Sains Malaysiana 52(6)(2023): 1879-1888

http://doi.org/10.17576/jsm-2023-5206-22

 

Bootstrap Methods for Estimating the Confidence Interval for the Parameter of the Zero-Truncated Poisson-Sujatha Distribution and Their Applications

(Kaedah Bootstrap untuk Menganggar Selang Keyakinan untuk Parameter Taburan Poisson-Sujatha Terpangkas Sifar dan Aplikasinya)

 

WARARIT PANICHKITKOSOLKUL*

 

Department of Mathematics and Statistics, Faculty of Science and Technology, Thammasat University, 12121 Pathumthani, Thailand

 

Received: 29 December 2022/Accepted: 12 June 2023

 

Abstract

Numerous phenomena involve count data containing non-zero values and the zero-truncated Poisson-Sujatha distribution can be used to model such data. However, the confidence interval estimation of its parameter has not yet been examined. In this study, confidence interval estimation based on percentile, simple, biased-corrected and accelerated bootstrap methods, as well as the bootstrap-t interval, was examined in terms of coverage probability and average interval length via Monte Carlo simulation. The results indicate that attaining the nominal confidence level using the bootstrap methods was not possible for small sample sizes regardless of the other settings. Moreover, when the sample size was large, the performances of the methods were not substantially different. Overall, the bias-corrected and accelerated bootstrap approach outperformed the others, even for small sample sizes. Last, the bootstrap methods were used to calculate the confidence interval for the zero-truncated Poisson-Sujatha parameter via three numerical examples, the results of which match those from the simulation study.

 

Keywords: Bootstrap interval; count data; interval estimation; Poisson-Sujatha distribution; simulation

 

Abstrak

Banyak fenomena melibatkan data bilangan yang mengandungi nilai bukan sifar dan taburan Poisson-Sujatha terpangkas sifar boleh digunakan untuk memodelkan data tersebut. Walau bagaimanapun, anggaran selang keyakinan parameternya masih belum diperiksa. Dalam kajian ini, anggaran selang keyakinan berdasarkan kaedah persentil, mudah, pembetulan berat sebelah dan dipercepatkan, serta selang bootstrap-t, telah diperiksa dari segi kebarangkalian liputan dan panjang selang purata melalui simulasi Monte Carlo. Keputusan menunjukkan bahawa mencapai tahap keyakinan nominal menggunakan kaedah bootstrap tidak mungkin untuk saiz sampel yang kecil tanpa mengira tetapan lain. Selain itu, apabila saiz sampel adalah besar, prestasi kaedah tidak jauh berbeza. Secara keseluruhannya, pendekatan bootstrap yang diperbetulkan berat sebelah dan dipercepatkan mengatasi prestasi yang lain, walaupun untuk saiz sampel yang kecil. Terakhir, kaedah bootstrap digunakan untuk mengira selang keyakinan bagi parameter Poisson-Sujatha terpangkas sifar melalui tiga contoh berangka, yang hasilnya sepadan dengan kajian simulasi.

 

Kata kunci: Anggaran selang; data bilangan; selang Bootstrap; simulasi; taburan Poisson-Sujatha

 

REFERENCES

Andrew, F.S. & Michael, R.W. 2022. Practical Business Statistics. 8th ed. San Diego: Academic Press.

Chernick, M.R. & Labudde, R.A. 2011. An Introduction to Bootstrap Methods with Applications to R. 1st ed. New Jersey: John Wiley and Sons.

David, F. & Johnson, N. 1952. The truncated Poisson. Biometrics 8(4): 275-285.

DiCiccio, T.J. & Efron, B. 1996. Bootstrap confidence intervals. Statistical Science.11(3): 189-212.

Efron, B. 1987. Better bootstrap confidence intervals. Journal of the American Statistical Association82(297): 171-185.

Efron, B. 1982. The jackknife, the bootstrap, and other resampling plans. CBMS-NSF Regional Conference Series in Applied Mathematics. Philadelphia: SIAM.

Efron, B. & Tibshirani, R.J. 1993. An Introduction to the Bootstrap. 1st ed. New York: Chapman and Hall.

