Comments on Taylor's see-saw approach to claims reserving
Abstract Taylor (1981) introduces the See-Saw (SS) model for claims reserving in order to make allowance for speed of finalization. The model is applied to live data given in Taylor (1981) and there is prima facie evidence to suggest that it does well, especially in the light of comparisons of actual versus expected payments for each payment year. The purpose of the present paper is to demonstrate that from the point of view of operational forecasting, which is the object of the claims reserving exercise, the fitting of the linear SS can be improved upon. Moreover, we employ the SS as a vehicle for indicating the kind of validation tests that ought to be carried out once the parameters of a proposed model have been estimated from the data. Essentially, we indicate how the properties of the residuals may be used for diagnostic checking of the model. Many researchers involved in the claims reserving area are of the view that the data are extremely noisy especially if the model put forward only explains a small proportion of the total variation. Why not test whether this is the case? It turns out that the particular linear SS used by Taylor does not explain the signal accurately so that the balance of the variation is not entirely due to noise.
Operational time and in-sample density forecasting
In this paper we consider a new structural model for in-sample density forecasting. In-sample density forecasting is to estimate a structured density on a region where data are observed and then re-use the estimated structured density on some region where data are not observed. Our structural assumption is that the density is a product of one-dimensional functions with one function sitting on the scale of a transformed space of observations. The transformation involves another unknown one-dimensional function, so that our model is formulated via a known smooth function of three underlying unknown one-dimensional functions. We present an innovative way of estimating the one-dimensional functions and show that all the estimators of the three components achieve the optimal one-dimensional rate of convergence. We illustrate how one can use our approach by analyzing a real dataset, and also verify the tractable finite sample performance of the method via a simulation study.
Normed units in abelian group rings
We compute explicitly the isomorphic structure of the normalized unit group of an abelian group ring under some minimal natural restrictions on the group basis and the coefficient ring. This enlarges affirmations due to Chatzidakis-Pappas (J. London Math. Soc., 1991) and Mollov (Publ. Math. Debrecen, 1971).