International Journal of Mathematical Sciences and Computing (IJMSC)
IJMSC Vol. 11, No. 1, 8 Apr. 2025
Cover page and Table of Contents: PDF (size: 837KB)
On the Noisy Four-Parameter Fisher‟s Z-Distribution of Bayesian Mixture Autoregressive (FZBMAR) Process via Mode as a Stable Location Parameter
PDF (837KB), PP.62-79
Views: 0 Downloads: 0
Author(s)
Index Terms
Autoregressive, Bayesian, FZBMAR Process, Share Price, Singly Truncated Student-t Distribution, Switching Time-Variant
Abstract
This paper aims at providing in-depth refinement to switching time-variant autoregressive processes via the mode as a stable location parameter in adopted noisy Fisher’s z-distribution that was impelled in a Bayesian setting. Explicitly, a four-parameter Fisher’s z-distribution of Bayesian Mixture Autoregressive (FZBMAR) process was proposed to congruous k-mixture components of Fisher’s z-switching mixture autoregressive processes that was based on shifting number of modes in the marginal density of any switching time-variant series of interest. The proposed FZBMAR process was not only used to seize what is term “most likely mode value” of the present conditional modal distribution given the immediate past but was also used to capture the conditional modal distribution of the observations given the immediate past that can either be perceived as an asymmetric or symmetric distributed varieties. The proposed FZBMAR process was compared with the existing Student-t Mixture Autoregressive (StMAR) and Gaussian Mixture Autoregressive (GMAR) processes with the demonstration of monthly average share prices (stock prices) of sixteen (16) swaying European economies. Based on the findings, the FZBMAR process outperformed the existing StMAR and GMAR processes in explaining the sixteen (16) swaying European economies share prices via a minimum Pareto-Smoothed Important Sampling Leave-One-Out Cross-Validation (PSIS-LOO) error process performance in comparison with AIC, HQIC by the latters. The same singly truncated student-t prior distribution was adopted for the noisy adoption of Fisher’s z hyper-parameters and the embedded autoregressive coefficients in the proposed FZBMAR process; such that their resulting posterior distributions gave the same singly truncated student-t distribution (conjugate) with an embedded Gamma variate.
Cite This Paper
Rasaki Olawale Olanrewaju, Sodiq Adejare Olanrewaju, "On the Noisy Four-Parameter Fisher’s Z-Distribution of Bayesian Mixture Autoregressive (FZBMAR) Process via Mode as a Stable Location Parameter", International Journal of Mathematical Sciences and Computing(IJMSC), Vol.11, No.1, pp. 62-79, 2025. DOI: 10.5815/ijmsc.2025.01.05
Reference
[1]D. Barber, A.T. Cemgil, and S. Chappa, Bayesian Time Series Models, Cambridge University Press, 978-0-521-19676, the Edinburgh Building, Cambridge CB2 8RU, UK, 2011. www.cambridge.org/9780521196765.
[2]G. J. McLachlan, S. X. Lee, and I. R. Suren, “Finite mixture models,” Annual Review of Statistics and its Application, vol. 6, no. 3, pp. 355–378, 2019.
[3]G. J. McLachlan and P. David, Finite Mixture Models, Hoboken: John Wiley & Sons, 2000.
[4]R. O. Olanrewaju, A. G. Waititu, and L. A. Nafiu, “Bull and bear dynamics of the Nigeria stock returns via mingled autoregressive random processes,” Open Journal of Statistics, vol. 11, no. 5, pp. 870–885, 2021, doi:10.4236/ojs.2021.115051.
[5]S. Virolainen, “A mixture autoregressive model on Gaussian and Student-t distribution,” Studies in Nonlinear Dynamics & Econometrics, vol. 26, no. 4 pp. 559–580, 2022, doi:10.1515/snde-2020-0060.
[6]J. Kumar, and V. Agiwal, “Bayesian Estimation for fully shifted panel AR(1) time series model,” Thia Journal of Mathematics, vol. 17, pp. 167–183, 2099, https://thaijmath.in.cmu.ac.th..
[7]N. D. Le, R. Martin, and A. E. Raftery, “Modeling flat stretches, bursts, and outliers in time series using mixture transition distribution models,” Journal of American Statistical Association, vol. 91, no. 436 pp. 1504–1515, 1996.
[8]C. S. Wong, W. S. Chan, and P. L. Kam, A Student-t-Mixture Autoregressive Model with Applications to Heavy-Tailed Financial Data, Singapore Economic Review Conference, pp. 1-10, 2009.
[9]R. O. Olanrewaju, A. G. Waititu, and L. A. Nafiu, “Kullback-Leibler divergence of mixture autoregressive random processes via Extreme-Value-Distributions (EVDs) noise with application of the processes to climate change,” Transactions on Machine and Artificial Intelligence, vol. 10, no. 1, pp. 1–18, 2022, doi:10.14738/tmlai.101.11544.
