The \(L_{2}\) rate of convergence for event history regression.

*(English)*Zbl 0927.62087Event history analysis is closely related to survival analysis and heavily used in economics, sociology and other social sciences. Typically, an “event history” can be characterized by a multi-state stochastic process moving among a finite number of states as time progresses with events corresponding to transitions between states. Censoring and time-dependent covariates are the two typical features among the event history data that create difficulties in the statistical analysis.

The authors consider events that occur according to a right-continuous semi-Markov process. Transitions are allowed from one state to itself. For example, the transitions could be job changes and the states could be employer types. The logarithms of the intensities of the transitions from one state to another are modelled as members of a linear function space, which may be finite- or infinite-dimensional. Maximum likelihood estimates are used, where the maximizations are taken over suitably chosen finite-dimensional approximating spaces. It is shown that the \(L_{2}\)-rates of convergence of the maximum likelihood estimates are determined by the approximation power and dimension of the approximating spaces.

The theory is applied to a functional ANOVA model. It is shown that the curse of dimensionality can be ameliorated if only main effects and low-order interactions are considered in functional ANOVA models.

The authors consider events that occur according to a right-continuous semi-Markov process. Transitions are allowed from one state to itself. For example, the transitions could be job changes and the states could be employer types. The logarithms of the intensities of the transitions from one state to another are modelled as members of a linear function space, which may be finite- or infinite-dimensional. Maximum likelihood estimates are used, where the maximizations are taken over suitably chosen finite-dimensional approximating spaces. It is shown that the \(L_{2}\)-rates of convergence of the maximum likelihood estimates are determined by the approximation power and dimension of the approximating spaces.

The theory is applied to a functional ANOVA model. It is shown that the curse of dimensionality can be ameliorated if only main effects and low-order interactions are considered in functional ANOVA models.

Reviewer: A.V.Swishchuk (Kyïv)

##### MSC:

62M10 | Time series, auto-correlation, regression, etc. in statistics (GARCH) |

62J10 | Analysis of variance and covariance (ANOVA) |

62P20 | Applications of statistics to economics |

62P25 | Applications of statistics to social sciences |

62G05 | Nonparametric estimation |

60K15 | Markov renewal processes, semi-Markov processes |