODUCTION
Uncertain phenomena or events can be modeled with a stochastic process. Such phenomena or events include determining the estimated total future claims Sakthivel & Rajitha (2017), the occurrence of natural disasters Tse (2014), risk theory Zhang et al. (2014) and queuing systems  (Nosova, 2019). Therefore, stochastic processes have an important role in solving problems in various fields in real life (Crescenzo et al., 2015). Stochastic process is a process to quantify the relationship of a set of random events at a given time interval Ross (2014), to describe the occurrence of uncertain phenomena or events in the future (Acuña-ureta et al., 2021). Stochastic process is categorized into two, namely discrete time stochastic processes and continuous time stochastic processes (Mangku, 2021). However, this research only focuses on one form of continuous-time stochastic processes, namely the Poisson process.
A Poisson process { ( ); ≥ 0} has a certain intensity function where the number of phenomena or occurrences at each time interval is Poisson distributed (Last & Penrose, 2018). Poisson process are categorizeded into homogeneous and non-homogeneous Poisson process. a process is called periodic when it has a periodic intensity function (Mangku, 2001). Periodic Poisson processes have been widely used to model phenomena or events including events in the field of communication Belitser et al. (2012), finance Engle (2000), seasonal extreme rainfall events Ngailo et al. (2016), flood events Barbier et al. (2013), and the occurrence of earthquakes (Shao & Lii, 2011). A compound Poisson process is a process that has a sequence independent of the Poisson process ( ( )) and has i.i.d. random variables (Mangku, 2021).
The study of compound homogeneous Poisson processes has been widely developed. However, this process cannot be used if there are events that have a higher probability to occuring at a certain time intervals compared to the other time intervals. With cases like this, it is necessary to consider that time is influential so that the process used is non-homogeneous Poisson process. a specific cases of compound non-homogeneous Poisson process is compound periodic Poisson process.
The study of compound periodic Poisson processes has been developed by many researchers. The first research was conducted by Ruhiyat et al. (2013), and Mangku et al. (2013), namely the estimation of the mean functions of compound periodic Poisson process. Furthermore, in Makhmudah et al. (2016) researched the estimation of the variance function in the compound periodic Poisson processuses the power function tren to estimate the mean function of the compound Poisson process. Fajri et al. (2022) that uses the trend of the power function to formulate the ̂, ( ) in the periodic Poisson process. Then the modification of the estimator by Utama et al. (2022) was carried out to determine the asymptotic distribution of ̂, ( ) . This research is focused on formulating confidence intervals on ̂, ( ) which has been discussed in (Utama et al., 2022). a related work on confidence interval can also be found in (Muhamad et al., 2022).

B. METHODS
This research focuses on theory development to formulate confidence intervals with the following research flow: 1. Preliminary study a. Studying theories relevant to the estimator of the variance function to be analyzed. b. Exploring the mathematical foundation to find interval estimators for existing models. c. Studying the formulations of estimators and the asymptotic distribution of the existing estimator.
2. Formulate confidence interval for the variance function and prove that probability of the parameter being in the confidence interval convergence to ( 1 − ), analytically and numerically. 3. Conduct computer simulations using R and Scilab software with the following parameters: a. The selected values were 1%, 5%, dan 10%. b. The selected values were 1 and 5, where = 1 representing small period, and = 5 representing large period. c. The length of the observation interval used was = 20 , 50 , dan 100 which respectively represent small, medium and large observation intervals, . d. The variance function estimators used were repeated 1000 times.
Assume { ( ), ≥ 0} is non-homogeneous Poisson random variable whose rate.( ) has two functions, namely a periodic function having a positive period , and a power function trend. Thus, its intensity function at each > 0 is formulated as ( ) denotes a function of periodicity with period , denotes the power function trend, and denotes the slope of the trend. The power is a real number in the 0 < < 1 2 , and isn't assumed to have parameters of any kind, except that it's a periodic function, which can be written as Assume { ( ), ≥ 0} is a process that states where ( ) denotes a Poisson process for which of intensity function is and { , = 1,2, … } are i.i.d random variable with < ∞ and 2 > 0 ( Kruczek et al., 2017). Suppose the variance functions of the process ( ) is denoted by ( ) which is formulated as Let = − ⌊ ⌋ , ⌊ ⌋ denotes the greatest integer that less than or equal to , for every ∈ ℝ.

