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State space mixed models for binary responses with scale mixture of normal distributions links. (English) Zbl 06975388
Summary: A state space mixed models for binary time series where the inverse link function is modeled to be a cumulative distribution function of the scale mixture of normal (SMN) distributions. Specific inverse links examined include the normal, Student-$$t$$, slash and the variance gamma links. The threshold latent approach to represent the binary system as a linear state space model is considered. Using a Bayesian paradigm, an efficient Markov chain Monte Carlo (MCMC) algorithm is introduced for parameter estimation. The proposed methods are illustrated with real data sets. Empirical results showed that the slash inverse link fits better over the usual inverse probit link.

##### MSC:
 62 Statistics
Scythe
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##### References:
 [1] Albert, J.; Chib, S., Bayesian analysis of binary and polychotomous response data, Journal of the American Statistical Association, 88, 669-679, (1993) · Zbl 0774.62031 [2] Basu, S.; Mukhopadhyay, S., Bayesian analysis of binary regression using symmetric and asymmetric links, Sankhyā: The Indian Journal of Statistics. Series B, 62, 372-379, (2000) · Zbl 0978.62017 [3] Basu, S.; Mukhopadhyay, S., Binary response regression with normal scale mixture links, (Dey, D. K.; Ghosh, S. K.; Mallick, B. K., Generalized Linear Models: A Bayesian Perspective, (2000), Marcell Decker New York), 231-239 · Zbl 1028.62015 [4] Carlin, B. P.; Polson, N. G., Monte Carlo Bayesian methods for discrete regression models and categorical time series, (Bernardo, J. M.; Berger, J. O.; Dawid, A. P.; Smith, A. F.M., Bayesian Statistics. Vol. 4, (1992), Clarendon Press Oxford, UK), 577-586 [5] Carvalho, C. M.; Johannes, M.; Lopes, H. F.; Polson, N. G., Particle learning and smoothing, Statistical Science, 25, 88-110, (2010) · Zbl 1328.62541 [6] Chow, S. T.B.; Chan, J. S.K., Scale mixtures distributions in statistical modelling, Australian & New Zealand Journal of Statistics, 50, 135-146, (2008) · Zbl 1145.62006 [7] Czado, C.; Song, P. X.-K., State space mixed models for longitudinal observations with binary and binomial responses, Statistical Papers, 49, 691-714, (2008) · Zbl 1312.62104 [8] de Jong, P.; Shephard, N., The simulation smoother for time series models, Biometrika, 82, 339-350, (1995) · Zbl 0823.62072 [9] Delatola, E.-I.; Griffin, J. E., Bayesian nonparametric modelling of the return distribution with stochastic volatility, Bayesian Analysis, 6, 901-926, (2011) · Zbl 1330.62116 [10] Fahrmeir, L., Posterior mode estimation by extended Kalman filtering for multivariate dynamic generalized linear models, Journal of the American Statistical Association, 87, 501-509, (1992) · Zbl 0781.62147 [11] Fonseca, T. C.O.; Ferreira, M. A.R.; Migon, H. S., Objective Bayesian analysis for the student-$$t$$ regression model, Biometrika, 95, 325-333, (2008) · Zbl 1400.62260 [12] Geweke, J., Evaluating the accuracy of sampling-based approaches to the calculation of posterior moments, (Bernardo, J. M.; Berger, J. O.; Dawid, A. P.; Smith, A. F.M., Bayesian Statistics. Vol. 4, (1992), Clarendon Press Oxford, UK), 577-586 [13] Gneiting, T.; Raftery, A. E., Strictly proper scoring rules, prediction and estimation, Journal of the American Statistical Association, 6, 901-926, (2007) · Zbl 1284.62093 [14] Good, I. J., Rational decisions, Journal of the Royal Statistical Society. Series B, 14, 107-114, (1952) [15] Kim, S.; Shepard, N.; Chib, S., Stochastic volatility: likelihood inference and comparison with ARCH models, Review of Economic Studies, 65, 361-393, (1998) · Zbl 0910.90067 [16] Lange, K. L.; Sinsheimer, J. S., Normal/independent distributions and their applications in robust regression, Journal of Computational and Graphical Statistics, 2, 175-198, (1993) [17] Lopes, H. F.; Carvalho, C. M.; Johannes, M.; Polson, N. G., Particle learning for sequential Bayesian computation (with discussion), (Bernardo, J. M.; Berger, J. O.; Dawid, A. P.; Heckerman, D.; Smith, A. F.M.; West, M., Bayesian Statistics. Vol. 9, (2011), Clarendon Press Oxford, UK), 317-360 [18] McCullagh, P.; Nelder, J. A., Generalized linear models, (1989), Chapman and Hall London · Zbl 0744.62098 [19] Naranjo, L., Martín, J., Pérez, C.J., 2012. Bayesian binary regression with exponential power link. Computational Statistics and Data Analysis. URL: http://dx.doi.org/10.1016/j.csda.2012.07.022 (forthcoming). · Zbl 06975401 [20] Pemstein, D.; Quinn, K. V.; Martin, A. D., The scythe statistical library: an open source C++ library for statistical computation, Journal of Statistical Software, 42, 1-26, (2011) [21] Smith, A. C.; Shah, S. A.; Hudsonc, A. E.; Purpurad, K. P.; Victor, J. D.; Brown, E. N.; Schiff, N. D., A Bayesian statistical analysis of behavioral facilitation associated with deep brain stimulation, Journal of Neuroscience Methods, 18, 267-276, (2009) [22] Song, P. X.-K., Monte Carlo Kalman filter and smoothing for multivariate discrete state space models, The Canadian Journal of Statistics, 28, 641-652, (2000) · Zbl 0958.62093 [23] Sttoffer, D. S.; Schert, M. S.; Richardson, G. A.; Day, N. L.; Coble, P. A., A Walsh-Fourier analysis of the effects of moderate maternal alcohol consumption on neonatal sleep-state cycling, Journal of the American Statistical Association, 83, 954-963, (1998) [24] West, M.; Harrison, J., Bayesian forecasting and dynamic models, (1997), Springer-Verlag New York · Zbl 0871.62026 [25] West, M.; Harrison, P. J.; Migon, H. S., Dynamic generalized linear models and Bayesian forecasting, Journal of the American Statistical Association, 136, 209-220, (1985), With discussion
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