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## Large infinitary languages: model theory by Dickmann M. A. By Dickmann M. A.

Dickmann M.A. huge infinitary languages (1975)(ISBN 0444106227)

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Example text

15) where, now, A( a) is the hat matrix associated with spline smoothing with smoothing parameter a. The equivalent degrees of freedom for noise increase from 0 when a= 0 (interpolation, hat matrix A the identity) to n - 2 when a = oo (linear regression). It follows immediately from the definitions that the GCV score can be written in the form GCV(a) = n x residual sum of squares (equivalent degrees of freedom)2 · The question of definition of equivalent degrees of freedom has been discussed at greater length by Buja, Hastie and Tibshirani (1989); see also Hastie and Tibshirani (1990, Appendix B).

The effect of the choice of smoothing parameter being only on lower order terms. 18) has asymptotic mean square error ~a4 n- 1 , almost twice as large. 17) has asymptotic mean square error 3a4n- 1 • The intuitive reason for the higher mean square error of the estimators based on local differencing is that they eliminate bias by placing emphasis on high-frequency effects. They do have the advantage of not requiring any choice of smoothing parameter, and having much smaller bias than vanishingly small in large samples.

This will give exactly the penalized sum of squares S(g), with smoothing parameter equal to the Lagrange multiplier a. 41) for a particular C, it is necessary to search on a until the optimizing function g satisfies the constraint Jg" 2 = C. Since Jg" 2 can easily be shown to be a decreasing function of a, this search is not prohibitively expensive since it can be carried out by a binary search procedure. However, it is unusual for the value C to be directly meaningful, and the usual practical approach in statistics is to regard the Lagrange multiplier a as the controlling parameter for the smoothing method.

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