Flowers-Cano, R.S., Ortiz-Gómez, R., León-Jiménez, J.E., Rivera, R.L. & Cruz, L.A.P. 2018. Comparison of bootstrap confidence intervals using Monte Carlo simulations. Water 10(2): 166. https://doi.org/10.3390/w10020166

Ghitany, M.E., Al-Mutairi, D.K. & Nadarajah, S. 2008. Zero-truncated Poisson-Lindley distribution and its application. Mathematics and Computers in Simulation 79(3): 279-287.

Henningsen, A. & Toomet, O. 2011. maxLik: A package for maximum likelihood estimation in R. Computational Statistics 26(3): 443-458.

Hussain, T. 2020. A zero truncated discrete distribution: Theory and applications to count data. Pakistan Journal of Statistics and Operation Research 16(1): 167-190.

Ihaka, R. & Gentleman, R. 1996. R: A language for data analysis and graphics. Journal of Computational and Graphical Statistics 5(3): 299-314.

Jung, K., Lee, J., Gupta, V. & Cho, G. 2019. Comparison of bootstrap confidence interval methods for GSCA using a Monte Carlo simulation. Frontiers in Psychology 10: 2215. https://doi.org/10.3389/fpsyg.2019.02215

Kissell, R. & Poserina, J. 2017. Optimal Sports Math, Statistics, and Fantasy. 1st ed. London: Academic Press.

Lindley, D.V. 1958. Fiducial distributions and Bayestheorem. Journal of the Royal Statistical Society: Series B 20(1): 102-107.

Manoharan, T., Arasan, J., Midi, H. & Adam, M.B. 2017. Bootstrap intervals in the presence of left-truncation, censoring and covariates with a parametric distribution. Sains Malaysiana 46(12): 2529-2539.

Meeker, W.Q., Hahn, G.J. & Escobar, L.A. 2017. Statistical Intervals: A Guide for Practitioners and Researchers. 2nd ed. New Jersey: John Wiley and Sons.

Reiser, M., Yao, L., Wang, X., Wilcox, J. & Gray, S. 2017. A comparison of bootstrap confidence intervals for multi-level longitudinal data using Monte-Carlo simulation. In Monte-Carlo Simulation-Based Statistical Modeling. ICSA Book Series in Statistics, edited by Chen, D.G. & Chen, J. Singapore: Springer.

Sangnawakij, P. 2021. Confidence interval for the parameter of the zero-truncated Poisson distribution. The Journal of Applied Science 20(2): 13-22.

Shanker, R. 2017a. A zero-truncated Poisson-Amarendra distribution and its application. International Journal of Probability and Statistics 6(4): 82-92.

Shanker, R. 2017b. Zero-truncated Poisson-Akash distribution and its applications. American Journal of Mathematics and Statistics 7(6): 227-236.

Shanker, R. 2016a. The discrete Poisson-Sujatha distribution. International Journal of Probability and Statistics 5(1): 1-9.

Shanker, R. 2016b. Sujatha distribution and its applications. Statistics in Transition-new Series 17(3): 391-410.

Shanker, R., Hagos, F., Selvaraj, S. & Yemane, A. 2015. On zero-truncation of Poisson and Poisson-Lindley distributions and their applications. Biometrics and Biostatistics International Journal 2(6): 168-181.

Shukla, K.K., Shanker, R. & Tiwari, M.K. 2020. Zero-truncated Poisson-Ishita distribution and its application.  Journal of Scientific Research 64(2): 287-294.

Siegel, A.F. 2016. Practical Business Statistics. 7th ed. London: Academic Press.

Singh, S.N. & Yadava, K.N. 1971. Trends in rural out-migration at household level. Rural Demography 8(1): 53-61.

Soetaert, K. 2021. Rootsolve: Nonlinear root finding, equilibrium and steady-state analysis of ordinary differential equations. R Package 1.8.2.3.

Turhan, N.S. 2020. Karl Pearsons chi-square tests. Educational Research Review 15(9): 575-580.

Wood, M. 2004. Statistical inference using bootstrap confidence intervals. Significance 1(4): 180-182.

 

*Corresponding author; email: wararit@mathstat.sci.tu.ac.th

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

previous