The L2 Rate of Convergence for Event History Regression with Time‐dependent Covariates

Consider repeated events of multiple kinds that occur according to a right-continuous semi-Markov process whose transition rates are influenced by one or more time-dependent covariates. 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 L2 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, where the logarithms of the intensities are approximated by functions having the form of a specified sum of a constant term, main effects (functions of one variable), and interaction terms (functions of two or more variables). 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.

[1]  Vicki A. Freedman,et al.  Event History Analysis , 1991 .

[2]  L. Schumaker.  Spline Functions: Basic Theory , 1981 .

[3]  C. J. Stone,et al.  Hazard Regression , 2022 .

[4]  Young K. Truong,et al.  The L2 rate of convergence for hazard regression , 1995 .

[5]  Young K. Truong,et al.  Polynomial splines and their tensor products in extended linearmodeling , 1997 .

[6]  Young K. Truong,et al.  Polynomial splines and their tensor products in extended linear modeling: 1994 Wald memorial lecture , 1997 .

[7]  J. Kalbfleisch,et al.  The Statistical Analysis of Failure Time Data , 1980 .

[8]  Finbarr O'Sullivan,et al.  Nonparametric Estimation in the Cox Model , 1993 .

[9]  S. Sardy,et al.  Triogram Models , 1996 .

[10]  A. Karr,et al.  Nonparametric Survival Analysis with Time-Dependent Covariate Effects: A Penalized Partial Likelihood Approach , 1990 .

[11]  Robert Gray,et al.  Flexible Methods for Analyzing Survival Data Using Splines, with Applications to Breast Cancer Prognosis , 1992 .

[12]  C. J. Stone,et al.  Optimal Global Rates of Convergence for Nonparametric Regression , 1982 .

[13]  A. Hamerle.  Multiple-spell regression models for duration data , 1989 .

[14]  A. Takemura.  Tensor Analysis of ANOVA Decomposition , 1983 .

[15]  Niels Keiding,et al.  Statistical Models Based on Counting Processes , 1993 .

[16]  D. Harrington,et al.  Regression Splines in the Cox Model with Application to Covariate Effects in Liver Disease , 1990 .

[17]  Joseph G. Voelkel,et al.  Nonparametric Inference for a Class of Semi-Markov Processes with Censored Observations , 1984 .

[18]  C. Kooperberg,et al.  Hazard regression with interval-censored data. , 1997, Biometrics.

[19]  Trevor Hastie,et al.  Polynomial splines and their tensor products in extended linear modeling. Discussion and rejoinder , 1997 .

[20]  Paul D. Allison,et al.  Event History Analysis : Regression for Longitudinal Event Data , 1984 .

[21]  Karl Ulrich Mayer,et al.  Event History Analysis in Life Course Research , 1990 .

[22]  C. J. Stone,et al.  The Use of Polynomial Splines and Their Tensor Products in Multivariate Function Estimation , 1994 .

[23]  Jianhua Z. Huang.  Projection estimation in multiple regression with application to functional ANOVA models , 1998 .