Rekursive Schätzverfahren in der Kredibilitätstheorie. (Recursive estimation procedures in credibility theory).

*(German)*Zbl 0653.62075Summary: Best affine-linear estimators based on observations \(\eta_ 1,...,\eta_ t\) are investigated with special regards to recursive computability. Under various specific model assumptions (linear dynamic model, linear model with recursive covariance structure, time series models), algorithms are presented and particularly their mutual correlations are shown. By identifying the classical credibility estimator as an \({\mathcal L}\) \(n_ 2\)-Hilbert space projection, easily all results can be transfered to a general credibility model.

Considering four, well-known, specific credibility models in more detail, both application of the algorithms and dissolving the normal equations leads to the solution of the credibility estimation problem. In case of C. A. Hachemeister’s regression model [Credibility, Theory Appl. 1981, Proc. Actuarial Res. Conf., Berkeley 1974, 129-163 (1975; Zbl 0354.62057)] and also in case of B. Sundt’s evolutionary model [Scand. Acturial J. 1981, 3-21 (1981; Zbl 0463.62093)] the recursive formulas are better suited for practical purpose.

Considering four, well-known, specific credibility models in more detail, both application of the algorithms and dissolving the normal equations leads to the solution of the credibility estimation problem. In case of C. A. Hachemeister’s regression model [Credibility, Theory Appl. 1981, Proc. Actuarial Res. Conf., Berkeley 1974, 129-163 (1975; Zbl 0354.62057)] and also in case of B. Sundt’s evolutionary model [Scand. Acturial J. 1981, 3-21 (1981; Zbl 0463.62093)] the recursive formulas are better suited for practical purpose.

##### MSC:

62P05 | Applications of statistics to actuarial sciences and financial mathematics |

##### Keywords:

Hachemeister’s regression model; Best affine-linear estimators; recursive computability; linear dynamic model; linear model with recursive covariance structure; time series models; algorithms; credibility estimator; Hilbert space projection; credibility models; evolutionary model; recursive formulas
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\textit{K.-P. Mangold}, Bl., Dtsch. Ges. Versicherungsmath. 18, No. 1, 27--44 (1987; Zbl 0653.62075)

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##### References:

[1] | Balakrishnan, A. V. (1984): Kaiman Filtering Theory, Optimization Software, Inc., New York. |

[2] | Box, G. E. P. andJenkins, G. M. (1970): Time Series Analysis. Forecasting and Control, Holden Day, San Francisco. · Zbl 0249.62009 |

[3] | Bühlmann, H. (1967): Experience rating and credibility, ASTIN-Bulletin 4, p. 199–207. |

[4] | Bühlmann, H. undStraub, E. (1970): Glaubwürdigkeit von SchadensÄtzen, Mitt. Schw. Vers.- math., S. 111–135. |

[5] | De Jong, P. andZehnwirth, B. (1983): Credibility theory and the Kaiman Filter, Insurance, North Holland, p. 281–286. |

[6] | De Vylder, F. (1976): Geometrical credibility, Scand. Act. Journal, p. 121–149. · Zbl 0345.62082 |

[7] | Hachemeister, C. A. (1974): Credibility for Regression Models with Application to Trend; Credibility: Theory and Applications, ed. by P. M. Kahn, New York. · Zbl 0354.62057 |

[8] | Kremer, E. (1982): Credibility Theory for Some Evolutionary Models, Scand. Act. J., p. 129–142. · Zbl 0503.62088 |

[9] | Kremer, E. (1984): The Linear Growth Credibility Model, Scand. Act. J., p. 143–149. · Zbl 0557.62087 |

[10] | Sundt, B. (1981): Recursive Credibility Estimation, Scand. Act. J., p. 25–32. · Zbl 0463.62093 |

[11] | Sundt, B. (1983): Finite Credibility Formulae in Evolutionary Models, Scand. Act. J., p. 239–255. · Zbl 0539.62110 |

[12] | Zehnwirth, B. (1985): Linear Filtering and Recursive Credibility Estimation, ASTIN-Bulletin 15, p. 19–36. |

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