Abstract
This paper shows that recent published mortality projections with unobserved exposure can be understood as structured density estimation. The structured density is only observed on a sub-sample corresponding to historical calendar times. The mortality forecast is obtained by extrapolating the structured density to future calendar times using that the components of the density are identified within sample. The new method is illustrated on the important practical problem of forecasting mesothelioma for the UK population. Full asymptotic theory is provided. The theory is given in such generality that it also introduces mathematical statistical theory for the recent continuous chain ladder model. This allows a modern approach to classical reserving techniques used every day in any non-life insurance company around the globe. Applications to mortality data and non-life insurance data are provided along with relevant small sample simulation studies.
Highlights
Let us assume that we have a structured density defined as a density that is a known function of one-dimensional densities, see Mammen and Nielsen (2003) for the equivalent definition of structured regression
The support with observations represents insurance claims until the current calendar time, and the support without observations represents future insurance claims. This forecast method has traditionally been called the chain ladder technique in actuarial science and the multiplicative density has been estimated as a structured histogram or equivalently from maximum likelihood assuming a multiplicative Poisson structure, see Wuthrich and Merz (2008) for and overview and Kuang et al (2009), Verrall et al (2010), Martınez-Miranda et al (2011, 2012, 2013a,b,c), for recent reformulations of classical chain ladder in mathematical statistical terms published in the actuarial literature
In this paper we propose to use our alternative approach based on structured non-parametric models and we will illustrate its power by applying it to mesothelioma mortality forecasts
Summary
Let us assume that we have a structured density defined as a density that is a known function of one-dimensional densities, see Mammen and Nielsen (2003) for the equivalent definition of structured regression. The support with observations represents insurance claims until the current calendar time, and the support without observations represents future insurance claims This forecast method has traditionally been called the chain ladder technique in actuarial science and the multiplicative density has been estimated as a structured histogram or equivalently from maximum likelihood assuming a multiplicative Poisson structure, see Wuthrich and Merz (2008) for and overview and Kuang et al (2009), Verrall et al (2010), Martınez-Miranda et al (2011, 2012, 2013a,b,c), for recent reformulations of classical chain ladder in mathematical statistical terms published in the actuarial literature. While these two applications rely on the multiplicative density structure, observations are available on very different underlying supports. All the calculations in the paper have been performed with R, R Development Core Team (2011)
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