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 BADARISAM^1,*, 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

Received: 24 January 2022/Accepted: 13 July 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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^*Corresponding author; email: m.syazwan@upnm.edu.my