The tails in moving average graduation

Moving average techniques provide a flexible tool for the smoothing of curves of vital rates and other curves based on sequences of initial parameter estimates. These techniques are very robust, but they also give rise to a perennial problem which is unique to moving averages: They leave the curve tails ungraduated. The smoothing of the tails must be based on considerations additional to those which guided the choice of a moving average formula. This paper offers a general solution to this problem and provides an account of our experience with its practical use in extensive empirical experiments. Our method can take care of heteroscedasticity, but the graduations seem little influenced by it. This surprising finding may help explain the practical success of classical moving average formulas, whose derivation has taken no real account of variance considerations. Our findings also suggest that minimum-R 1 solutions may have better properties than the min-R 3 or min-R 4 moving averages usually recommended in the actuarial literature. Furthermore, when we are not interested in periodic components of the true curve, formulas exact for straight lines may often be more adequate than the conventional reliance on cubics.