Sains Malaysiana 51(11)(2022): 3819-3827
http://doi.org/10.17576/jsm-2022-5111-25
Confidence
Interval for Parameters Estimates in Circular Simultaneous Functional
Relationship Model (CSFRM) for Equal Variances using Normal Asymptotic and
Bootstrap Confidence Intervals
(Selang Keyakinan Anggaran Parameter untuk Model Hubungan Fungsian Membulat Serentak (CSFRM) dengan Andaian Ralat Varians sama menggunakan Pendekatan Asimptot dan Pembutstrapan)
FATIN NAJIHAH BADARISAM1,*,
MOHD SYAZWAN MOHAMAD ANUAR2, ABDUL GHAPOR HUSSIN1, ADZHAR
RAMBLI3 & NURUL RAUDHAH ZULKIFLI3
1Faculty of Defence Science and Technology, National Defence
University of Malaysia, Kem Sungai Besi, 57000 Kuala Lumpur, Federal Territory, Malaysia
2Centre for Defence Foundation Studies, National Defence
University of Malaysia, Kem Sungai Besi, 57000 Kuala Lumpur, Federal Territory, Malaysia
3Faculty of Computer and Mathematical Sciences, Universiti Teknologi MARA, 40450
Shah Alam, Selangor Darul Ehsan, Malaysia
Diserahkan: 24 Januari 2022/Diterima: 13 Julai 2022
Abstract
Few
studies have considered the functional relationship model for circular
variables. Anuar has proposed a new model of Circular
Simultaneous Functional Relationship Model for equal variances. However, the
confidence interval for all parameter estimates in this model has not received
any consideration in any literature. This paper proposes the confidence
interval for all parameter estimates of von Mises distribution in this model.
The parameters are estimated using minimum sum (ms) and polyroot function provided in (built-in package) Splus statistical software. The parameters confidence may
be obtained from parameter estimation. Those estimation values are obtained by
minimizing the negative value of the log-likelihood function. Then, the
confidence interval for all parameters based on the bootstrap method will be
compared with the normal asymptotic confidence interval via simulation studies.
It is found that bootstrap method is the superior method by measuring the
performance using coverage probability and expected length. The confidence
intervals are illustrated using real wind direction data of Bayan Lepas that collected at 16.3 m above ground level, latitude
05°18’N and longitude 100°16’E. The results showed that the estimate parameters
fall between the estimate interval, and we note that the method works well for
this model.
Keywords: Bootstrap confidence
interval; circular simultaneous functional relationship model; normal
asymptotic confidence interval; parameters estimate; von Mises distribution
Abstrak
Beberapa kajian telah mempertimbangkan model hubungan fungsian untuk pemboleh ubah membulat. Anuar telah mencadangkan model baru iaitu Model Hubungan Fungsian Membulat Serentak dengan Andaian Ralat Varians Sama. Walau bagaimanapun, selang keyakinan semua anggaran parameter untuk model ini tidak mendapat pertimbangan di mana-mana kepustakaan. Kajian ini mencadangkan selang keyakinan untuk semua anggaran parameter taburan von Mises dalam model ini. Parameter dianggarkan menggunakan fungsi minimum sum (ms) dan fungsi polyroot yang dibekalkan (built-in) dalam perisian statistik Splus. Keyakinan parameter boleh didapati daripada anggaran parameter. Nilai anggaran tersebut boleh diperoleh dengan meminimumkan nilai negatif fungsi kemungkinan log. Kemudian, selang keyakinan terhadap semua anggaran parameter berdasarkan kaedah pembustrapan dibandingkan dengan kaedah normal asimptot melalui kajian simulasi. Didapati kaedah pembustrapan adalah kaedah unggul dengan mengukur prestasi menggunakan kebarangkalian liputan dan jangkaan panjang. Kaedah ini diilustrasikan menggunakan data arah angin Bayan Lepas yang dikumpul pada 16.3 m di atas paras tanah, latitud 05°18’N dan longitud 100°16’E. Hasil kajian menunjukkan bahawa semua anggaran parameter jatuh antara selang anggaran dan kaedah tersebut berfungsi dengan baik untuk model ini.
Kata kunci: Anggaran parameter; model hubungan fungsian membulat serentak; selang keyakinan pembutstrapan; selang keyakinan normal asimptot; taburan von Mises
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*Pengarang untuk surat-menyurat; email: m.syazwan@upnm.edu.my