Bayesian inference of double seasonal autoregressive models under scale-mixtures of normal errors with applications

Ayman A. Amin; Farouq Mohammad A. Alam; Ayman A. Amin; Farouq Mohammad A. Alam

doi:10.3934/math.2026993

AIMS Mathematics

2026, Volume 11, Issue 8: 24667-24690. doi: 10.3934/math.2026993

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Bayesian inference of double seasonal autoregressive models under scale-mixtures of normal errors with applications

Ayman A. Amin ^1,2,
Farouq Mohammad A. Alam ^{3
,
,}

1.
Department of Mathematics, College of Science and Arts, Najran University, Najran, Saudi Arabia
2.
Science and Engineering Research Center, Najran University, Najran, Saudi Arabia
3.
Department of Statistics, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia

Received: 29 April 2026 Revised: 03 July 2026 Accepted: 22 July 2026 Published: 12 August 2026
MSC : 37M10, 62F15

Double seasonal autoregressive (DSAR) models have been developed for modeling time series characterized by two layers of seasonal behavior. A prevalent limitation in existing Bayesian analysis of these models is their reliance on the Gaussian errors assumption, which is routinely violated by empirical data. To overcome this restriction, we develop a Bayesian inferential framework for DSAR models in which the error term is governed by the scale-mixtures of normal (SMN) family, which is a flexible class of symmetric, heavy-tailed distributions. By assigning carefully chosen prior distributions to the DSAR parameters, we establish closed-form full conditional posteriors, i.e., a multivariate normal for the DSAR coefficient vectors and an inverse-gamma for the scale parameter. The conditional posteriors associated with the SMN-specific parameters are likewise obtained in closed form, although certain members of this collection do not correspond to standard distributional families. Exploiting these tractable conditional posteriors, we construct a hybrid Markov chain Monte Carlo (MCMC) algorithm—the Gibbs sampler interlaced with Metropolis-Hastings steps—to approximate the marginal posterior distributions of all DSAR parameters. The performance of the proposed algorithm is assessed through an extensive Monte Carlo simulation experiment and through real-data analyses of hourly electricity load records in Croatia and Italy.
- multiplicative DSAR,
- posterior analysis,
- MCMC methods,
- Gibbs sampler,
- metropolis-Hastings
Citation: Ayman A. Amin, Farouq Mohammad A. Alam. Bayesian inference of double seasonal autoregressive models under scale-mixtures of normal errors with applications[J]. AIMS Mathematics, 2026, 11(8): 24667-24690. doi: 10.3934/math.2026993

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Abstract

Double seasonal autoregressive (DSAR) models have been developed for modeling time series characterized by two layers of seasonal behavior. A prevalent limitation in existing Bayesian analysis of these models is their reliance on the Gaussian errors assumption, which is routinely violated by empirical data. To overcome this restriction, we develop a Bayesian inferential framework for DSAR models in which the error term is governed by the scale-mixtures of normal (SMN) family, which is a flexible class of symmetric, heavy-tailed distributions. By assigning carefully chosen prior distributions to the DSAR parameters, we establish closed-form full conditional posteriors, i.e., a multivariate normal for the DSAR coefficient vectors and an inverse-gamma for the scale parameter. The conditional posteriors associated with the SMN-specific parameters are likewise obtained in closed form, although certain members of this collection do not correspond to standard distributional families. Exploiting these tractable conditional posteriors, we construct a hybrid Markov chain Monte Carlo (MCMC) algorithm—the Gibbs sampler interlaced with Metropolis-Hastings steps—to approximate the marginal posterior distributions of all DSAR parameters. The performance of the proposed algorithm is assessed through an extensive Monte Carlo simulation experiment and through real-data analyses of hourly electricity load records in Croatia and Italy.

