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 SAFARI1,2,*, 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
Diserahkan: 22 Julai2023/Diterima: 16 Januari2024
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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*Pengarang untuk surat-menyurat; email:
aslam.safari@upm.edu.my
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