Sains Malaysiana 46(8)(2017): 1347–1353

http://dx.doi.org/10.17576/jsm-2017-4608-22

 

Estimation of Concentration Parameter for Simultaneous Circular Functional Relationship Model Assuming Unequal Error Variance

(Anggaran Parameter Kepekatan untuk Model Hubungan Fungsian Membulat Serentak dengan Andaian Ralat Varians tak Sama)

 

NURKHAIRANY AMYRA MOKHTAR¹, YONG ZULINA ZUBAIRI^2* & ABDUL GHAPOR HUSIN¹

 

¹Faculty of Defence Sciences and Technology, National Defence University of Malaysia, Kem Sungai Besi, 57000 Kuala Lumpur, Federal Territory, Malaysia

 

²Centre for Foundation Studies in Science, University of Malaya, 50603 Kuala Lumpur, Federal Territory, Malaysia

 

Diserahkan: 28 September 2016/ Diterima: 26 Januari 2017

ABSTRACT

In this study, we propose the estimation of the concentration parameter for simultaneous circular functional relationship model. In this case, the variances of the error term are not necessarily equal and the ratio of the concentration parameter λ = is not necessarily 1. The modified Bessel function was expended by using the asymptotic power series and it became a cubic equation of κ. From the cubic equation of κ, the roots were obtained by using the polyroot function in SPlus software. Simulation study was done to study the mean, estimated bias, absolute relative estimated bias, estimated standard error and estimated root mean square error of the estimation of the concentration parameter. From the simulation study, large concentration parameter and sample size show that the estimated concentration parameter has smaller bias. Also, an illustration to a real wind and wave data set is given to show its practical applicability.

 

Keywords: Circular variables; concentration parameter; simultaneous circular functional relationship model; unequal error variance

 

ABSTRAK

Dalam kajian ini, kami ingin mencadangkan anggaran parameter kepekatan untuk model hubungan fungsian membulat serentak. Dalam kes ini, perbezaan tempoh ralat tidak semestinya sama dan nisbah parameter λ = kepekatan tidak semestinya 1. Fungsi Bessel yang diubah suai telah dimajukan dengan menggunakan siri kuasa asimptot dan ia menjadi satu persamaan kubik. Daripada persamaan kubik, punca diperoleh dengan menggunakan fungsi polyroot dalam perisian statistic SPlus. Kajian simulasi telah dilakukan untuk mengkaji purata, anggaran berat sebelah, mutlak relatif berat sebelah dianggarkan, anggaran ralat piawai dan dianggarkan punca purata ralat kuasa dua daripada anggaran parameter kepekatan. Daripada kajian simulasi, parameter kepekatan dan sampel saiz yang besar menunjukkan anggaran parameter kepekatan mempunyai berat sebelah yang lebih kecil. Ilustrasi menggunakan data sebenar angin dan gelombang diberikan untuk menunjukkan kesesuaian praktikal.

 

Kata kunci: Model hubungan fungsian membulat serentak; parameter kepekatan; pemboleh ubah membulat; ralat varians tak sama

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