[10]C. S. Wong, and W. K. Li, “On a logistic mixture autoregressive models,” Biometrika, vol. 88, pp. 833–846, 2001.
[11]C. S. Wong, and W. K. Li, “On a mixture autoregressive model,” Journal of Royal Statistics Society: Series B Statistical Methodology, vol. 62, pp. 95–115, 2000.
[12]N. Nguyen “Hidden markov model for stock trading,” International of Financial Studies, vol. 6, no. 36, pp. 2–17, 2018.
[13]M. Maleki, and A. R. Nematollahi, “Autoregressive models with mixture of scale mixtures of Gaussian innovations,” Iran Journal of Science and Technology: Transactions A Science, vol. 41, no. 4, pp. 1099–1107, 2017, https://doi.org/10.1007/s40995-017-0237-6.
[14]R. O. Olanrewaju, A. G. Waititu, and L. A. Nafiu, “Fréchet random noise for k-regime- switching mixture autoregressive,” American Journal of Mathematics and Statistics, vol. 11, no. 1, pp. 1–10, 2021. doi.10.5923/j.ajms.20211101.01.
[15]M. Maleki, J. E. Contreras-Reyes, and M. R. Mahmoud, “Robust mixture modeling based on two-piece scale mixtures of normal family,” Axioms, vol. 8, no. 38, pp. 2-14, 2019. doi:10.3390/axioms8020038.
[16]D. Ravagli, and G. N. Boshnakov, “Bayesian analysis of mixture autoregressive models covering the complete parameter space,” Computational Statistics, vol. 37, pp. 1399-1433, 2022. https://doi.org/10.1007/s00180-021-01162-8.
[17]O. Papaspiliopoulos, and G. O. Roberts, “Retrospective markov chain monte carlo method for dirichlet process hierarchical models,” Biometrika, vol. 95, no. 1, pp. 169–186, 2008.
[18]M. Maleki, H. Arezo, and R. B. Arellano-Valle, “Symmetrical and asymmetrical mixture autoregressive processes,” Brazilian Journal of Probability and Statistics, vol. 34, pp. 273–290, 2020.
[19]R. M. Neal, MCMC using Hamiltonian Dynamics. In Handbook of Markov Chain Monte Carlo. Edited by Steve Brooks, Andrew Gelman, Galin L. Jones and Xiao-Li Meng. Boca Raton: CRC Press, pp. 116–162, 2011.
[20]G. N. Boshnakov, and D. Ravagli, MixAR: Mixture Autoregressive Models. Rpackage Version 0.22.4., 2020. https://CRAN.R-project.org/package=mixAR.
[21]A.S. Hossain, Complete Bayesian Analysis of some Mixture Time Series Models. PhD thesis, Probability and Statistics Group, School of Mathematics, University of Manchester, 2012.
[22]D. Ghouil, and M. Ourbih-Tari, “Bayesian autoregressive adaptive refined descriptive sampling algorithm in the monte carlo simulation,” Statistical Theory and Related Fields, vol. 7, no. 3, pp. 177-187, 2023. doi:10.1080/24754269.2023.2180225.
[23]R. A. Fisher, On a Distribution Yielding the Error Functions of Several Well-Known Statistics. Paper presented at the International Congress of Mathematics, Toronto, ON, Canada, April 11–16; pp. 805–813, 1924.
[24]A. Solikhah, K. Heri, I. Nur, F. Kartika, and S. C. Achmad, Extending Runjags: A Tutorial on Adding Fisher’s z Distribution to Runjags. AIP Conference Proceedings 2329: 060005, 2021.
[25]L. A. Aroian, “A study of RA Fisher’s z-distribution and the related F-distribution,” The Annals of Mathematical Statistics, vol. 12, pp. 429–948, 1941.
[26]A. P. Dempster, N. M. Laird, and D. B. Rubin, “Maximum likelihood from incomplete data via the EM-algorithm,” Journal of Royal Statistical Society B, vol. 39, pp. 1–38, 1977.
[27]G. Huerta, and W. Mike, “Priors and component structures in autoregressive time series models,” Journal of the Royal Statistical Society: Series B Statistical Methodology, vol. 61, pp. 881–99, 1999.
[28]H. J. Kim, “Moments of truncated student-t distribution,” Journal of the Korean Statistical Society, vol. 37, pp. 81–87, 2008.
[29]S. Richardson, and P. J. Green, “On Bayesian analysis of mixtures with an unknown number of components,” Journal Royal Statistics Society: Series B Statistical Methodology, vol. 59, no. 4, pp. 731–792, 1997.
[30]A. Vehtari, G. Andrew, S. Daniel, C. Bob, and C. B. Paul, “Rank-normalization, folding and localization: an improved rhat for assessing convergence of mcmc,” Bayesian Analysis, vol. 1, pp. 1–28, 2020.