Abstract
This paper presents a theoretical and empirical study of likelihood inference for the autoregressive models with finite (m-component) mixture of scale mixtures of normal (Gaussian) (SMN) innovations. This model involves autoregressive models with single and mixture component of innovations, which are frequently used in time series data analysis. An EM-type algorithm for the maximum likelihood estimation is developed and the observed information matrix is obtained. The performance of the proposed model through a simulation study is also evaluated. The model is then applied on a real time series data set.
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References
Akaike H (1974) A new look at the statistical model identification. IEEE Trans Autom Control 19:716–723
Akkaya A, Tiku ML (2001) Estimating parameters in autoregressive models in non-normal situations: asymmetric innovations. Commun Stat Theory Methods 30:517–536
Andel J (1988) On AR(1) processes with exponential white noise. Commun Stat Theory Methods 17:1481–1495
Andrews DR, Mallows CL (1974) Scale mixture of normal distribution. J Roy Stat Soc B 36:99–102
Bache K, Lichman M (2013) UCI machine learning repository. Irvine, CA: University of California, School of Information and Computer Science. (http://archive.ics.uci.edu/ml)
Bandon P (2009) Estimation of autoregressive models with epsilon-skew-normal innovations. J Multivar Anal 100:1761–1776
Besbeas P, Morgan BTJ (2004) Integrated squared error estimation of normal mixtures. Comput Stat Data Anal 44(3):517–526
Böhning D, Seidel W, Alfó M, Garel B, Patilea V, Walther G (2007) Advances in mixture models. Comput Stat Data Anal 51:5205–5210
Böhning D, Hennig C, McLachlan GJ, McNicholas PD (2014) The 2nd special issue on advances in mixture models. Comput Stat Data Anal 71:1–2
Brockwell PJ, Davis RA (1991) Time series: theory and methods, 2nd edn. Springer, New York
Carvalho AX, Tanner MA (2007) Modelling nonlinear count time series with local mixtures of Poisson autoregressions. Comput Stat Data Anal 51:5266. doi:10.1016/j.csda.2006.09.032
Christmas J, Everson R (2011) Robust autoregression: Student’s t innovations using variational Bayes. IEEE Trans Signal Process 59:48–57
Dempster AP, Laird NM, Rubin DB (1977) Maximum likelihood from incomplete data via the EM algorithm. J R Stat Soc Ser B Methodol 39:1–22
Glasbey CA (2001) Non-linear autoregressive time series with multivariate Gaussian mixtures as marginal distributions. J R Stat Soc Ser C 50:143–154
Kay S, Salisbury S (1990) Improved active sonar detection in reverberation using autoregressive prewhiteners. J Acoust Soc Am 87(4):1603–1611
Khorshidi S, Karimi M, Nematollahi AR (2011) New autoregressive (AR) order selection criteria based on the prediction error estimation. Sig Process 91(10):2359–2370
Hodgkiss WS, Hansen DS (1985) Maximum-entropy and Bayesian methods in inverse problems. Springer, Dordrecht
Janacek GJ, Swift AL (1990) A class of models for non-normal time series. J Time Ser Anal 11:19–31
Li WK, Mcleod AI (1988) ARMA modeling with non-Gaussian innovations. J Time Ser Anal 9:155–168
Liu C, Rubin DB (1994) The ECME algorithm: a simple extension of EM and ECM with faster monotone convergence. Biometrika 81:633–648
Liu CH, Rubin DB (1995) ML estimation of the t distribution using EM and its extensions, ECM and ECME. Statist Sinica 5:19–40
Meng X, Rubin DB (1993) Maximum likelihood estimation via the ECM algorithm: A general framework. Biometrika 80:267–278
McLachlan GJ, Peel D (2000) Finite mixture models. Wiley, New York
Ogawa T, Sonoda H, Ishiwa S, Shigeta Y (1992) An application of autoregressive model to pattern discrimination of brain electrical activity mapping. J Brain Topogr 6(1):3–11
Penny WD, Roberts SJ (2002) Variational Bayes for generalized autoregressive models. IEEE Trans Signal Process 50(9):2245–2257
Ramdane-Cherif Z, Nait-Ali A, Motsch JF, Krebs MO (2004) An autoregressive (AR) model applied to eye tremor movement, clinical application in schizophrenia. J Med Syst 28(5):489–495
Roberts S, Husmeier JD, Rezek I, Penny W (1998) Bayesian approaches to Gaussian mixture modeling. IEEE Trans Pattern Anal Mach Intell 20(11):1133–1142
Schwarz G (1978) Estimating the dimension of a model. Ann Stat 6:461–464
Subasi A, Alkan A, Koklukaya E, Kiymik MK (2005) Wavelet neural network classification of EEG signals by using AR model with MLE preprocessing. Neural Netw 18:985–997
Tiku ML, Wong WK, Bian G (1999) Estimating parameters in autoregressive models in non-normal situations; symmetric innovations. Commun Stat Theory Methods 28:315–341
Wong CS, Li WK (2001) On a mixture autoregressive conditional heteroscedastic model. J Am Stat Assoc Ser B 96:982–995
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The authors would like to thank the anonymous reviewers for their suggestions, corrections and encouragement, which helped us to improve the earlier versions of the manuscript.
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Maleki, M., Nematollahi, A.R. Autoregressive Models with Mixture of Scale Mixtures of Gaussian Innovations. Iran J Sci Technol Trans Sci 41, 1099–1107 (2017). https://doi.org/10.1007/s40995-017-0237-6
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DOI: https://doi.org/10.1007/s40995-017-0237-6