Sains Malaysiana 48(4)(2019): 887–892

http://dx.doi.org/10.17576/jsm-2019-4804-21

 

A Half-Circular Distribution on a Circle

(Taburan Separa-Bulat dalam Bulatan)

 

ADZHAR RAMBLI¹*, IBRAHIM MOHAMED², KUNIO SHIMIZU³ & NORLINA MOHD RAMLI⁴

 

¹Centre of Statistical & Decision Science Studies, Faculty of Computer and Mathematical Sciences, Universiti Teknologi MARA, 40450 Shah Alam, Selangor Darul Ehsan, Malaysia

 

²Institute of Mathematical Sciences, University of Malaya, 50603 Kuala Lumpur, Federal Territory, Malaysia

 

³School of Statistical Thinking, The Institute of Statistical Mathematics, Tokyo, Japan

 

⁴Department of Ophtalmology, University of Malaya, 50603 Kuala Lumpur, Federal Territory, Malaysia

 

Received: 7 April 2017/Accepted: 18 January 2019

 

ABSTRACT

Up to now, circular distributions are defined in [0,2 π), except for axial distributions on a semicircle. However, some circular data lie within just half of this range and thus may be better fitted by a half-circular distribution, which we propose and develop in this paper using the inverse stereographic projection technique on a gamma distributed variable. The basic properties of the distribution are derived while its parameters are estimated using the maximum likelihood estimation method. We show the practical value of the distribution by applying it to an eye data set obtained from a glaucoma clinic at the University of Malaya Medical Centre, Malaysia.

 

Keywords: Gamma distribution; inverse stereographic projection; maximum likelihood estimation; trigonometric moments; unimodality

 

ABSTRAK

Sehingga kini, taburan bulatan ditakrifkan dalam [0,2 π), kecuali untuk taburan paksi aksial pada semi-bulatan. Walau bagaimanapun, terdapat data bulatan berada hanya separuh daripada julat ini dan ia lebih sesuai dengan taburan separuh-bulatan, maka kami mencadang dan membangunkan dalam kertas ini menggunakan teknik unjuran stereografik songsang pada pemboleh ubah taburan gamma. Sifat asas taburan diperoleh manakala parameter dinilai menggunakan kaedah anggaran kebolehjadian maksimum. Nilai praktikal taburan ini dipraktiskan pada set data mata yang diperoleh daripada klinik glaukoma di Pusat Kesihatan, Universiti Malaya, Malaysia.

 

Kata kunci: Anggaran kebolehjadian maksimum; momen trigonometrik; taburan Gamma; unimodaliti; unjuran stereografik songsang

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*Corresponding author; email: adzfranc@gmail.com