Asymptotic Integrated Mean Square Error Using Least Squares and Bias Minimizing Splines

Let Sᵈₖ be the set of dth order splines on 0, 1 having k knots ₁ < ₂ < ₖ. 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 ₁ < < ₖ and (iii) number of knots.