Fitting Moments with Weights

This note investigates ways to fit individual claim loss data to a prior known "'underlying severity level" by adjusting the relative importance, or weight, assigned to each claim. Here, "'underlying severity level" is measured by the weighted mean cost per case. The paper also generalizes the approach to accommodate fitting higher moments o f the loss distribution, especially the variance. It establishes the existence of an optimal reweighing, but whose calculation may be too difficult for practical application. To address this, the paper describes two easier calculations, one designed to fit only the mean and another to fit both mean and variance.