Essays on Boundaries Effects and Practical Considerations for Univariate Graduation of Mortality by Local Likelihood Models - Tomas, Julien (Assurances et gestion des risques / Insurance and Risk Management, 2012)
Local regression is a popular form of non-parametric regression, combining excellent theoretical properties with conceptual simplicity and flexibility to find structure in many datasets. Local regression smoothers fit low-order polynomials locally in the points surrounding a target point. The estimate, at each target point, is a weighted mean taken from the polynomial with observations close to the target point receiving the largest weights. Unfortunately this simplicity has flaws. At the boundary, the weight function is asymmetric and the estimate may have substantial bias. Bias can be a problem if the regression function has relatively high curvature in the boundary. It leads to a disturbing nuisance affecting applications as well as global measures of performance of the estimators like mean squared error or deviations between the true curve and estimated curve.In this article, we consider the alleviation of this boundary problem for the context of univariate graduation of mortality by local likelihood models. We consider three specifc treatments to reduce the impact of these boundary effects including symmetric and asymmetric weight systems. We analyze local statistical properties of smoothers subject to an a priori fixed bandwidth restriction. The weighting systems of these estimators depend on smoothing parameters that traditionally are estimated by means of data dependent optimization criteria. However by imposing to all of them the condition of a fixed bandwidth, we can measure the performance of each smoother and study where the contribution to these criteria are coming from the design space. Apart from statistical considerations, the choice of the parameters could be refined by taking into account the nature of the risk considered. The results are compared to the Whittaker-Henderson model for which it is not necessary to give specific treatment at the boundary.