The Annals of Statistics

Polynomial splines and their tensor products in extended linear modeling: 1994 Wald memorial lecture

Mark H. Hansen, Charles Kooperberg, Young K. Truong, and Charles J. Stone

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Analysis of variance type models are considered for a regression function or for the logarithm of a probability function, conditional probability function, density function, conditional density function, hazard function, conditional hazard function or spectral density function. Polynomial splines are used to model the main effects, and their tensor products are used to model any interaction components that are included. In the special context of survival analysis, the baseline hazard function is modeled and nonproportionality is allowed. In general, the theory involves the $L_2$ rate of convergence for the fitted model and its components. The methodology involves least squares and maximum likelihood estimation, stepwise addition of basis functions using Rao statistics, stepwise deletion using Wald statistics and model selection using the Bayesian information criterion, cross-validation or an independent test set. Publicly available software, written in C and interfaced to S/S-PLUS, is used to apply this methodology to real data.

Article information

Ann. Statist., Volume 25, Number 4 (1997), 1371-1470.

First available in Project Euclid: 9 September 2002

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Primary: 62G07: Density estimation
Secondary: 62J12: Generalized linear models

ANOVA density estimation generalized additive models generalized linear models least squares logistic regression optimal rates of convergence proportional hazards model spectral estimation survival analysis


Stone, Charles J.; Hansen, Mark H.; Kooperberg, Charles; Truong, Young K. Polynomial splines and their tensor products in extended linear modeling: 1994 Wald memorial lecture. Ann. Statist. 25 (1997), no. 4, 1371--1470. doi:10.1214/aos/1031594728.

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