Sains Malaysiana 48(5)(2019): 1151–1156

http://dx.doi.org/10.17576/jsm-2019-4805-24

 

A Comparison of Asymptotic and Bootstrapping Approach in Constructing Confidence Interval of the Concentration Parameter in von Mises Distribution

(Perbandingan Pendekatan Asimptot dan Pembutstrapan dalam Membina Selang Keyakinan Parameter Menumpu bagi Taburan von Mises)

 

NOR HAFIZAH MOSLIM^1,2, YONG ZULINA ZUBAIRI³*, ABDUL GHAPOR HUSSIN⁴, SITI FATIMAH HASSAN³ & NURKHAIRANY AMYRA MOKHTAR⁴

 

¹Institute of Graduate Studies, Universiti Malaya, 50603 Kuala Lumpur, Federal Territory, Malaysia

 

²Faculty of Industrial Sciences & Technology, Universiti Malaysia Pahang, Lebuhraya Tun Razak, 26300 Gambang, Pahang Darul Makmur, Malaysia

 

³Centre of Foundation Studies for Sciences, University of Malaya, 50603 Kuala Lumpur, Federal Territory, Malaysia

 

⁴Faculty of Defense Sciences and Technology, National Defense University of Malaysia, Kem Sungai Besi, 57000 Kuala Lumpur, Federal Territory, Malaysia

 

Diserahkan: 23 September 2018/Diterima: 6 Mac 2019

 

ABSTRACT

Bootstrap is a resampling procedure for estimating the distributions of statistics based on independent observations. Basically, bootstrapping has been established for the use of parameter estimation of linear data. Thus, the used of bootstrap in confidence interval of the concentration parameter, κ in von Mises distribution which fitted the circular data is discussed in this paper. The von Mises distribution is the ’natural’ analogue on the circle of the Normal distribution on the real line and widely used to describe circular variables. The distribution has two parameters, namely mean direction, μ and concentration parameter, κ, respectively. The confidence interval based on the calibration bootstrap method will be compared with the existing method, confidence interval based on the asymptotic to the distribution of κ. Simulation studies were conducted to examine the empirical performance of the confidence intervals. Numerical results suggest the superiority of the proposed method based on measures of coverage probability and expected length. The confidence intervals were illustrated using daily wind direction data recorded at maximum wind speed for seven stations in Malaysia. From point estimates of the concentration parameter and the respective confidence interval, we note that the method works well for a wide range of κ values. This study suggests that the method of obtaining the confidence intervals can be applied with ease and provides good estimates.

 

Keywords: Calibration bootstrap; circular variable; concentration parameter; von Mises distribution

 

ABSTRAK

Kaedah pembutstrapan adalah proses persampelan semula data bagi menganggarkan taburan statistik berdasarkan pemerhatian bebas. Kebelakangan ini, kaedah pembutstrapan telah digunakan secara meluas untuk menganggar parameter data linear. Oleh itu, dalam kajian ini, kami menggunakan kaedah pembutstrapan dalam membina selang keyakinan terhadap parameter menumpu, κ bagi taburan von Mises. Taburan von Mises dikenali sebagai taburan normal membulat dan ia merupakan taburan yang menyerupai taburan normal seperti yang biasa digunakan dalam statistik linear. Taburan ini mempunyai dua parameter, iaitu min berarah, μ dan parameter menumpu, κ. Selang keyakinan berdasarkan kaedah pembutstrapan penentukuran akan dibandingkan dengan kaedah sedia ada, selang keyakinan berdasarkan asimptotik κ. Kajian simulasi dan penilaian bagi saiz selang dan kebarangkalian menumpu telah dijalankan bagi menilai ketepatan empirik selang keyakinan tersebut. Kaedah ini diilustrasikan menggunakan data arah angin harian yang dirakamkan pada kelajuan angin maksimum bagi tujuh stesen di Malaysia. Titik penganggaran bagi parameter menumpu dan selang keyakinan, masing-masing menunjukkan kaedah pembutstrapan penentukuran ini berfungsi dengan baik untuk pelbagai nilai κ. Kajian ini menunjukkan bahawa kaedah mendapatkan selang keyakinan boleh digunakan dengan mudah dan memberikan anggaran yang baik.

 

Kata kunci: Parameter menumpu; pemboleh ubah membulat; pembutstrapan penentukuran; taburan von Mises

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*Pengarang untuk surat-menyurat; email: yzulina@um.edu.my