Sains Malaysiana 44(7)(2015): 1033–1039

A Comparison between Bayesian and Maximum Likelihood Estimations in Estimating

Finite Mixture Model for Financial Data

(Perbandingan antara Bayesian dan Anggaran Kebolehjadian Maksimum dalam Menganggar

Model Campuran Terhingga untuk Data Kewangan)

SEUK-YEN PHOONG* & MOHD TAHIR ISMAIL

School of Mathematical Sciences, Universiti Sains Malaysia, 11800 Penang, Malaysia

Diserahkan: 22 Mei 2013/Diterima: 5 Februari 2015

ABSTRACT

Over the years, maximum likelihood estimation and Bayesian method became popular statistical tools in which applied to fit finite mixture model. These trends begin with the advent of computer technology during the last decades. Moreover, the asymptotic properties for both statistical methods also act as one of the main reasons that boost the popularity of the methods. The difference between these two approaches is that the parameters for maximum likelihood estimation are fixed, but unknown meanwhile the parameters for Bayesian method act as random variables with known prior distributions. In the present paper, both the maximum likelihood estimation and Bayesian method are applied to investigate the relationship between exchange rate and the rubber price for Malaysia, Thailand, Philippines and Indonesia. In order to identify the most plausible method between Bayesian method and maximum likelihood estimation of time series data, Akaike Information Criterion and Bayesian Information Criterion are adopted in this paper. The result depicts that the Bayesian method performs better than maximum likelihood estimation on financial data.

Keywords: Akaike information criterion; Bayesian information criterion; Bayesian method; finite mixture model; maximum likelihood estimation

ABSTRAK

Sejak beberapa tahun, anggaran kebolehjadian maksimum dan kaedah Bayesian menjadi alat statistik popular yang sesuai digunakan untuk model campuran terhingga. Trend ini bermula dengan adanya teknologi komputer sejak sedekad yang lalu. Selain itu, sifat asimptot bagi kedua-dua kaedah statistik juga menjadi salah satu daripada faktor utama dalam meningkatkan populariti kaedah ini. Perbezaan antara kedua-dua kaedah ini adalah parameter untuk anggaran kebolehjadian maksimum adalah tetap tetapi tidak diketahui manakala parameter bagi kaedah Bayesian bertindak sebagai pemboleh ubah rawak dengan taburan yang dikenali sebelum ini. Dalam kertas ini, kedua-dua anggaran kebolehjadian maksimum dan kaedah Bayesian digunakan untuk mengkaji hubungan antara kadar pertukaran wang dan harga getah bagi Malaysia, Thailand, Filipina dan Indonesia. Untuk mengenal pasti kaedah yang paling munasabah antara kaedah Bayesian dan anggaran kebolehjadian maksimum untuk data siri masa, kriteria maklumat Akaike dan kriteria maklumat Bayesian diguna pakai dalam kertas ini. Kesimpulannya, keputusan menunjukkan bahawa kaedah Bayesian mempunyai prestasi yang lebih baik daripada anggaran kebolehjadian maksimum dalam menganalisis data kewangan.

Kata kunci: Anggaran kebolehjadian maksimum; kaedah Bayesian; kriteria maklumat Akaike; kriteria maklumat Bayesian; model campuran terhingga

RUJUKAN

Avdis, E. & Wachter, J.A. 2013. Maximum likelihood estimation of the equity premium. Working paper.

Burger, K., Smith, H. & Vogelvang, B. 2002. Exchange rates and natural rubber prices, the effect of the Asian crisis. Spain: 2002 International Congress Zaragoza, no. 24958.

Cipriani, M., Costantini, R. & Guarino, A. 2012. A Bayesian approach to experimental analysis: Trading in a laboratory financial market. Review of Economic Design 16(2): 175-191.

Coles, S.G., Sisson, S.A. & Pericchi, L.R. 2005. A case for a reassessment of the risks of extreme hydrological hazards in the Caribbean. SERRA 20: 296-306.

Duan, J.C. & Simonato, J.G. 2001. Maximum likelihood estimation of deposit insurance value with interest rate risk. Journal of Empirical Finance 9: 109-132.

Durham, G.B. & Gallant, A.R. 2002. Numerical techniques for maximum likelihood estimation of continuous-time diffusion processes. Journal of Business and Economic Statistics 20(3): 297-316.

Feng, X.X. & Xie, D.J. 2012. Bayesian estimation of CIR model. Journal of Data Science 10: 271-280.

Fisher, R.A. 1922. On the mathematical foundations of theoretical statistics. Philosophical Transactions of the Royal Society of London A 222: 309-368.

Hosmer, D.W. 1973. A comparison of iterative maximum likelihood estimates of the parameters of a mixture of two normal distributions under three different types of sample. Biometrics 29: 761-770.

Jacquier, E. & Polson, N.G. 2012. Asset allocation in finance: A Bayesian perspective. In Hierarchinal Models and MCMC: A tribute to Adrian Smith, edited by Dellaportas, D., Polson, N. & Stephen, G. Oxford: Oxford University Press.

Johnson, P.H. Jr., Qi, Y.X. & Chueh, Y.C. 2011. Bias-corrected maximum likelihood estimation in actuarial science. Proceedings of 46th Actuarial Research Conference.

Kladivko, K. 2007. Maximum likelihood estimation of the Cox-Ingersoll-Ross process: The Matlab implementation. Technical Computing Prague.

Laplace, P.S. 1986. Memoir on the probability of the causes of events. Statistical Science 1 3: 364-378.

Lepage, G. 2012. Maximum likelihood estimation for conditionally heteroscedastic models when the innovation process is in the domain of attraction of a stable law. Parallel Meetings, 27-31 August 2012, Malaga, Spain.

Love, K.R., Ye, K.Y., Smith, E.P. & Prisley, S.P. 2007. Error models in geographic information systems vector data using Bayesian methods. International Journal of Geographical Information Science technical report No. 07-1.

Martina, M.L.V., Todini, E. & Libralon, A. 2008. Rainfall thresholds for flood warning systems: A Bayesian decision approach. Water Science and Technology Library 63(3): 203-227.

McLachlan, G.J. & Peel, D. 2000. Finite Mixture Models. New York: John Wiley & Sons.

Monahan, J.F. 1983. Fully Bayesian analysis of ARMA time series models. Journal of Econometrics 21: 307-331.

Newcomb, S. 1886. A generalized theory of the combination of observations so as to obtain the best result. American Journal of Mathematics 8: 343-366.

Newton, M.A., Kenziorski, C.M., Richmond, C.S., Blattner, F.R. & Tsui, K.W. 2001. On differential variability of expression ratios: Improving statistical inference about gene expression changes from microarray data. Journal of Computational Biology 8: 37-52.

Olson, D.A., Junker, N.W. & Korty, B. 1995. Evaluation of 33 years of quantitative precipitation forecasting at the NMC. Weather Forecasting 10: 498-511.

Pandey, B.N., Dwivedi, N. & Bandyopadhyay, P. 2011. Comparison between Bayesian and maximum likelihood estimation of scale parameter in Weibull distribution with known shape under linex loss function. Journal of Scientific Research 55: 163-172.

Safaa Nasir & Nashaat Al-Anber. 2012. A comparison of the Bayesian and other methods for estimation of reliability function for Burr-XII distribution. Journal of Mathematics and Statistics 8(1): 42-48.

Titterington, D.M., Smith, A.F.M. & Markov, U.E. 1985. Statistical Analysis of Finite Mixture Distributions. New York: Wiley.

*Pengarang untuk surat-menyurat; email: yen_phoong@hotmail.com