References

[1]	A. A. Amin, W. Emam, Y. Tashkandy, C. Chesneau, Bayesian subset selection of seasonal autoregressive models, Math., 11 (2023), 1–13. https://doi.org/10.3390/math11132878 doi: 10.3390/math11132878
[2]	A. A. Amin, Full bayesian analysis of double seasonal autoregressive models with real applications, J. Appl. Stat., 51 (2024), 1524–1544. https://doi.org/10.1080/02664763.2023.2211754 doi: 10.1080/02664763.2023.2211754
[3]	S. Shaarawy, M. Ismail, Bayesian inference for seasonal arma models, Egypt. Stat. J., 31 (1987), 323–336. https://doi.org/10.21608/esju.1987.428903 doi: 10.21608/esju.1987.428903
[4]	A. A. Amin, S. A. Alghamdi, Bayesian identification procedure for triple seasonal autoregressive models, Mathematics, 11 (2023), 3823. https://doi.org/10.3390/math11183823 doi: 10.3390/math11183823
[5]	G. Barnett, R. Kohn, S. Sheather, Bayesian estimation of an autoregressive model using markov chain monte carlo, J. Econom., 74 (1996), 237–254. https://doi.org/10.1016/0304-4076(95)01744-5 doi: 10.1016/0304-4076(95)01744-5
[6]	G. Barnett, R. Kohn, S. Sheather, Robust bayesian estimation of autoregressive–moving-average models, J. Time Ser. Anal., 18 (1997), 11–28. https://doi.org/10.1111/1467-9892.00036 doi: 10.1111/1467-9892.00036
[7]	J. Vermaak, M. Niranjan, S. J. Godsill, Markov chain Monte Carlo estimation for the seasonal autoregressive process with application to pitch modelling, Technical report, Department of Engineering, University of Cambridge, 1998.
[8]	M. A. Ismail, A. A. Amin, Gibbs sampling for sarma models, Pak. J. Stat., 30 (2014), 153–168.
[9]	A. A. Amin, Gibbs sampling for bayesian prediction of sarma processes, Pak. J. Stat. Oper. Res., 15 (2019), 397–418. https://doi.org/10.18187/pjsor.v15i2.2174 doi: 10.18187/pjsor.v15i2.2174
[10]	A. A. Amin, M. A. Ismail, Gibbs sampling for double seasonal autoregressive models, Commun. Stat. Appl. Method., 22 (2015), 557–573. https://doi.org/10.5351/CSAM.2015.22.6.557 doi: 10.5351/CSAM.2015.22.6.557
[11]	A. A. Amin, Bayesian analysis of double seasonal autoregressive models, Sankhya B, 82 (2020), 328–352. https://doi.org/10.1007/s13571-019-00192-z doi: 10.1007/s13571-019-00192-z
[12]	G. Ferreira, L. M. Castro, V. H. Lachos, R. Dias, Bayesian modeling of autoregressive partial linear models with scale mixture of normal errors, J. Appl. Stat., 40 (2013), 1796–1816. https://doi.org/10.1080/02664763.2013.796349 doi: 10.1080/02664763.2013.796349
[13]	M. R. Mahmoudi, M. Maleki, D. Baleanu, V. T. Nguyen, K. H. Pho, A bayesian approach to heavy-tailed finite mixture autoregressive models, Symmetry, 12 (2020), 929. https://doi.org/10.3390/sym12060929 doi: 10.3390/sym12060929
[14]	H. Ryu, D. H. Kim, Robust bayesian analysis for autoregressive models, J. Korean Data Inf. Sci. Soc., 26 (2015), 487–493. https://doi.org/10.7465/jkdi.2015.26.2.487 doi: 10.7465/jkdi.2015.26.2.487
[15]	M. Agarwal, P. K. Tripathi, Classical and bayes analyses of autoregressive model with heavy-tailed error, Am. J. Math. Manag. Sci., 43 (2024), 40–60. https://doi.org/10.1080/01966324.2024.2309387 doi: 10.1080/01966324.2024.2309387
[16]	A. A. Amin, S. A. Alghamdi, Bayesian estimation of autoregressive models with exogenous variables under scale-mixtures of normal errors, Mathematics, 14 (2026), 2188. https://doi.org/10.3390/math14122188 doi: 10.3390/math14122188
[17]	A. A. Amin, Bayesian modeling and forecasting of seasonal autoregressive models with scale-mixtures of normal errors, Comput. Stat., 40 (2025), 3453–3475. https://doi.org/10.1007/s00180-025-01617-2 doi: 10.1007/s00180-025-01617-2
[18]	A. A. Amin, Full bayesian analysis of seasonal autoregressive models under scale-mixtures of normal errors, J. Stat. Comput. Simul., 96 (2026), 184–204. https://doi.org/10.1080/00949655.2025.2556269 doi: 10.1080/00949655.2025.2556269
[19]	D. F. Andrews, C. L. Mallows, Scale mixtures of normal distributions, J. R. Stat. Soc. Ser. B, 36 (1974), 99–102. https://doi.org/10.1111/j.2517-6161.1974.tb00989.x doi: 10.1111/j.2517-6161.1974.tb00989.x
[20]	M. West, On scale mixtures of normal distributions, Biometrika, 74 (1987), 646–648.
