A Bayesian Approach to HeavyTailed Finite Mixture Autoregressive Models
Abstract
:1. Introduction
2. The SMSN Distributions
 (a)
 If $E\left[{U}^{1/2}\right]<\infty $, then ${\mu}_{Y}=E\left[Y\right]=\mu b\Delta $,
 (b)
 If $E\left[{U}^{1}\right]<\infty $, then ${\sigma}_{Y}^{2}=Var\left[Y\right]={\sigma}^{2}{k}_{2}{b}^{2}{\Delta}^{2}$,
 (c)
 $Y$ has a stochastic representation given by$Y=\mu +\Delta W+\gamma {U}^{1/2}{W}_{1},$where$b=\sqrt{2/\pi}E\left[{U}^{1/2}\right]$, $\delta =\lambda /\sqrt{1+{\lambda}^{2}}$, $\Delta =\sigma \delta $, ${\gamma}^{2}={\sigma}^{2}{\Delta}^{2}$, $W={U}^{1/2}\left{w}_{0}\right$, and ${W}_{0}$ and ${W}_{1}$ are independent standard normal random variables.
3. The SMSN–MAR Model and Bayesian Estimates
3.1. The SMSN–MAR Model
 g is a known positive integer which indicates the number of components in the model;
 each component occurs with probabilities${\pi}_{i}0,i=1,\dots ,\mathrm{g},{{\displaystyle \sum}}_{i=1}^{\mathrm{g}}{\pi}_{i}=1,$ which obeys a discrete distribution $\mathit{\pi}$;
 for each $i=1,\dots ,\mathrm{g}$,
 ➢
 the ith autoregressive component is of order ${p}_{i}\ge 1$;
 ➢
 ${\phi}_{i,j},j=0,\dots ,{p}_{i}$, are the autoregressive coefficients of the ith components;
 ➢
 each ith innovation’s component${\epsilon}_{i,t}$ distributed as following:$${\epsilon}_{i,t}{\begin{array}{c}\begin{array}{c}\\ i.i.d.\end{array}\\ ~\\ \end{array}}^{}SMSN\left({b}_{i}{\Delta}_{i},{\sigma}_{i}^{2},{\lambda}_{i},{\mathit{\nu}}_{i}\right);i=1,\dots ,\mathrm{g};t=0,\pm 1,\pm 2,\dots $$
3.2. Bayesian Approach
Algorithm 1: MCMC 
For$i=1,\dots ,\mathrm{g}$, and$t=1,\dots ,n$,

 ∎
 ST–MAR:$${U}_{ti}\mathit{\theta},{Z}_{ti}=1,\mathit{D}~\mathrm{Gamma}\left(\frac{{\nu}_{i}+2}{2},\frac{{\nu}_{i}+{c}_{ti}}{2}\right),i=1,\dots ,\mathrm{g},t=1,\dots ,n,$$$$\begin{array}{cc}\hfill \pi \left({\nu}_{i}{\mathit{\theta}}_{\left({\nu}_{i}\right)},{Z}_{ti}=1,\mathit{D}\right)& \propto {\left({2}^{{\nu}_{i}/2}\mathsf{\Gamma}\left({\nu}_{i}/2\right)\right)}^{n}\hfill \\ & \times \mathrm{Gamma}\left(\frac{n{\nu}_{i}}{2}1,\frac{1}{2}\left[{\displaystyle {\displaystyle \sum}_{t=1}^{n}}\left({U}_{ti}\mathrm{log}{U}_{ti}\right)+{\varsigma}_{i}\right]\right){I}_{\left(2,\infty \right)}\left({\nu}_{i}\right).\hfill \end{array}$$
 ∎
 SSL–MAR:$${U}_{ti}\mathit{\theta},{Z}_{ti}=1,\mathit{D}~\text{}\mathrm{Gamma}\left({\nu}_{i}+1,\frac{{c}_{ti}}{2}\right){I}_{\left(0,1\right)}\left({U}_{ti}\right),i=1,\dots ,\mathrm{g},$$$${\nu}_{i}{\mathit{\theta}}_{\left({\nu}_{i}\right)},\mathit{U},{Z}_{ti}=1,\mathit{D}~\text{}\mathrm{Gamma}\left(n+{a}_{i},{b}_{i}{{\displaystyle \sum}}_{t=1}^{n}\mathrm{log}{U}_{ti}\right),i=1,\dots ,\mathrm{g}.$$
 ∎
 SCN–MAR:$$\pi \left({u}_{ti}\mathit{\theta},{Z}_{ti}=1,\mathit{D}\right)=\frac{{A}_{ti}}{{A}_{ti}+{B}_{ti}}I\left({U}_{ti}={\gamma}_{i}\right)+\frac{{B}_{ti}}{{A}_{ti}+{B}_{ti}}I\left({U}_{ti}=1\right),i=1,\dots ,\mathrm{g},$$$${\nu}_{i}{\mathit{\theta}}_{\left({\nu}_{i}\right)},{Z}_{ti}=1,\mathit{D}~\mathrm{Beta}\left(\frac{n{{\displaystyle \sum}}_{t=1}^{n}{U}_{ti}}{1{\gamma}_{i}}+1,\frac{{{\displaystyle \sum}}_{t=1}^{n}{U}_{ti}n{\gamma}_{i}}{1{\gamma}_{i}}+1\right),i=1,\dots ,\mathrm{g},$$$$\pi \left({\gamma}_{i}{\mathit{\theta}}_{\left({\gamma}_{i}\right)},{Z}_{ti}=1,\mathit{D}\right)\propto {\nu}_{i}^{\frac{n{{\displaystyle \sum}}_{t=1}^{n}{U}_{ti}}{1{\gamma}_{i}}}{\left(1{\nu}_{i}\right)}^{\frac{{{\displaystyle \sum}}_{t=1}^{n}{U}_{ti}n{\gamma}_{i}}{1{\gamma}_{i}}},i=1,\dots ,\mathrm{g}.$$
4. Numerical Studies
4.1. First Scheme
4.2. Second Scheme
4.3. Real Data
5. Conclusions
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
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Model  SN–MAR  ST–MAR  SSL–MAR  SCN–MAR  

