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

Faisal Muhammad, I Wayan Mangku, Bib Paruhum Silalahi

#### Abstract

We consider the problem of estimating 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 , (ii) to prove that the probability that the mean function contained in the confidence interval converges to , 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. The main results are 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 Trend; Mean Function; Confidence Interval; Poisson Process.

PDF

#### References

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

S. Udayabaskaran and V. T. Dora Pravina, "Transient Analysis of an M/M/1 Queueing System with Server Operating in Three Models," Far East Journal of Mathematical Science, vol. 101, pp. 1395-1418, 2017.

A. Chadidjah, "Proses Poisson dalam Estimasi Total Klaim," in Prosiding Seminar Nasional Matematika dan Pendidikan Matematika dengan Tema Peran Matematika dan Pendidikan Matematika dalam Menghadapi Isu-Isu Global, 2015, pp. 325-336.

S. M. Ross, Introduction to Probaility Models, Ninth ed. Florida: Academic Press Inc, 2010.

I. W. Mangku, "Estimating the intensity function of a Cyclic Posson Process," Univ of Amsterdam, 2001.

T. A. Walls and J. L. Schafer, "Models for Intensice Longitudinal Data," Oxford Univ Pr, 2006.

J. Geng, W. Shi, and G. Hu, "Bayesian nonparametric nonhomogeneous Poisson process with applications to USGS earthquake data," Eslevier, vol. 41, p. 100495, March 2021.

D. Munandar, S. Supian, and Subiyanto, "Probability distributions of COVID-19 tweet posted trends use a nonhomogeneous Poisson process," International Journal of Quantitative Research and Modeling, vol. 1, no.4, pp. 229-238, 2020.

E. Lawrence, S. Vander Wiel, C. Law, S. B. Spolar, and G. C. Bower, "the nonhomogeneous Poisson process for fast radio burst rates," Astron. J., vol. 154, no.3, p. 117, 2017.

F. Grabski, "Nonhomogeneous Poisson process anf compound Poisson process in the modelling of random process related to road accidents," J. KONES, vol. 26, no.1, pp. 39-46, 2019.

R. Ruhiyat, I. W. Mangku, and I. G. P. Purnaba, "Consistent Estimation of the Mean Function of Compound Cyclic Poisson Process," Far East J. Math. Sci, vol. 77, no. 2, pp. 183-194, 2013.

F. I. Makhmudah, I. W. Mangku, and H. Sumarno, "Estimating the Variance Function of A Compound Cyclic Poisson Process," Far East Journal of Mathematical Science (FJMS), vol. 100, no. 6, pp. 911-922, Sep 2016.

I. F Sari, I. W. Mangku, and H. Sumarno, "Estimating the Mean Function of a Compound Cyclic Poissom Process in the Presence od Power Function Trend," Far East J. Math. Sci, vol. 100, no. 11, pp. 1825-1840, 2016.

A. Fajri, "Pendugaan Ragam pada Proses Poisson Periodik Majemuk dengan Tren Fungsi Pangkat," IPB University, 2018.

N. I. Safitri, "Sebaran Asimtotik Penduga Fungsi Nilai Harapan Proses Poisson Periodik Majemuk dengan Tren Fungsi Pangkat," IPB University, 2022.

A. Fajri, "Selang Kepercayaan Fungsi Nilai Harapan dan Fungsi Ragam Proses Poisson Majemuk dengan Intensitas Fungsi Pangkat," IPB University, 2017.

S. Utami, "Interval Kepercayaan Fungsi Nilai Harapan dan Fungsi Ragam Proses Poisson Majemuk degan Intensitas Eksponensial Fungsi Linear," IPB University, 2018.

DOI: https://doi.org/10.18860/ca.v7i3.15989

### Refbacks

• There are currently no refbacks.