[21]	M. A. Tanner, W. H. Wong, The calculation of posterior distributions by data augmentation, J. Am. Stat. Assoc., 82 (1987), 528–540. https://doi.org/10.2307/2289457 doi: 10.2307/2289457
[22]	K. L. Lange, R. J. A. Little, J. M. G. Taylor, Robust statistical modeling using the $t$ distribution, J. Am. Stat. Assoc., 84 (1989), 881–896. https://doi.org/10.1080/01621459.1989.10478852 doi: 10.1080/01621459.1989.10478852
[23]	W. H. Rogers, J. W. Tukey, Understanding some long-tailed symmetrical distributions, Stat. Neerl., 26 (1972), 211–226. https://doi.org/10.1111/j.1467-9574.1972.tb00191.x doi: 10.1111/j.1467-9574.1972.tb00191.x
[24]	J. W. Tukey, A survey of sampling from contaminated distributions, in Contrib. Probab. Stat. (ed. I. Olkin). Stanford Univ. Press, Stanford, CA, 1960,448–485.
[25]	A. Gelman, J. B. Carlin, H. S. Stern, D. B. Rubin, Bayesian Data Analysis, 2nd edition, Chapman & Hall/CRC, Boca Raton, FL, 2004, https://doi.org/10.1201/9780429258480
[26]	A. Zellner, On assessing prior distributions and Bayesian regression analysis with $g$-prior distributions, in Bayesian Inference Decision Tech.: Essays Honor Bruno de Finetti (eds. P. Goel and A. Zellner). North-Holland/Elsevier, Amsterdam, 1986,233–243.
[27]	C. Fernandez, E. Ley, M. F. Steel, Benchmark priors for bayesian model averaging, J. Econom., 100 (2001), 381–427. https://doi.org/10.1016/S0304-4076(00)00076-2 doi: 10.1016/S0304-4076(00)00076-2
[28]	A. A. Amin, Kullback-leibler divergence to evaluate posterior sensitivity to different priors for autoregressive time series models, Commun. Stat. Simul. Comput., 48 (2019), 1277–1291. https://doi.org/10.1080/03610918.2017.1410709 doi: 10.1080/03610918.2017.1410709
[29]	H. Jeffreys, Theory of Probability, 3rd edition, Oxford Univ. Press, Oxford, 1961.
[30]	C. R. B. Cabral, V. H. Lachos, M. R. Madruga, Bayesian analysis of skew-normal independent linear mixed models with heterogeneity in the random-effects population, J. Stat. Plann. Inference, 142 (2012), 181–200. https://doi.org/10.1016/j.jspi.2011.07.007 doi: 10.1016/j.jspi.2011.07.007
[31]	J. Geweke, Evaluating the accuracy of sampling-based approaches to the calculations of posterior moments, Bayesian Stat., 4 (1992), 641–649. https://doi.org/10.1093/oso/9780198522669.003.0010 doi: 10.1093/oso/9780198522669.003.0010
[32]	A. E. Raftery, S. M. Lewis, The number of iterations, convergence diagnostics and generic metropolis algorithms, Pract. Markov Chain Monte Carlo, 7 (1995), 763–773.
[33]	J. P. LeSage, Applied econometrics using MATLAB, Technical report, Department of Economics, University of Toronto, 1999.
[34]	C. P. Robert, G. Casella, The metropolis—hastings algorithm, in Monte Carlo Stat. Methods. Springer, 2004,267–320. https://doi.org/10.1007/978-1-4757-4145-2_7
[35]	W. A. Link, M. J. Eaton, On thinning of chains in mcmc, Methods Ecol. Evol., 3 (2012), 112–115. https://doi.org/10.1111/j.2041-210X.2011.00131.x doi: 10.1111/j.2041-210X.2011.00131.x
[36]	B. Carpenter, A. Gelman, M. D. Hoffman, D. Lee, B. Goodrich, M. Betancourt, et al., Stan: A probabilistic programming language, J. Stat. Softw., 76 (2017), 1–32.
[37]	O. Abril-Pla, V. Andreani, C. Carroll, L. Dong, C. J. Fonnesbeck, M. Kochurov, et al., Pymc: A modern, and comprehensive probabilistic programming framework in python, PeerJ Comput. Sci., 9 (2023), e1516.
[38]	A. Gelman, W. R. Gilks, G. O. Roberts, Weak convergence and optimal scaling of random walk metropolis algorithms, Ann. Appl. Probab., 7 (1997), 110–120.
[39]	J. W. Taylor, Short-term electricity demand forecasting using double seasonal exponential smoothing, J. Oper. Res. Soc., 54 (2003), 799–805, https://doi.org/10.1057/palgrave.jors.2601589 doi: 10.1057/palgrave.jors.2601589
[40]	G. Schwarz, Estimating the dimension of a model, Ann. Stat., 6 (1978), 461–464. https://doi.org/10.1214/aos/1176344136 doi: 10.1214/aos/1176344136
[41]	D. J. Spiegelhalter, N. G. Best, B. P. Carlin, A. Van Der Linde, Bayesian measures of model complexity and fit, J. R. Stat. Soc. Ser. B Stat. Methodol., 64 (2002), 583–639.
[42]	J. Geweke, Bayesian treatment of the independent student-$t$ linear model, J. Appl. Econom., 8 (1993), S19–S40. https://doi.org/10.1002/jae.3950080504 doi: 10.1002/jae.3950080504

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