Parameters (Values)  Mean  SD  Mean  SD  Mean  SD  Mean  SD 
${\phi}_{1,0}\left(0.1\right)$  0.1003  0.02131  0.1001  0.02743  0.1004  0.02309  0.1010  0.02301 
${\phi}_{2,0}\left(0.1\right)$  0.1002  0.02242  0.1007  0.02436  0.1005  0.02419  0.1012  0.02502 
${\phi}_{1,1}\left(0.5\right)$  0.5017  0.01483  0.5011  0.01738  0.5010  0.01333  0.5033  0.01501 
${\phi}_{1,2}\left(0.3\right)$  0.3011  0.01746  0.3009  0.01846  0.3024  0.01963  0.3032  0.01758 
${\phi}_{2,1}\left(0.6\right)$  0.6032  0.01592  0.6048  0.01977  0.6024  0.01846  0.6101  0.01610 
${\phi}_{2,2}\left(0.2\right)$  0.2101  0.02032  0.2097  0.02006  0.2008  0.01926  0.2112  0.02094 
${\sigma}_{1}^{2}\left(1\right)$  1.1023  0.02064  1.0969  0.31741  1.1021  0.27493  1.1026  0.02080 
${\sigma}_{2}^{2}\left(2\right)$  2.1978  0.03027  2.2019  0.28950  2.1992  0.31171  2.2011  0.03311 
${\lambda}_{1}\left(2\right)$  2.0184  0.81651  2.0025  0.94561  1.9367  0.82846  2.0221  0.82331 
${\lambda}_{2}\left(4\right)$  4.0038  1.09817  3.9014  0.95928  3.9930  0.90563  4.0112  1.09831 
${\mathit{\nu}}_{1}\left(3\right)$      3.8957  0.56842  3.9473  1.14587  0.5211  0.01918 
${\mathit{\nu}}_{2}\left(3\right)$      3.8957  0.46877  3.9473  1.24587  0.5192  0.02111 
${\gamma}_{1}\left(0.5\right)$              0.5103  0.02013 
${\gamma}_{2}\left(0.5\right)$              0.5201  0.02321 
${\pi}_{1}\left(0.4\right)$  0.4011  0.04113  0.4008  0.02795  0.3957  0.01758  0.4011  0.04113 
Model  Number of Component (g)  EAIC  EBIC 

NMAR  1  3290.6352  3297.8210 
2  3067.3456  3291.9571  
3  3175.4758  3210.9882  
SNMAR  1  3262.7364  3271.7441 
2  3038.7464  3165.0641  
3  3175.7387  3183.7251  
STMAR  1  2979.3748  2986.2951 
2  2816.8374  2887.3601  
3  2910.6364  2977.6781  
SSLMAR  1  3176.3647  3240.2990 
2  2997.0694  3099.8811  
3  3146.7564  3157.6441  
SCNMAR  1  3164.7564  3191.3452 
2  3009.3745  3097.3383  
3  3165.7564  3182.6013 
Component  Parameters  Bayesian Estimates  S.E. 

First AR Component  ${\pi}_{1}$  $0.5802$  $0.0156$ 
${\phi}_{1,0}$  $0.0046$  $0.0100$  
${\phi}_{1,1}$  $0.1478$  $0.0304$  
${\phi}_{1,2}$  $0.0487$  $0.0102$  
${\phi}_{1,3}$  $0.0238$  $0.0011$  
${\sigma}_{1}^{2}$  $0.0036$  $0.0032$  
${\lambda}_{1}$  $1.0312$  $0.0193$  
$\mathit{\nu}$  $2.0034$  $0.0722$  
Second AR Component  ${\pi}_{2}$  $0.4198$  $0.0121$ 
${\phi}_{2,0}$  $0.0030$  $0.0013$  
${\phi}_{2,1}$  $0.2256$  $0.0345$  
${\sigma}_{2}^{2}$  $1.0679$  $0.0094$  
${\lambda}_{2}$  $0.9017$  $0.0435$  
$\mathit{\nu}$  $2.0042$  $0.0847$ 
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Mahmoudi, M.R.; Maleki, M.; Baleanu, D.; Nguyen, V.T.; Pho, K.H. A Bayesian Approach to HeavyTailed Finite Mixture Autoregressive Models. Symmetry 2020, 12, 929. https://doi.org/10.3390/sym12060929
Mahmoudi MR, Maleki M, Baleanu D, Nguyen VT, Pho KH. A Bayesian Approach to HeavyTailed Finite Mixture Autoregressive Models. Symmetry. 2020; 12(6):929. https://doi.org/10.3390/sym12060929
Chicago/Turabian StyleMahmoudi, Mohammad Reza, Mohsen Maleki, Dumitru Baleanu, VuThanh Nguyen, and KimHung Pho. 2020. "A Bayesian Approach to HeavyTailed Finite Mixture Autoregressive Models" Symmetry 12, no. 6: 929. https://doi.org/10.3390/sym12060929