In-sample forecasting with local linear survival densities

In this paper, in-sample forecasting is defined as forecasting a structured density to sets where it is unobserved. The structured density consists of one-dimensional in-sample components that identify the density on such sets. We focus on the multiplicative density structure, which has recently been seen as the underlying structure of non-life insurance forecasts. In non-life insurance, the in-sample area is defined as one triangle and the forecasting area as the triangle which, added to the first triangle, completes a square. In recent approaches, two one-dimensional components are estimated by projecting an unstructured two-dimensional density estimator onto the space of multiplicatively separable functions. We show that time-reversal reduces the problem to two one-dimensional problems, where the one-dimensional data are left-truncated and a one-dimensional survival density estimator is needed. We then use the local linear density smoother with weighted crossvalidated and do-validated bandwidth selectors. Full asymptotic theory is provided, with and without time-reversal. Finite-sample studies and an application to non-life insurance are included.