Nonparametric Regression and Generalized Linear Models: A by Peter J. Green, Bernard W. Silverman (auth.)

By Peter J. Green, Bernard W. Silverman (auth.)

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The first key result in the development of an economical way of calculating the cross-validation score is as follows. 5) where g is the spline smoother calculated from the full data set { (t;, Y;)} with smoothing parameter a. This theorem shows that, provided the diagonal entries A,;(a) are known, the cross-validation score can be calculated from the residuals Y; - g(t,) about the spline smoother calculated from the full data set. Therefore no additional smoothing problems have to be solved. However, it would appear from (3.

A refinement of the approximation near the boundary of the interval on which the data lie (see Silverman, 1984a) shows that the weight function is distorted there, though its exponential decay away from the boundary is not affected. • Altering the smoothing parameter a affects the local bandwidth in the same multiplicative way everywhere. Note that the local bandwidth is proportional to the fourth root of a. We should not be surprised if the appropriate numerical value of a varies widely between different problems, particularly if the scale of the design variable is different.

The B-splines form a set of natural cubic splines that are non-negative and have only limited support: for 3 ::; j ::; q- 2 the function /31 is zero outside (sj-2. s1+2), whilst f3t, /32, /3q-! and /3q are similar, but linear outside (st. Sq). Another possible approach, particularly appropriate if the function g is naturally required to be periodic, is to expand g in terms of a basis of trigonometric functions. 1 Details of the calculations We can now set out the details of the calculations required for the basis functions approach.

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