Sains Malaysiana 52(1)(2023): 295-304

http://doi.org/10.17576/jsm-2023-5201-24

 

Approximation of the Sum of Independent Lognormal Variates using Lognormal Distribution by Maximum Likelihood Estimation Approached

(Penghampiran terhadap Jumlah Variat Tak Bersandar menggunakan Taburan Lognormal Berdasarkan Pendekatan Penganggaran Kebolehjadian Maksimum)

 

ABDUL RAHMAN OTHMAN1,  LAI CHOO HENG2,  SONIA AÏSSA3 & NORA MUDA4,*

 

1School of Distance Education, Universiti Sains Malaysia, 11800 Pulau Pinang, Malaysia

2Kolej Vokasional Nibong Tebal, Jalan Bukit Panchor, 14300 Nibong Tebal, Pulau Pinang, Malaysia 

3Institut National de la Recherche Scientifique, Énergie Matériaux Télécommunications Research Centre, 800, De La Gauchetière Ouest, Bureau 6900, Montréal, Québec H5A 1K6, Canada

4Department of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, 43600 UKM Bangi, Selangor Darul Ehsan, Malaysia

 

Received: 11 March 2022/Accepted: 10 October 2022

 

Abstract

Three methods of approximating the sum of lognormal variates to a lognormal distribution were studied. They were the Wilkinson approximation, the Monte Carlo version of the Wilkinson approximation and the approximation using estimated maximum likelihood lognormal parameters. The lognormal variates were generated empirically using Monte Carlo simulation based on several conditions such as number of lognormal variates in the sum, number of sample points in the variates, the variates are independent and identically distributed (IID) and also not identically distributed (NIID) with lognormal parameters. Evaluation of all three lognormal approximation methods was performed using the Anderson Darling test. Results show that the approximation using estimated maximum likelihood lognormal parameters produced Type I errors close to the 0.05 target and is considered the best approximation.

 

Keywords: Anderson-Darling test;  lognormal approximation; maximum likelihood; sum of lognormal variates; Wilkinson

 

Abstrak

Tiga kaedah penghampiran bagi jumlah variat lognormal terhadap taburan lognormal telah dikaji. Tiga kaedah penghampiran tersebut adalah kaedah penghampiran Wilkinson, kaedah versi Monte Carlo bagi penghampiran Wilkinson dan kaedah penghampiran dengan penganggaran kebolehjadian maksimum bagi parameter lognormal. Pemboleh ubah lognormal dijana secara empirik melalui simulasi Monte Carlo dengan beberapa keadaan simulasi iaitu bilangan jumlah pemboleh ubah lognormal, bilangan sampel bagi pemboleh ubah lognormal, pemboleh ubah lognormal tak bersandar dan tertabur secara secaman mengikut taburan (IID) dan juga tidak secaman mengikut taburan (NIID) berdasarkan parameter lognormal. Penilaian bagi ketiga-tiga kaedah penghampiran lognormal tersebut dijalankan menggunakan ujian Anderson Darling. Hasil menunjukkan penghampiran menggunakan penganggaran kebolehjadian maksimum terhadap parameter lognormal telah menghasilkan ralat Jenis 1 menghampiri nilai sasaran ralat 0.05 dan dikatakan sebagai penghampiran terbaik.

 

Kata kunci: Jumlah variat lognormal; kebolehjadian maksimum; penghampiran lognormal; ujian Anderson-Darling; Wilkinson

 

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*Corresponding author; email: noramuda@ukm.edu.my

   

   

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