Summary
Dynamic exponential family regression provides a framework for nonlinear regression analysis with time dependent parametersβ 0,β 1, …,β t, …, dimβ t=p. In addition to the familiar conditionally Gaussian model, it covers e.g. models for categorical or counted responses. Parameters can be estimated by extended Kalman filtering and smoothing. In this paper, further algorithms are presented. They are derived from posterior mode estimation of the whole parameter vector (β′0, …,β′t) by Gauss-Newton resp. Fisher scoring iterations. Factorizing the information matrix into block-bidiagonal matrices, algorithms can be given in a forward-backward recursive form where only inverses of “small”p×p-matrices occur. Approximate error covariance matrices are obtained by an inversion formula for the information matrix, which is explicit up top×p-matrices.
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References
Ameen JRM, Harrison PJ (1985) Normal discount Bayesian models. In: Bernardo JM, DeGroot MH, Lindley DV, Smith AFM (eds) Bayesian Statistics 2:271–294
Anderson BDO, Moore JB (1979) Optimal Filtering. Prentice Hall, Englewood Cliffs
Baker RJ, Thompson R (1981) Composite link functions in generalized linear models. Appl Stat 30:125–131
Fahrmeir L (1988) Extended Kalman filtering for dynamic generalized linear models and survival data. Regensburger Beiträge zur Statistik und Ökonometrie 10
Fahrmeir L, Kaufmann H (1985) Consistency and asymptotic normality of the maximum likelihood estimator in generalized linear models. Ann Statist 13:342–368
Fahrmeir L, Kaufmann H (1987) Regression models for nonstationary categorical time series. J Time Ser Anal 8:147–160
Jörgensen B (1983) Maximum likelihood estimation and large sample inference for generalized linear and nonlinear regression models. Biometrika 70:19–28
Kaufmann H (1987) Regression models for nonstationary categorical time series: asymptotic estimation theory. Ann Statist 15:79–98
Kitagawa G (1987) Non-Gaussian state-space modelling of nonstationary time series (with comments). JASA 82:1032–1063
Nelder JA, Wedderburn RWM (1972) Generalized linear models. J Roy Statist Soc Ser A 135:370–384
Sage AP, Melsa JL (1971) Estimation Theory with Applications to Communication and Control. McGraw Hill, New York
West M (1985) Generalized linear models: scale parameters, outlier accommodation and prior distributions. In: Bernardo JM, DeGroot MH, Lindley DV, Smith AFM (eds) Bayesian Statistics 2:531–538
West M, Harrison RJ, Migon HS (1985) Dynamic generalized linear models and Bayesian forecasting. JASA 80:73–83
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Heinz Leo Kaufmann, my friend and coauthor for many years, died in a tragical rock climbing accident in August 1989. This paper is dedicated to his memory.
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Fahrmeir, L., Kaufmann, H. On kalman filtering, posterior mode estimation and fisher scoring in dynamic exponential family regression. Metrika 38, 37–60 (1991). https://doi.org/10.1007/BF02613597
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DOI: https://doi.org/10.1007/BF02613597