Deficiencies in the Theory of Free-Knot and Variable-Knot Spline Graduation

Abstract

This paper revisits the theory and practical implementation of graduation of mortality rates using spline functions, and in particular, variable-knot cubic spline graduation. The paper contrasts the actuarial literature on free-knot splines with the mathematical literature. It finds that the practical difficulties of implementing free-knot spline graduation are not recognised in the actuarial literature reviewed. The paper also revisits the results of the graduation of the English Life Tables no. 14 (ELT 14) experience for male lives using a \'multistart\' optimisation approach for the free-knot graduation. Application of this technique results in the finding that the chi-squared values reported in the ELT 14 graduation for male lives for 10, 11 and 12 knots were not optimal values. The multistart optimisation results appear to show that McCutcheon\'s t-statistic, which is used in variable-knot spline graduation to select the optimal number of knots, may not in fact result in an optimal choice. Free-knot spline graduation should be used with caution and variable-knot spline graduation, in the form that employs McCutcheon\'s t-statistic, should not be used. KEYWORDS Mortality; graduation; splines; free-knot spline; variable-knot spline


(South African Actuarial Journal: 2002 2: 69-82)