Sains Malaysiana 52(1)(2023): 281-294

http://doi.org/10.17576/jsm-2023-5201-23

A New Exponentiated Beta Burr Type X Distribution: Model, Theory, and Applications

(Taburan Beta Burr Jenis X Baru yang Dipertingkatkan: Model, Teori dan Aplikasi)

YIT LENG OH^1,2, FONG PENG LIM^1,*, CHUEI YEE CHEN¹, WENDY SHINYIE LING¹ & YUE FANG LOH³

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

²Faculty of Business, Multimedia University, 75450 Melaka, Malaysia

³Faculty of Business and Management, UCSI University, 56000 Kuala Lumpur, Federal Territory, Malaysia

Received: 20 May 2022/Accepted: 10 October 2022

Abstract

In recent years, many attempts have been carried out to develop the Burr type X distribution, which is widely used in fitting lifetime data. These extended Burr type X distributions can model the hazard function in decreasing, increasing and bathtub shapes, except for unimodal. Hence, this paper aims to introduce a new continuous distribution, namely exponentiated beta Burr type X distribution, which provides greater flexibility in order to overcome the deficiency of the existing extended Burr type X distributions. We first present its density and cumulative function expressions. It is then followed by the mathematical properties of this new distribution, which include its limit behaviour, quantile function, moment, moment generating function, and order statistics. We use maximum likelihood approach to estimate the parameters and their performance is assessed via a simulation study with varying parameter values and sample sizes. Lastly, we use two real data sets to illustrate the performance and flexibility of the proposed distribution. The results show that the proposed distribution gives better fits in modelling lifetime data compared to its sub-models and some extended Burr type X distributions. Besides, it is very competitive and can be used as an alternative model to some nonnested models. In summary, the proposed distribution is very flexible and able to model various shaped hazard functions, including the increasing, decreasing, bathtub, and unimodal.

Keywords: Beta generalized; Burr type X; exponentiated; survival analysis; unimodal

Abstrak

Dalam beberapa tahun kebelakangan ini, banyak percubaan telah dijalankan untuk membangunkan taburan Burr jenis X yang digunakan secara meluas dalam model sepanjang hayat yang sesuai. Taburan lanjutan Burr jenis X ini boleh memodelkan fungsi hazard dalam bentuk menurun, meningkat dan bathtub, kecuali bagi unimod. Kertas ini bertujuan untuk memperkenalkan taburan berterusan baharu, iaitu taburan Burr jenis X beta eksponen, yang lebih keluwesan, bagi mengatasi kekurangan taburan lanjutan Burr jenis X sedia ada. Kami bermula dengan membentangkan ketumpatan dan ungkapan fungsi terkumpulnya. Ia kemudiannya diikuti dengan sifat matematik taburan baharu ini, yang merangkumi kelakuan hadnya, fungsi kuantil, momen, fungsi penjanaan momen dan statistik pesanan. Kami menggunakan pendekatan kemungkinan maksimum untuk menganggarkan parameter dan prestasinya dinilai melalui kajian simulasi dengan nilai parameter dan saiz sampel yang berbeza-beza. Akhir sekali, kami menggunakan dua set data sebenar untuk menggambarkan prestasi dan berkefleksibelan taburan yang dicadangkan. Keputusan menunjukkan bahawa taburan yang dicadangkan memberikan kesesuaian yang lebih baik dalam pemodelan data sepanjang hayat berbanding dengan sub-modelnya dan beberapa taburan lanjutan Burr jenis X. Selain itu, ia sangat bersaing dan boleh digunakan sebagai model alternatif kepada beberapa model tidak bersarang. Secara ringkasnya, taburan yang dicadangkan adalah sangat fleksibel dan boleh memodelkan pelbagai bentuk fungsi hazard, termasuk peningkatan, penurunan, bathtub dan unimod.

Kata kunci: Analisis kemandirian; beta teritlak; Burr jenis X; eksponen; unimod

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^*Corresponding author; email: fongpeng@upm.edu.my