Sains Malaysiana 55(3)(2026): 503-515

http://doi.org/10.17576/jsm-2026-5503-12
 

Robust Estimation of Inverse Pareto Distribution: Insights into Malaysian Lower-Income Groups

(Anggaran Teguh Taburan Pareto Songsang: Pandangan tentang Kumpulan Berpendapatan Rendah di Malaysia)

 

MUHAMMAD FAHEM MUSA¹, MOHD AZMI HARON^1,*, MUHAMMAD ASLAM MOHD SAFARI^2,3 & ZAILAN SIRI¹ 

 

¹Institute of Mathematical Sciences, Faculty of Science, Universiti Malaya, 50603 Kuala Lumpur, Malaysia

 ²Institute for Mathematical Research, Universiti Putra Malaysia, 43400 UPM Serdang, Selangor, Malaysia

³Department of Mathematics and Statistics, Faculty of Science, Universiti Putra Malaysia, 43400 UPM Serdang, Selangor, Malaysia

 

Received: 22 January 2025/Accepted: 19 February 2026

 

Abstract

The lower tail data of the income distribution can be well described by the inverse Pareto distribution, which is a useful statistical model. Researchers often use the maximum likelihood approach in evaluating the model’s shape parameter. However, this approach might be unreliable as extreme values data points can easily affect it negatively, which may cause inaccurate estimates. This paper introduces a robust estimator known as repeated median estimator for the inverse Pareto distribution. We also demonstrate its efficiency by measuring its asymptotic relative efficiency. The robustness of the method is assessed by the breakdown point and the influence function. The Monte Carlo simulation studies are then conducted to measure its performance as compared to its counterparts. It is found that our proposed estimator outperforms the maximum likelihood, method of moments, and method of product spacings based on the simulation studies. We then apply the inverse Pareto distribution, leveraging the repeated median estimator, to model the lower-income data from the Household Income Surveys in Malaysia for the years 2012, 2014, 2016, 2019, and 2022. To compute the income disparity of low-income households in Malaysia, the parametric Lorenz curve is fitted using the inverse Pareto model, and the Gini coefficient is estimated. This confirms that the proposed approach achieves meaningful practical performance improvements over conventional estimation techniques in the presence of outliers.

Keywords: Inverse Pareto distribution; Malaysian lower income; Monte Carlo simulation; repeated median estimator; robust estimation

 

Abstrak

Taburan Pareto songsang ialah model statistik yang berkesan dalam menerangkan data ekor bawah pengagihan pendapatan. Penyelidik kebiasaannya bergantung kepada penganggar kebolehjadian maksimum dalam menganggarkan parameter bentuk model ini. Walau bagaimanapun, kaedah ini mungkin tidak berkesan kerana ia boleh dipengaruhi dengan mudah oleh nilai melampau yang boleh menyebabkan anggaran yang tidak tepat. Dalam kajian ini, kami memperkenalkan penganggar teguh yang dikenali sebagai penganggar median berulang. Kami juga menunjukkan kecekapannya dengan mengukur kecekapan relatif asimtotik. Keteguhan penganggar ini dinilai oleh nilai pecahan dan fungsi pengaruh. Kajian simulasi Monte Carlo kemudiannya dijalankan untuk mengukur prestasinya berbanding dengan penganggar lain. Didapati bahawa penganggar yang kami cadangkan mengatasi kebolehjadian maksimum, kaedah momen dan kaedah jarak produk berdasarkan kajian simulasi. Kajian ini kemudiannya menggunakan taburan Pareto songsang dengan memanfaatkan penganggar median berulang untuk memodelkan data pendapatan rendah daripada Tinjauan Pendapatan Isi Rumah di Malaysia bagi tahun 2012, 2014, 2016, 2019 dan 2022. Untuk mengukur jurang pendapatan isi rumah berpendapatan rendah di Malaysia, keluk Lorenz dipadankan dan pekali Gini dianggarkan berdasarkan model Pareto songsang. Ini mengesahkan bahawa pendekatan yang dicadangkan mencapai peningkatan prestasi praktikal yang signifikan berbanding dengan teknik anggaran konvensional walaupun terdapat pencilan.

Kata kunci: Anggaran teguh; pendapatan rendah Malaysia; penganggar median berulang; simulasi Monte Carlo; taburan Pareto songsang

 

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*Corresponding author; email: azmiharon@um.edu.my