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Uniform error bounds involving logspline models. (English) Zbl 0729.62035
Probability, statistics, and mathematics, Pap. in Honor of Samuel Karlin, 335-355 (1989).
[For the entire collection see Zbl 0679.00013.]
Splines are of increasing importance in statistical theory and methodology. In particular, the author and C.-Y. Koo [Function estimates, Proc. Conf. Arcata/Calif. 1985, Contemp. Math. 59, 1-15 (1986; Zbl 0618.62038)] considered exponential families of densities in which the logarithm of the density is a spline. Such exponential families are the subject of the present paper, as are corresponding exponential response models. In each context we use an extension of a key result of C. de Boor [Math. Comput. 30, 765-771 (1976; Zbl 0345.65004)] to obtain a bound on the $$L_{\infty}$$ norm of the approximation error associated with maximizing the associated expected log-likelihood.

##### MSC:
 62G07 Density estimation 41A15 Spline approximation