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Unformatted text preview: The Annals of Statistics 1995, Vol . 23, No . 6, 1865 ] 1895 THE 1994 NEYMAN MEMORIAL LECTURE SMOOTHING SPLINE ANOVA FOR EXPONENTIAL FAMILIES, WITH APPLICATION TO THE WISCONSIN EPIDEMIOLOGICAL STUDY OF DIABETIC RETINOPATHY 1 B Y G RACE W AHBA , 2 Y UEDONG W ANG , 3 C HONG G U , 4 R ONALD K LEIN 5 AND B ARBARA K LEIN 6 University of Wisconsin ] Madison, University of Michigan, Purdue University, University of Wisconsin ] Madison and University of Wisconsin ] Madison Let y , i s 1, . . . , n , be independent observations with the density of i . w . . x y of the form h y , f s exp y f y b f q c y , where b and c are i i i i i i i given functions and b is twice continuously differentiable and bounded .. . 1 . d . away from 0 . Let f s f t i , where t s t , . . . , t g T m ??? m T s T , i 1 d the T a . are measurable spaces of rather general form and f is an unknown function on T with some assumed smoothness properties . . 4 . Given y , t i , i s 1, . . . , n , it is desired to estimate f t for t in some i region of interest contained in T . We develop the fitting of smoothing . . spline ANOVA models to this data of the form f t s C q f t q a a a . f t , t q ??? . The components of the decomposition satisfy side a- b a b a b conditions which generalize the usual side conditions for parametric ANOVA . The estimate of f is obtained as the minimizer, in an appropriate . . . function space, of L y , f q l J f q l J f q ??? , a a a a a- b a b a b a b . . where L y , f is the negative log likelihood of y s y , . . . , y 9 given f , 1 n the J , J , . . . are quadratic penalty functionals and the ANOVA decom- a a b position is terminated in some manner . There are five major parts re- . quired to turn this program into a practical data analysis tool: 1 methods for deciding which terms in the ANOVA decomposition to include model . . selection , 2 methods for choosing good values of the smoothing parame- . ters l , l , . . . , 3 methods for making confidence statements concern- a a b . ing the estimate, 4 numerical algorithms for the calculations and, finally, . 5 public software . In this paper we carry out this program, relying on earlier work and filling in important gaps . The overall scheme is applied Received January 1995; revised May 1995 . 1 This work formed the basis for the Neyman Lecture at the 57th Annual Meeting of the Institute of Mathematical Statistics, Chapel Hill, North Carolina, June 23, 1994, presented by Grace Wahba . 2 Research supported in part by NIH Grant EY09946 and NSF Grant DMS-91-21003 . 3 Research supported in part by NIH Grants EY09446, P60-DK20572 and P30-HD18258 ....
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This note was uploaded on 08/07/2008 for the course ARCH 4015 taught by Professor Deegger during the Fall '08 term at Virginia Tech.
- Fall '08