Sains Malaysiana 50(7)(2021): 2047-2058

http://doi.org/10.17576/jsm-2021-5007-19

Composite Pareto Distributions for Modelling Household Income Distribution in Malaysia

(Taburan Komposit Pareto untuk Pemodelan Taburan Pendapatan Isi Rumah di Malaysia)

MUHAMMAD HILMI ABDUL MAJID* & KAMARULZAMAN IBRAHIM

Department of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, 43600 UKM Bangi, Selangor Darul Ehsan, Malaysia

Diserahkan: 19 Jun 2020/Diterima: 19 November 2020

ABSTRACT

Composite Pareto distributions are flexible as the models allow for data to be described by two distributions: a Pareto distribution for the data above a threshold value and another separate distribution for data below the threshold value. It is noted in some previous literatures that the Paretian tail behaviour can be observed in the distribution of Malaysian household income. In this paper, the composite Pareto models are fitted to the Malaysian household income data of several years. These fitted composite Pareto models are then compared to several univariate models for describing income distribution using pseudo-likelihood based AIC, BIC and Kolmogorov-Smirnov goodness-of-fit test. It is found that the income distributions in Malaysia can be best described by the lognormal-Pareto (II) model as compared to other candidate models.

Keywords: Composite model; goodness-of-fit; income distribution; Pareto distribution; pseudo-likelihood

ABSTRAK

Taburan komposit Pareto adalah luwes kerana model ini boleh menerangkan sesuatu data menggunakan dua taburan: taburan Pareto untuk data di atas suatu nilai ambang dan taburan yang berasingan untuk data di bawah nilai ambang tersebut. Kajian sebelum ini telah menyatakan bahawa ciri-ciri ekor Pareto dapat diperhatikan pada taburan pendapatan isi rumah di Malaysia. Dalam kajian ini, model komposit Pareto disuaikan ke atas data pendapatan isi rumah di Malaysia. Model komposit Pareto ini akan dibandingkan dengan model univariat lain untuk menerangkan taburan pendapatan dengan menggunakan AIC, BIC dan ujian kebagusan penyuaian Kolmogorov-Smirnov berasaskan pseudo-kebolehjadian. Kajian mendapati taburan pendapatan di Malaysia boleh diterangkan menggunakan model lognormal-Pareto (II) lebih baik berbanding calon model lain.

Kata kunci: Kebagusan penyuaian; model komposit; pseudo-kebolehjadian; taburan Pareto; taburan pendapatan

RUJUKAN

Aminzadeh, M.S. & Deng, M. 2019. Bayesian predictive modeling for Inverse Gamma-Pareto composite distribution. Communications in Statistics-Theory and Methods 48(8): 1938-1954.

Arnold, B.C. 2008. Pareto and generalized Pareto distributions. In Modeling Income Distributions and Lorenz Curves, edited by Chotikapanich, D. New York: Springer Science & Business Media. pp. 119-145.

Bakar, S.A.A., Nadarajah, S., Adzhar, Z.A.A.K. & Mohamed, I. 2016. Gendist: An R package for generated probability distribution models. PLoS ONE 11(6): e0156537.

Bakar, S.A.A., Hamzah, N.A., Maghsoudi, M. & Nadarajah, S. 2015. Modeling loss data using composite models. Insurance: Mathematics and Economics 61: 146-154.

Bee, M. 2015. Estimation of the lognormal-Pareto distribution using probability weighted moments and maximum likelihood. Communications in Statistics-Simulation and Computation 44(8): 2040-2060.

Ciumara, R. 2006. An actuarial model based on the composite Weibull-Pareto distribution. Mathematical Reports 8(4): 401-414.

Congressional Budget Office. 2019. Projected changes in the distribution of household income, 2016 to 2021.

Cooray, K. & Ananda, M. 2005. Modeling actuarial data with a composite lognormal-Pareto model. Scandinavian Actuarial Journal 2005(5): 321-334.

Gunawan, D., Panagiotelis, A., Griffiths, W. & Chotikapanich, D. 2020. Bayesian weighted inference from surveys. Australian & New Zealand Journal of Statistics 62(1): 71-94.

Hu, Y. & Scarrott, C. 2018. evmix: An R package for extreme value mixture modeling, threshold estimation and boundary corrected kernel density estimation. Journal of Statistical Software 84(5): 1-18.

Janczura, J. & Weron, R. 2010. Goodness-of-fit testing for regime-switching models. MPRA Paper. https://mpra.ub.uni-muenchen.de/id/eprint/22871.

Masseran, N., Yee, L.H., Safari, M.A.M. & Ibrahim, K. 2019. Power law behavior and tail modeling on low income distribution. Mathematics and Statistics 7(3): 70-77.

Nadarajah, S. & Bakar, S.A.A. 2013. CompLognormal: An R package for composite lognormal distributions. The R Journal 5(2): 97-103.

OECD. 2015. In it Together: Why Less Inequality Benefits All. Paris: OECD Publishing. pp. 19-58.

Pfeffermann, D. 1993. The role of sampling weights when modeling survey data. International Statistical Review/Revue Internationale de Statistique 61(2): 317-337.

Ragayah, H.M.Z. 2008. Income inequality in Malaysia. Asian Economic Policy Review 3(1): 114-132.

Razak, F.A. & Shahabuddin, F.A. 2018. Malaysian household income distribution: A fractal point of view. Sains Malaysiana 47(9): 2187-2194.

Reynkens, T., Verbelen, R., Bardoutsos, A., Cornilly, D., Geogebeur, Y. & Herrmann, K. 2020. ReIns: Functions from “reinsurance: actuarial and statistical aspects.” https://CRAN.R-project.org/package=ReIns.

Sablica, L. & Hornik, K. 2020. mistr: A Computational framework for mixture and composite distributions. The R Journal 12(1): 283-299.

Safari, M.A.M., Masseran, N. & Ibrahim, K. 2018a. Optimal threshold for Pareto tail modelling in the presence of outliers. Physica A: Statistical Mechanics and its Applications 509: 169-180.

Safari, M.A.M., Masseran, N. & Ibrahim, K. 2018b. A robust semi-parametric approach for measuring income inequality in Malaysia. Physica A: Statistical Mechanics and its Applications 512: 1-13.

Scollnik, D.P.M. 2007. On composite lognormal-Pareto models. Scandinavian Actuarial Journal 2007(1): 20-33.

Scollnik, D.P.M. & Sun, C. 2012. Modeling with Weibull-Pareto models. North American Actuarial Journal 16(2): 260-272.

Skinner, C.J., Holt, D. & Smith, T.M.F. 1989. Analysis of Complex Surveys. Chichester, New York: Wiley.

Teodorescu, S. & Vernic, R. 2013. On composite Pareto models. Mathematical Reports 15(65): 11-29.

Teodorescu, S. & Vernic, R. 2009. Some composite exponential-Pareto models for actuarial prediction. Journal for Economic Forecasting 12(4): 82-100.

*Pengarang untuk surat-menyurat; email: hilmi.majid@ukm.edu.my