C. RESULT AND DISCUSSION 1. Confidence Interval of the Variance Function
A confidence interval is an interval estimate with a confidence coefficient (Hogg et al., 2019). Variance is used to measure the average amount of fluctuation of a random variable from an expected value (Ghahramani, 2016). Based on Theorem 1, we obtain = 0 and 2 = (1 + , ) 2 (1 − ) 2 2 + , 2 (1 − )( − ) 2 2 . From the expected value and variance above, the confidence interval of the variance function with a confidence level of 1 − for 0 < < 1 can be written as follows with Φ denotes the standardized normal distribution function and

Simulation of Confidence Intervals for Variance Functions
In this research, the simulation process aims to strengthen the results of analytical confidence interval formulation. The simulation process was carried out using R and Scilab software. R software's used to realize the data of , for variance function, while Scilab software's used to show the illustration image of , for variance function with 1000 times iterations. The results for confidence interval simulation are shown in Table 1. Based on Table 1, the simulation results show that the percentage of , that does not contain parameters for = 0.05 and = 1 with 1000 repetitions converges to . For = 1% the percentage range is from 0.9% − 1.4%, for = 5% the percentage range is from 5.0% − 5.6%, while for = 10% the percentage range is from 9.9% − 10.5%. Therefore, the absolute errors tend to be relatively small, in the range of 0.0% − 0.6%. Likewise, the percentage of , that contains parameters will corverge to 1 − , with = 1% in the range 98.6% − 99.1%, = 5% in the range 94.4% − 95.0%, and = 10% in the range 89.5% − 90.1%, The results from Table 1 also show that if is larger, then the number of , parameters is smaller, i.e. for = 1% ranges from 986 − 991, = 5% ranges from 944 − 950, and = 10% ranges from 895 − 901. The illustrative results of the confidence interval ( , ) using only 100 repetitions for = 1 can be viewed in Figure 1.  Figure 1 is an illustration result using Scilab software by inputting 100 repetitions of the , with = 0.05 and = 1. In Figure 1 there is a horizontal line and vertical lines. The vertical line is the confidence intervals of the variance function, while horizontal line represents the variance function value. The simulation results of the , above show 2 vertical lines that dont intersect with the horizontal line. This shows that a variance function value isn't in the 2 , . Table 1 shows that for = 1% and = 100 there are 9 , that dont contain parameters. Thus, in the 101 st to 1000 th repetitions there are 7 , that dont contain a variance function value. The simulation results using = 0.05 and = 5 for 1000 repetitions are shown in Table 2. Based on Table 2, the simulation results show that the percentage of , that does not contain parameters for = 0.05 and = 1 with 1000 repetitions converges to . For = 1% the percentage range is from 0.9% − 1.6%, for = 5% the percentage range is from 4.9% − 5.2%, while for = 10% the percentage range is from 10.0% − 10.2%. Therefore, the absolute errors tend to be relatively small, within the range of 0.0% − 0.6%. Likewise, the percentage of , that contains parameters will corverge to 1 − , with = 1% in the range 98.4% − 99.1%, = 5% in the range 94.8% − 95.1%, and = 10% in the range 89.8% − 90.0%, The results from Table 2 also show that if is larger, then the number of , parameters is smaller, i.e. for = 1% ranges from 984 − 991, = 5% ranges from 948 − 951, and = 10% ranges from 898 − 900. The illustrative results of the confidence interval ( , ) using only 100 repetitions with = 5 can be viewed in Figure 2. 896 | JTAM (Jurnal Teori dan Aplikasi Matematika) | Vol. 7, No. 3, July 2023, pp. 889-898 Table 2 shows that for = 1% and = 100 there are 16 , that dont contain parameters. Thus, for the 101 st to 1000 th repetitions there are 14 , that dont contain a variance function value.
Based on the simulation results of , given in Table 1 and 2 show that the probability of the variance function ( ) covered by the , is consistant with the analytical result, that is converges to 1 − , as → ∞. The result of this research are in accordance with research conducted by (Muhamad et al., 2022) which obtained that the probability of parameters being covered in the confidence interval is getting nearer to 1 − . If the real level used is 1%, 5%, and 10%, then the confidence interval obtained will approach 1 − namely 99% for = 1%, 95% for = 5%, and 90% for = 10% at a finite observation time [0, ].

D. CONCLUSION AND SUGGESTIONS
The results formulation show the confidence interval for the ( ) of the periodic Poisson process by adding a power function trend is obtained The convergence of the probability that the ( ) of a compound periodic Poisson process included in the confidence interval is