Let Sᵈₖ be the set of dth order splines on 0, 1 having k knots ξ₁ < ξ₂ ⋯ < ξₖ. We consider the estimation of a sufficiently smooth response function g, using n uncorrelated observations, by an element s of Sᵈₖ. For large n and k we have discussed the asymptotic behavior of the integrated mean square error (IMSE) for two types of estimators: (i) the least squares estimator and (ii) a bias minimizing estimator. The asymptotic expression for IMSE is minimized with respect to three variables. (i) the allocation of observation (ii) the displacement of knots ξ₁ < ⋯ < ξₖ and (iii) number of knots.
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Agarwal et al. (1980) studied this question.