Abstract
In this paper, we adopt a Bayesian point of view for predicting real continuous-time processes. We give two equivalent definitions of a Bayesian predictor and study some properties: admissibility, non-unbiasedness, prediction sufficiency, comparison with efficient predictors. Prediction of Poisson process and prediction of Ornstein-Uhlenbeck process in the continuous and sampled situations are considered. Various simulations illustrate comparison with non-Bayesian predictors.
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References
Adke, S R and Ramanathan, T V (1997). On optimal prediction for stochastic processes. J. Stat. Plann Infer. 63, 1, 1–7.
Blanke, D and Bosq, D (2012) Bayesian prediction for stochastic processes. arXiv: 1211.2300v1, math.ST/1211.2300v1.
Bosq, D (2012) A note on Bayesian prediction. Rend Circ Mat Palermo, Special issue, Series II, Suppl. 84, 81-90.
Bosq, D and Blanke, D (2007) Prediction and inference in large dimensions. Wiley series in probability and statistics, Wiley-Dunod.
Cox, D D (1991). Gaussian likelihood estimation for nearly nonstationary AR(1) processes. Ann. Stat. 19, 3, 1129–1142.
Díaz J (1990). Bayesian forecasting for AR(1) models with normal coefficients. Comm. Stat. Theory Methods 19, 6, 2229–2246.
Grenander, U (1981). Abstract inference. Wiley, New York.
Kutoyants, YA (2004) Statistical Inference for Ergodic Diffusion Processes. Springer Series In Statistics, Springer.
Lehmann, E L and Casella, G (1998). Theory of point estimation, 2nd edn. Springer, New-York.
Liptser, R S and Shiryaev, A N (2001). Statistics of random processes I, II, 2nd edn. Springer, New-York.
Onzon, E (2011). Multivariate Cramér-Rao inequality for prediction and efficient predictors. Stat. Probab. Lett. 81, 3, 429–437.
Sáfadi, T and Morettin, P A (2000). Bayesian analysis of threshold autoregressive moving average models. Sankhyā Ser. B 62, 3, 353–371.
Sáfadi, T and Morettin, P A (2003). A Bayesian analysis of autoregressive models with random normal coefficients. J. Stat. Comput. Simul. 73, 8, 563–573.
Thompson, B and Vladimirov, I (2005). Bayesian parameter estimation and prediction in mean reverting stochastic diffusion models. Nonlinear Anal. 63, 5–7, e2367–e2375.
Yatracos, Y G (1992). On prediction and mean-square error. Canad J Statist 20, 2, 187–200.
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Blanke, D., Bosq, D. Bayesian Prediction for Stochastic Processes: Theory and Applications. Sankhya A 77, 79–105 (2015). https://doi.org/10.1007/s13171-014-0059-y
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DOI: https://doi.org/10.1007/s13171-014-0059-y