proposal number: DMS-9504463 PI: Charles Stone Institution: UC Berkeley Title: Extending Linear Modeling with Splines An abstract for the project In many statistical models of current theoretical and practical interest, the log-likelihood function depends linearly on the one or more unknown functions. The theory and methodology of such models, referred to as extended linear models, are particularly tractable when the models are concave; that is, when the log-likelihood depends concavely on the unknown function or functions. Among the class of concave extended linear models are ordinary, logistic, probit, Poisson and other generalized linear models, multiple logistic regression (which is useful in multiple classification), hazard regression (for survival analysis), and models for the estimation of density and conditional density functions. In the context of such models, polynomial splines and their tensor products are natural building blocks for constructing finite-dimensional estimates of infinite-dimensional main effects and loworder interactions, and the resulting ANOVA decompositions provide an insightful tool for data analysis. In this research, the corresponding nonadaptive theory and closely related adaptive methodology will further be developed and refined in a variety of settings. In particular, an attempt will be made to establish uniform rates of convergence first in the regression context and then for other extended linear models, where the maximum likelihood estimates are intrinsically nonlinear. The theory for spectral density estimation in the context of a stationary time series will be extended to handle mixed spectra. The theory and methodology for survival analysis will be extended to include time-dependent covariates and directly to fit a flexible proportional hazards (Cox) model without modeling the dependence of the hazard function on time. The methodology and theory that have already been develo ped for extended linear models will be modified to handle neural spike train processes that occur in neurophysiology and event history analysis, which is commonly used in sociology and other social sciences. This research is part of the investigator's long-term research program in statistics, which began two decades ago. Prior to that time, the emphasis on statistics had been on parametric modeling, which involves fitting models with a fixed, finite number of unknown parameters. At that time, the investigator joined the small cadre of statisticians working in the field of nonparametric modeling. A half-dozen years later, the investigator further specialized his research program to what essentially amounts to the synthesis of parametric and nonparametric modeling; specifically, flexible parametric models are employed that have increasingly many parameters as more and more data become available. A decade ago, the investigator began another long-term project: teaching and simultaneously writing a textbook for an upper-division or graduate-level first course in probability and statistics in which the statistics portion is presented in an innovative manner suggested by this synthesis of parametric and nonparametric modeling. (The textbook will be published this summer.) This educational project, in turn, has had a catalytic effect on the investigator's research program. Moreover, in the direction of human resource development to improve the civil infrastructure, the investigator has used the positions of Teaching Assistant for the course and Research Assistant on previous NSF grants to recruit Ph. D. students, train them in teaching and in the investigator's research program, and place them in academic and industrial research positions. Recently one such former student, who currently has a tenure-track position at a leading American statistics department, assisted an American software company in winning a small SBIR contract to create a commercial software implementation of the state-of-the-art survival analysis methodology that was developed by the investigator and this and another former student. The resulting product should prove very useful in biotechnology. Another former student, who has a permanent position at one of America's leading research labs, has continued a methodological development initiated in his Ph. D. Dissertation and successfully applied this methodology to improve the quality of the IC manufacturing process of the Lab's parent company. Hopefully, the investigator's continued research and training of Ph. D. students and his related collaborations with former Ph. D. students will have more such worthwhile consequences. ??
