Sains Malaysiana 53(2)(2024): 461-476

http://doi.org/10.17576/jsm-2024-5302-18

Examining Tail Index Estimators in New Pareto Distribution: Monte Carlo Simulations and Income Data Applications

(Menyemak Penganggar Indeks Ekor dalam Taburan Pareto Baharu: Simulasi Monte Carlo dan Aplikasi Data Pendapatan)

MUHAMMAD ASLAM MOHD SAFARI^1,2,*, NURULKAMAL MASSERAN³ & MOHD AZMI HARON⁴

¹Department of Mathematics and Statistics, Faculty of Science, Universiti Putra Malaysia, 43400 UPM Serdang, Selangor, Malaysia

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

³Department of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, 43600 UKM Bangi, Selangor, Malaysia

⁴Institute of Mathematical Sciences, Faculty of Science, Universiti Malaya, 50603 Kuala Lumpur, Malaysia

Received: 22 July 2023/Accepted: 16 January 2024

Abstract

An evolved form of Pareto distribution, the new Pareto-type distribution, offers an alternative model for data with heavy-tailed characteristics. This investigation examines and discusses fourteen diverse estimators for the tail index of the new Pareto-type, including estimators such as maximum likelihood, method of moments, maximum product of spacing, its modified version, ordinary least squares, weighted least squares, percentile, Kolmogorov-Smirnov, Anderson-Darling, its modified version, Cramér-von Mises, and Zhang's variants of the previous three. Using Monte Carlo simulations, the effectiveness of these estimators is compared both with and without the presence of outliers. The findings show that, without outliers, the maximum product of spacing, its modified version, and maximum likelihood are the most effective estimators. In contrast, with outliers present, the top performers are Cramér-von Mises, ordinary least squares, and weighted least squares. The study further introduces a graphical method called the new Pareto-type quantile plot for validating the new Pareto-type assumptions and outlines a stepwise process to identify the optimal threshold for this distribution. Concluding the study, the new Pareto-type distribution is employed to model the high-end household income data from Italy and Malaysia, leveraging all the methodologies proposed.

Keywords: Estimation techniques; heavy-tailed data; income data modelling; Monte Carlo analysis; Pareto distribution; robustness

Abstrak

Satu taburan Pareto yang berkembang iaitu taburan jenis Pareto baharu, menawarkan model alternatif untuk data dengan ciri ekor berat. Kajian ini meneliti dan membincangkan empat belas penganggar yang pelbagai bagi indeks ekor jenis Pareto baharu, termasuk penganggar seperti kebolehjadian maksimum, kaedah momen, produk jarak maksimum bersama versi yang diubah suai, kuasa dua terkecil biasa, kuasa dua terkecil berwajaran, persentil, Kolmogorov-Smirnov, Anderson-Darling Bersama versi yang diubah suai, Cramér-von Mises, dan varian Zhang bagi Kolmogorov-Smirnov, Anderson-Darling serta Cramér-von Mises. Dengan menggunakan simulasi Monte Carlo, keberkesanan penganggar ini dibandingkan dengan kehadiran dan tanpa kehadiran titik terpencil. Hasil kajian menunjukkan bahawa, tanpa titik terpencil, produk jarak maksimum bersama versi yang diubah suai dan kebolehjadian maksimum adalah penganggar yang paling berkesan. Sebaliknya, dengan kehadiran titik terpencil, penganggar terbaik adalah Cramér-von Mises, kuasa dua terkecil biasa dan kuasa dua terkecil berwajaran. Kajian ini seterusnya memperkenalkan kaedah grafik yang disebut sebagai plot kuantil jenis Pareto baharu untuk mengesahkan andaian jenis Pareto baharu dan menggariskan proses bertahap untuk mengenal pasti ambang optimum untuk taburan ini. Mengakhiri kajian, taburan jenis Pareto baharu digunakan untuk memodelkan data pendapatan isi rumah kelas atas dari Itali dan Malaysia, memanfaatkan semua kaedah yang dicadangkan.

Kata kunci: Analisis Monte Carlo; data ekor berat; kaedah penganggaran; keteguhan; pemodelan data pendapatan; taburan Pareto

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*Corresponding author; email: aslam.safari@upm.edu.my