, Volume 38, Issue 1, pp 37–60 | Cite as

On kalman filtering, posterior mode estimation and fisher scoring in dynamic exponential family regression

  • L. Fahrmeir
  • H. Kaufmann
I. Publications


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 (β0, …,βt) 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.


Kalman Filter Extended Kalman Filter Information Matrix Exponential Family Correction Step 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. Ameen JRM, Harrison PJ (1985) Normal discount Bayesian models. In: Bernardo JM, DeGroot MH, Lindley DV, Smith AFM (eds) Bayesian Statistics 2:271–294Google Scholar
  2. Anderson BDO, Moore JB (1979) Optimal Filtering. Prentice Hall, Englewood CliffszbMATHGoogle Scholar
  3. Baker RJ, Thompson R (1981) Composite link functions in generalized linear models. Appl Stat 30:125–131zbMATHCrossRefMathSciNetGoogle Scholar
  4. Fahrmeir L (1988) Extended Kalman filtering for dynamic generalized linear models and survival data. Regensburger Beiträge zur Statistik und Ökonometrie 10Google Scholar
  5. Fahrmeir L, Kaufmann H (1985) Consistency and asymptotic normality of the maximum likelihood estimator in generalized linear models. Ann Statist 13:342–368zbMATHMathSciNetGoogle Scholar
  6. Fahrmeir L, Kaufmann H (1987) Regression models for nonstationary categorical time series. J Time Ser Anal 8:147–160zbMATHMathSciNetGoogle Scholar
  7. Jörgensen B (1983) Maximum likelihood estimation and large sample inference for generalized linear and nonlinear regression models. Biometrika 70:19–28zbMATHMathSciNetGoogle Scholar
  8. Kaufmann H (1987) Regression models for nonstationary categorical time series: asymptotic estimation theory. Ann Statist 15:79–98zbMATHMathSciNetGoogle Scholar
  9. Kitagawa G (1987) Non-Gaussian state-space modelling of nonstationary time series (with comments). JASA 82:1032–1063zbMATHMathSciNetGoogle Scholar
  10. Nelder JA, Wedderburn RWM (1972) Generalized linear models. J Roy Statist Soc Ser A 135:370–384CrossRefGoogle Scholar
  11. Sage AP, Melsa JL (1971) Estimation Theory with Applications to Communication and Control. McGraw Hill, New YorkGoogle Scholar
  12. 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 2:531–538Google Scholar
  13. West M, Harrison RJ, Migon HS (1985) Dynamic generalized linear models and Bayesian forecasting. JASA 80:73–83zbMATHMathSciNetGoogle Scholar

Copyright information

© Physica-Verlag Ges.m.b.H 1991

Authors and Affiliations

  • L. Fahrmeir
    • 1
  • H. Kaufmann
    • 1
  1. 1.Universität Regensburg, Lehrstuhl für StatistikRegensburg

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