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On construction and estimation of stationary mixture transition distribution models. (English) Zbl 07546476

Summary: Mixture transition distribution (MTD) time series models build high-order dependence through a weighted combination of first-order transition densities for each one of a specified number of lags. We present a framework to construct stationary MTD models that extend beyond linear, Gaussian dynamics. We study conditions for first-order strict stationarity which allow for different constructions with either continuous or discrete families for the first-order transition densities given a prespecified family for the marginal density, and with general forms for the resulting conditional expectations. Inference and prediction are developed under the Bayesian framework with particular emphasis on flexible, structured priors for the mixture weights. Model properties are investigated both analytically and through synthetic data examples. Finally, Poisson and Lomax examples are illustrated through real data applications. Supplementary files for this article are available online.

MSC:

62-XX Statistics
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[1] Arnold, B. C.; Castillo, E.; Sarabia, J.-M.; Sarabia, J. M., Conditional Specification of Statistical Models (1999), Springer Science & Business Media · Zbl 0932.62001
[2] Azzalini, A., The Skew-Normal and Related Families, 3 (2013), Cambridge: Cambridge University Press, Cambridge
[3] Bartolucci, F.; Farcomeni, A., “A Note on the Mixture Transition Distribution and Hidden Markov Models, Journal of Time Series Analysis, 31, 132-138 (2010) · Zbl 1224.62030
[4] Berchtold, A., “Estimation in the Mixture Transition Distribution Model, Journal of Time Series Analysis, 22, 379-397 (2001) · Zbl 0973.62066
[5] ———, “Mixture Transition Distribution (MTD) Modeling of Heteroscedastic Time Series, Computational Statistics & Data Analysis, 41, 399-411 (2003) · Zbl 1256.62048
[6] Berchtold, A.; Raftery, A., “The Mixture Transition Distribution Model for High-Order Markov Chains and Non-Gaussian Time Series, Statistical Science, 17, 328-356 (2002) · Zbl 1013.62088
[7] Bolano, D.; Berchtold, A., “General Framework and Model Building in the Class of Hidden Mixture Transition Distribution Models, Computational Statistics & Data Analysis, 93, 131-145 (2016) · Zbl 1468.62029
[8] Cervone, D.; Pillai, N. S.; Pati, D.; Berbeco, R.; Lewis, J. H., “A Location-Mixture Autoregressive Model for Online Forecasting of Lung Tumor Motion, The Annals of Applied Statistics, 8, 1341-1371 (2014) · Zbl 1303.62058
[9] Connor, R. J.; Mosimann, J. E., “Concepts of Independence for Proportions With a Generalization of the Dirichlet Distribution, Journal of the American Statistical Association, 64, 194-206 (1969) · Zbl 0179.24101
[10] Dai, B.; Ding, S.; Wahba, G., “Multivariate Bernoulli Distribution, Bernoulli, 19, 1465-1483 (2013) · Zbl 1440.62227
[11] Dunn, P. K.; Smyth, G. K., “Randomized Quantile Residuals,”, Journal of Computational and Graphical Statistics, 5, 236-244 (1996)
[12] Eltoft, T.; Kim, T.; Lee, T.-W., “On the Multivariate Laplace Distribution, IEEE Signal Processing Letters, 13, 300-303 (2006)
[13] Escarela, G.; Mena, R. H.; Castillo-Morales, A., “A Flexible Class of Parametric Transition Regression Models Based on Copulas: Application to Poliomyelitis Incidence, Statistical Methods in Medical Research, 15, 593-609 (2006) · Zbl 1109.62076
[14] Ferguson, T. S., “A Bayesian Analysis of Some Nonparametric Problems, The Annals of Statistics, 1, 209-230 (1973) · Zbl 0255.62037
[15] Fong, P. W.; Li, W. K.; Yau, C.; Wong, C., “On a Mixture Vector Autoregressive Model, Canadian Journal of Statistics, 35, 135-150 (2007) · Zbl 1124.62059
[16] Hassan, M. Y.; Lii, K.-S., “Modeling Marked Point Processes Via Bivariate Mixture Transition Distribution Models, Journal of the American Statistical Association, 101, 1241-1252 (2006) · Zbl 1120.62324
[17] Heiner, M.; Kottas, A., “Autoregressive Density Modeling With the Gaussian Process Mixture Transition Distribution, Journal of Time Series Analysis, to appear (2021) · Zbl 1484.60080
[18] ———, “Estimation and Selection for High-Order Markov Chains With Bayesian Mixture Transition Distribution Models, Journal of Computational and Graphical Statistics, to appear (2021)
[19] Heiner, M.; Kottas, A.; Munch, S., “Structured Priors for Sparse Probability Vectors With Application to Model Selection in Markov Chains, Statistics and Computing, 29, 1077-1093 (2019) · Zbl 1430.62194
