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

Ade Irawan; I Wayan Mangku; Retno Budiarti

doi:10.31764/jtam.v7i3.14836

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

Ade Irawan, I Wayan Mangku, Retno Budiarti

Abstract

This research is a follow-up research of Utama (2022) on asymptotic distribution of an estimator for variance function of a compound periodic Poisson with the power function trend. The objectives of this research are (i) to formulate a confidence interval for the variance function of a compound periodic Poisson process with a power function trend and (ii) to prove the convergence to 1-α probability of the parameter included in the confidence interval. This research process begins with a review of the existing formulation of the variance function estimator and its asymptotic distribution. Next, the confidence interval for the variance function of the compound periodic Poisson process with a power function trend is formulated and the convergence to 1-α is determined. After obtaining the confidence interval, the research continued by conducting computer simulations to confirmed the results obtained analytically. The results obtained show that the confidence interval for the variance function of a compound periodic Poisson process with a power function trend converges to 1-α both analytically and numerically for different finite time intervals.

Keywords

Convidence Interval; Variance Function; Power Function Trend.

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References

Acuña-ureta, D. E., Orchard, M. E., & Wheeler, P. (2021). Computation of Time Probability Distributions for the Occurrence of Uncertain Future Events. Mechanical Systems and Signal Processing, 150(9), 107332. https://doi.org/10.1016/j.ymssp.2020.107332

Barbier, E. B., Georgiou, I. Y., Enchelmeyer, B., & Reed, D. J. (2013). The Value of Wetlands in Protecting Southeast Louisiana from Hurricane Storm Surges. PLos ONE, 8(3), 1–6. https://doi.org/10.1371/journal.pone.0058715

Belitser, E., Serra, P., & Zanten, H. V. A. N. (2012). Estimating the Period of a Cyclic Non-Homogeneous Poisson Process. The Scandinavian Journal of Statistics, 1982, 204–218. https://doi.org/10.1111/j.1467-9469.2012.00806.x

Crescenzo, A. Di, Martinucci, B., & Shelemyahu Zacks. (2015). Compound Poisson Process with a Poisson Subordinator. Journal of Applied Probability, 52(2), 360–374. https://doi.org/10.1239/jap/1437658603

Engle, R. (2000). The Econometrics of Ultra-High-Frequency Data. Econometrica, 68(1), 1–22. https://doi.org/10.1111/1468-0262.00091

Fajri, A., Mangku, I., & Sumarno, H. (2022). Kajian Penduga Fungsi Ragam Proses Poisson. MILANG, 18(2), 87–97. https://doi.org/10.29244/milang.18.2.87-97

Ghahramani, S. (2016). Fundamental of Probability (3rd ed.). Florida: CRC Press Taylor & Francis Group.

Hogg, R. V., McKean, J. W., & Allen T. Craig. (2019). Introduction to Mathematical Statistics (7th ed.). New Jersey (US): Prentice Hall.

Kruczek, P., Polak, M., Wyłomańska, A., & Kawalec, W. (2017). Application of Compound Poisson Process for Modelling of ore Flow in a Belt Conveyor System with Cyclic Loading. International Journal of Mining, Reclamation and Environment, 0930(11), 1–16. https://doi.org/10.1080/17480930.2017.1388335

Last, G., & Penrose, M. (2018). Lectures on the Poisson Process. New York: Sheridan Books, Inc.

Li, C., Zhong, S. J., Wang, L., Su, W., & C Fang. (2014). Waiting Time Distribution of Solar Energetic Particle Events Modeled with Non-stationary Poisson Process. The Astrophysical Jurnal Letters, 792(2), 121–124. https://doi.org/10.1088/2041-8205/792/2/L26

Makhmudah, F., Mangku, I., & Sumarno, H. (2016). Estimating The Variance Function of a Compound Cyclic Poisson Process. Far East Journal of Mathematics Sciences, 100(6), 911–922. https://doi.org/dx.doi.org/10.17654/MS100060911

Mangku, I. W. (2021). Proses Stokastik Dasar. Bogor: IPB Press.

Mangku, I. W., Ruhiyat, & Purnaba, I. G. P. (2013). Statistics Properties of an Estimator for the Mean Function of a Compound Cyclic Poisson Process. Far East Journal of Mathematical Sciences, 82(2), 227–237.

Muhamad, F., Mangku, I. W., & Silalahi, B. P. (2022). 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, 7(3), 411–419. https://doi.org/10.18860/ca.v7i3.15989

Ngailo, T., Shaban, N., Reuder, J., Rutalebwa, E., & Mugume, I. (2016). Non Homogeneous Poisson Process Modelling of Seasonal Extreme Rainfall Events in Tanzania. International Journal of Science and Research, 5(10), 1858–1868. https://doi.org/10.21275/ART20162322

Nosova, M. (2019). Research of a three-phase autonomous queuing system with a Markov Modulated Poisson process‏. Researchgate.Net‏, January. www.researchgate.net/publication/330116702

Ross, S. M. (2014). Introduction to Probability Models (11st ed.). Orlando, Florida: Academic Press Inc. https://doi.org/10.1016/c2013-0-11417-1

Ruhiyat, R., Mangku, I., & Sumarno, H. (2013). Consistent Estimation of the Mean Function of a Compound Cyclic Poisson Process. Far East J. Math. Sci., 77(2), 183–194.

Sakthivel, K. M., & Rajitha, C. S. (2017). Artificial Intelligence for Estimation of Future Claim Frequency in Non-Life Insurance. Global Journal of Pure and Applied Mathematics, 13(6), 1701–1710. www.ripublication.com/gjpam17/gjpamv13n6_19.pdf

Shao, N., & Lii, K. (2011). Modelling non-homogeneous Poisson processes with Almost Periodic Intensity Functions. Journal of The Royal Statistical Society, 73(1), 99–122. https://doi.org/10.1111/j.1467-9868.2010.00758.x

Tse, K. (2014). Some Applications of the Poisson Process. Applied Mathematics, 05(19), 3011–3017. https://doi.org/10.4236/am.2014.519288

Utama, M. W., Mangku, I. W., & Silalahi, B. P. (2022). Asymptotic Distribution of an Estimator for Variance Function of a Compound Periodic Poisson Process with Power Function Trend. Jurnal Teori Dan Aplikasi Matematika, 6(4), 1081–1095. https://doi.org/10.31764/jtam.v6i4.10213

Zhang, H., Liu, Y., & Li, B. (2014). Notes on Discrete Compound Poisson Model with Applications to Risk Theory. Insurance: Mathematics and Economics, 59, 325–336. https://doi.org/10.1016/j.insmatheco.2014.09.012

DOI: https://doi.org/10.31764/jtam.v7i3.14836