Abstract: Structures such as independence of random variables in probability densities and hazard proportionality in covariate dependent hazard functions have important interpretations in statistical analysis. Such structures can be characterized by term eliminations from an analysis of variance (ANOVA) decomposition in log density or log hazard. Nonparametric estimation of these functions with an ANOVA decomposition built in can be achieved by using tensor product splines in a penalized likelihood approach. In this article, a feasible algorithm with automatic multiple smoothing parameters is described to implement this approach, and examples are presented to illustrate some applications of the technique. For density estimation, a novel feature is the possibility of assessing/enforcing independence when data are truncated to a non rectangular domain. For hazard estimation, models more general than but reducible to proportional hazard models are available, and model terms are estimated simultaneously via penalized full likelihood.
Key words and phrases: Analysis of variance decomposition, penalized likelihood method, performance-oriented iteration, smoothing parameter, tensor product spline.