[20] Holgate, P., “Estimation for the Bivariate Poisson Distribution, Biometrika, 51, 241-287 (1964) · Zbl 0133.11802
[21] Joe, H., Dependence Modeling With Copulas (2014), Boca Raton, FL: CRC Press, Boca Raton, FL · Zbl 1346.62001
[22] Kalliovirta, L.; Meitz, M.; Saikkonen, P., “A Gaussian Mixture Autoregressive Model for Univariate Time Series, Journal of Time Series Analysis, 36, 247-266 (2015) · Zbl 1320.62201
[23] ———, “Gaussian Mixture Vector Autoregression, Journal of Econometrics, 192, 485-498 (2016) · Zbl 1420.62389
[24] Khalili, A.; Chen, J.; Stephens, D. A., “Regularization and Selection in Gaussian Mixture of Autoregressive Models, Canadian Journal of Statistics, 45, 356-374 (2017) · Zbl 1474.62054
[25] Kocherlakota, S.; Kocherlakota, K., Encyclopedia of Statistical Sciences, Bivariate Discrete Distributions (2006), Hoboken, NJ: Wiley, Hoboken, NJ · Zbl 0618.62049
[26] Kotz, S.; Kozubowski, T.; Podgorski, K., The Laplace Distribution and Generalizations: A Revisit With Applications to Communications, Economics, Engineering, and Finance (2012), Springer Science & Business Media
[27] Lanne, M.; Saikkonen, P., “Modeling the US short-term Interest Rate by Mixture Autoregressive Processes, Journal of Financial Econometrics, 1, 96-125 (2003)
[28] Lau, J. W.; So, M. K., “Bayesian Mixture of Autoregressive Models, Computational Statistics & Data Analysis, 53, 38-60 (2008) · Zbl 1452.62655
[29] Le, N. D.; Martin, R. D.; Raftery, A. E., “Modeling Flat Stretches, Bursts Outliers in Time Series Using Mixture Transition Distribution Models, Journal of the American Statistical Association, 91, 1504-1515 (1996) · Zbl 0881.62096
[30] Li, C.-S.; Lu, J.-C.; Park, J.; Kim, K.; Brinkley, P. A.; Peterson, J. P., “Multivariate Zero-Inflated Poisson Models and Their Applications, Technometrics, 41, 29-38 (1999)
[31] Li, G.; Zhu, Q.; Liu, Z.; Li, W. K., “On Mixture Double Autoregressive Time Series Models,”, Journal of Business & Economic Statistics, 35, 306-317 (2017)
[32] Luo, J.; Qiu, H.-b., “Parameter Estimation of the WMTD Model, Applied Mathematics-A Journal of Chinese Universities, 24, 379 (2009) · Zbl 1211.62054
[33] MacDonald, I. L.; Zucchini, W., Hidden Markov and Other Models for Discrete-Valued Time Series, 110 (1997), Boca Raton, FL: CRC Press, Boca Raton, FL · Zbl 0868.60036
[34] Meitz, M.; Preve, D.; Saikkonen, P., “A Mixture Autoregressive Model Based on Student’s t-Distribution, Communications in Statistics-Theory and Methods, 1-76 (2021) · Zbl 07649581
[35] Mena, R. H.; Walker, S. G., “Stationary Mixture Transition Distribution (MTD) Models Via Predictive Distributions, Journal of Statistical Planning and Inference, 137, 3103-3112 (2007) · Zbl 1114.62091
[36] Nguyen, H. D.; McLachlan, G. J.; Ullmann, J. F.; Janke, A. L., “Laplace Mixture Autoregressive Models,”, Statistics & Probability Letters, 110, 18-24 (2016) · Zbl 1419.62240
[37] Pitt, M. K.; Chatfield, C.; Walker, S. G., “Constructing First Order Stationary Autoregressive Models Via Latent Processes, Scandinavian Journal of Statistics, 29, 657-663 (2002) · Zbl 1035.62086
[38] Raftery, A.; Tavaré, S., “Estimation and Modelling Repeated Patterns in High Order Markov Chains With the Mixture Transition Distribution Model, Journal of the Royal Statistical Society, Series C, 43, 179-199 (1994) · Zbl 0825.62667
[39] Raftery, A. E., “A Model for High-Order Markov Chains, Journal of the Royal Statistical Society, Series B, 47, 528-539 (1985) · Zbl 0593.62091
[40] ———, “Change Point and Change Curve Modeling in Stochastic Processes and Spatial Statistics, Journal of Applied Statistical Science, 1, 403-423 (1994) · Zbl 0876.62072
[41] Sethuraman, J., “A Constructive Definition of Dirichlet Priors, Statistica Sinica, 4, 639-650 (1994) · Zbl 0823.62007
[42] Vitolo, C., “hddtools: Hydrological Data Discovery Tools, The Journal of Open Source Software, 2, 56 (2017)
[43] Wong, C.; Chan, W.; Kam, P., “A Student t-Mixture Autoregressive Model With Applications to Heavy-Tailed Financial Data, Biometrika, 96, 751-760 (2009) · Zbl 1170.62065
[44] Wong, C. S.; Li, W. K., “On a Mixture Autoregressive Model,”, Journal of the Royal Statistical Society, Series B, 62, 95-115 (2000) · Zbl 0941.62095
[45] ———, “On a Logistic Mixture Autoregressive Model, Biometrika, 88, 833-846 (2001) · Zbl 0985.62074
[46] ———, “On a Mixture Autoregressive Conditional Heteroscedastic Model, Journal of the American Statistical Association, 96, 982-995 (2001) · Zbl 1051.62091
[47] Zhu, F.; Li, Q.; Wang, D., “A Mixture Integer-Valued ARCH Model, Journal of Statistical Planning and inference, 140, 2025-2036 (2010) · Zbl 1184.62159
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