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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 ZUBAIRI2* & 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
Received: 28 September
2016/ Accepted: 26 January 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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*Corresponding author; email: yzulina@um.edu.my |
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