# Geometrically Designed, Variable Knot Regression Splines: Asymptotics and Inference

Kaishev, V. K., Dimitrova, D. S., Haberman, S. and Verrall, R. J. (2006). Geometrically Designed, Variable Knot Regression Splines: Asymptotics and Inference (Statistical Research Paper No. 28). Cass Business School, City University, London.

## Abstract

A new method for Computer Aided Geometric Design of least squares (LS) splines with variable knots, named GeDS, is presented. It is based on the property that the spline regression function, viewed as a parametric curve, has a control polygon and, due to the shape preserving and convex hull properties, closely follows the shape of this control polygon. The latter has vertices, whose x-coordinates are certain knot averages, known as the Greville sites and whose y-coordinates are the regression coefficients. Thus, manipulation of the position of the control polygon and hence of the spline curve may be interpreted as estimation of its knots and coefficients. These geometric ideas are implemented in the two stages of the GeDS estimation method. In stage A, a linear LS spline fit to the data is constructed, and viewed as the initial position of the control polygon of a higher order (n>2) smooth spline curve. In stage B, the optimal set of knots of this higher order spline curve is found, so that its control polygon is as close to the initial polygon of stage A as possible and finally, the LS estimates of the regression coefficients of this curve are found. To implement stage A, an automatic adaptive knot location scheme for generating linear spline fits is developed. At each step of stage A, a knot is placed where a certain bias dominated measure is maximal. This stage is equipped with a novel stopping rule which serves as a model selector. The optimal knots defined in stage B ensure that the higher order spline curve is nearly a variation diminishing (i.e., shape preserving) spline approximation to the linear fit of stage A. Error bounds for this approximation are derived in Kaishev et al. (2006). The GeDS method produces simultaneously linear, quadratic, cubic (and possibly higher order) spline fits with one and the same number of B-spline regression functions. Large sample properties of the GeDS estimator are also explored, and asymptotic normality is established. Asymptotic conditions on the rate of growth of the knots with the increase of the sample size, which ensure that the bias is of negligible magnitude compared to the variance of the GeD estimator, are given. Based on these results, pointwise asymptotic confidence intervals with GeDS are also constructed and shown to converge to the nominal coverage probability level for a reasonable number of knots and sample sizes.

Publication Type: Report spline regression, B-splines, Greville abscissae, variable knot splines, control polygon, asymp- totic confidence interval, coverage probability, asymptotic normality H Social Sciences > HA StatisticsH Social Sciences > HD Industries. Land use. Labor > HD61 Risk ManagementH Social Sciences > HG Finance Cass Business School > Actuarial Science & InsuranceCass Business School > Actuarial Science & Insurance > Statistical Research Reports http://openaccess.city.ac.uk/id/eprint/16797
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