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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Abstract

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

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

Dates
First available in Project Euclid: 9 September 2002

Permanent link to this document
http://projecteuclid.org/euclid.aos/1031594728

Digital Object Identifier
doi:10.1214/aos/1031594728

Mathematical Reviews number (MathSciNet)
MR1463561

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

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

Citation

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. http://projecteuclid.org/euclid.aos/1031594728.


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