Automatic, computer aided geometric design of free-knot, regression splines

A new algorithm for Computer Aided Geometric Design of least squares (LS) splines with variable knots, named GeDS, is presented. It is based on interpreting functional spline regression as a parametric B-spline curve, and on using the shape preserving property of its control polygon. The GeDS algorithm includes two major stages. For the first stage, an automatic adaptive, knot location algorithm is developed. By adding knots, one at a time, it sequentially "breaks" a straight line segment into pieces in order to construct a linear LS B-spline fit, which captures the "shape" of the data. A stopping rule is applied which avoids both over and under fitting and selects the number of knots for the second stage of GeDS, in which smoother, higher order (quadratic, cubic, etc.) fits are generated. The knots appropriate for the second stage are determined, according to a new knot location method, called the averaging method. It approximately preserves the linear precision property of B-spline curves and allows the attachment of smooth higher order LS B-spline fits to a control polygon, so that the shape of the linear polygon of stage one is followed. The GeDS method produces simultaneously linear, quadratic, cubic (and possibly higher order) spline fits with one and the same number of B-spline regression functions. The GeDS algorithm is very fast, since no deterministic or stochastic knot insertion/deletion and relocation search strategies are involved, neither in the first nor the second stage. Extensive numerical examples are provided, illustrating the performance of GeDS and the quality of the resulting LS spline fits. The GeDS procedure is compared with other existing variable knot spline methods and smoothing techniques, such as SARS, HAS, MDL, AGS methods and is shown to produce models with fewer parameters but with similar goodness of fit characteristics, and visual quality

Bootstrap Estimation of the Predictive Distributions of Reserves using Paid and Incurred Claims

This paper presents a bootstrap approach to estimate the prediction distributions of reserves produced by the Munich chain ladder (MCL) model. The MCL model was introduced by Quarg and Mack (2004) and takes into account both paid and incurred claims information. In order to produce bootstrap distributions, this paper addresses the application of bootstrapping methods to dependent data, with the consequence that correlations are considered. Numerical examples are provided to illustrate the algorithm and the prediction errors are compared for the new bootstrapping method applied to MCL and a more standard bootstrapping method applied to the chain-ladder technique

Entropy, longevity and the cost of annuities

This paper presents an extension of the application of the concept of entropy to annuity costs. Keyfitz (1985) introduced the concept of entropy, and analysed this in the context of continuous changes in life expectancy. He showed that a higher level of entropy indicates that the life expectancy has a greater propensity to respond to a change in the force of mortality than a lower level of entropy. In other words, a high level of entropy means that further reductions in mortality rates would have an impact on measures like life expectancy. In this paper, we apply this to the cost of annuities and show how it allows the sensitivity of the cost of a life annuity contract to changes in longevity to be summarized in a single figure index

Generalized Log-Normal Chain-Ladder

We propose an asymptotic theory for distribution forecasting from the log normal chain-ladder model. The theory overcomes the difficulty of convoluting log normal variables and takes estimation error into account. The results differ from that of the over-dispersed Poisson model and from the chain-ladder based bootstrap. We embed the log normal chain-ladder model in a class of infinitely divisible distributions called the generalized log normal chain-ladder model. The asymptotic theory uses small $\sigma$ asymptotics where the dimension of the reserving triangle is kept fixed while the standard deviation is assumed to decrease. The resulting asymptotic forecast distributions follow t distributions. The theory is supported by simulations and an empirical application

Modelling Claims Run-off with Reversible Jump Markov Chain Monte Carlo Methods

In this paper we describe a new approach to modelling the development of claims run-off triangles. This method replaces the usual adhoc practical process of extrapolating a development pattern to obtain tail factors with an objective procedure. An example is given, illustrating the results in a practical context, and the WinBUGS code is supplied

Double Chain Ladder

By adding the information of reported count data to a classical triangle of reserving data, we derive a suprisingly simple method for forecasting IBNR and RBNS claims. A simple relationship between development factors allows to involve and then estimate the reporting and payment delay. Bootstrap methods provide prediction errors and make possible the inference about IBNR and RBNS claims, separately

Double Chain Ladder and Bornhuetter-Ferguson

In this article we propose a method close to Double Chain Ladder (DCL) introduced by Martínez-Miranda, Nielsen, and Verrall (2012a). The proposed method is motivated by the potential lack of stability of the DCL method (and of the classical Chain ladder method [CLM] itself). We consider the implicit estimation of the underwriting year inflation in the CLM method and the explicit estimation of it in DCL. This may represent a weak point for DCL and CLM because the underwriting year inflation might be estimated with significant uncertainty. A key feature of the new method is that the underwriting year inflation can be estimated from the less volatile incurred data and then transferred into the DCL model. We include an empirical illustration that illustrates the differences between the estimates of the IBNR and RBNS cash flows from DCL and the new method. We also apply bootstrap estimation to approximate the predictive distributions

Geometrically Designed Variable Knot Splines in Generalized (Non-)Linear Models

In this paper we extend the GeDS methodology, recently developed by Kaishev et al. (2016) for the Normal univariate spline regression case, to the more general GNM (GLM) context. Our approach is to view the (non-)linear predictor as a spline with free knots which are estimated, along with the regression coefficients and the degree of the spline, using a two stage algorithm. In stage A, a linear (degree one) free-knot spline is fitted to the data applying iteratively re-weighted least squares. In stage B, a Schoenberg variation diminishing spline approximation to the fit from stage A is constructed, thus simultaneously producing spline fits of second, third and higher degrees. We demonstrate, based on a thorough numerical investigation that the nice properties of the Normal GeDS methodology carry over to its GNM extension and GeDS favourably compares with other existing spline methods. The proposed GeDS GNM(GLM) methodology is extended to the multivariate case of more than one independent variable by utilizing tensor product splines and their related shape preserving variation diminishing property