Estimation Of Actuarial Loss Functions And The Tail Index Using Transformations In Kernel Density Estimation

In this paper we concentrate on the estimation of loss functions using nonparametric methods. We focus on the parametric transformation approach to kernel smoothing introduced by Wand, Marron and Ruppert (1991) and compare it with the standard kernel estimator and the multiplicative bias correction method (Hjort and Glad, 1995 and Jones, Linton and Nielsen, 1995). We advocate in this paper that the transformation method behaves excellently when it comes to estimating actuarial and financial loss functions. It is a very suitable approach when estimating functions with a lognormal type of shape. Loss functions have typically one mode for the low loss values and then a long heavy tail (Klugman, Panjer and Willmot, 1998),. We show by a simulation study that the method is able to estimate all three possible kind of tails, as defined in Embrechts, Kl*ppelberg and Mikosch (1999), namely the Fr’chet type, the Weibull type and the Gumbel type. This makes this method extremely powerful specially for actuaries at all levels, i.e. the non-life actuary calculating the risk of his auto claims, when a life actuary consider the risk of a group life insurance, or when these actuaries consider the relevant price for a reinsurance contract, where the reinsurer takes over the risk corresponding to the tail of the distribution. Besides, when the actuary calculates the total risk of a portfolio, perhaps using the famous recursion formula of Panjer (1981) or some more recent generalisation of this formula, then the loss function of the individual claims is needed. We show that our version of the transformation principle of Wand, Marron and Ruppert (1991) is able to estimate the risk of a heavy-tailed distribution beyond the data. A heavy-tailed distribution is here of Pareto shape. When we use our non- parametric estimator to fit the closest possible Pareto distribution in the tail we see, that at least for the estimator we tried out in our simulations, then our method performs much better than the Hill estimator (Hill, 1975), that is widely used in actuarial science and finance, see Embrechts, Kl*ppelberg and Mikosch (1997, p. 330) and Danielson and Vries (1997). On top of this, our estimation technique does not have to bother about where the tail begins such as the Hill estimator has to. This is often a difficult question and fatal errors leading to massive economic losses may happen. The beginning of the tail can be estimated by a methodology that can be compared to the trade-off between bias and variance in kernel density estimation, but the only (to our knowledge) published method on this (see Hall,1990) involves sub-bootstrapping, that is hard to understand and implement for the practitioner (see, Danielson and Vries,1997). Recently another method has appeared, namely the semiparametric robust estimator of Feuerverger and Hall (1999). Compared to this estimator, our proposal has the advantage of being directly linked to the non-parametrically estimated loss function. The connection between the loss-function and the tail index is therefore immediate in our estimator of the tail index. We consider this to be a considerable competitive advantage of our method when it comes to the application in the fields of actuarial and financial loss functions. We believe that our method is a big jump forward for the practitioner, whether it is an actuary or a financial analyst, to get a quick and easily understood estimator of all the things he needs to know about his loss distribution, using a moderate computational effort. The organisation of the paper is as follows: in section one there is an introduction to the estimation of loss functions. Section two presents the theoretical properties when using a fixed transformation in kernel density estimation, and afterwards, section three explains the transformation algorithm that is proposed in this paper. Section four is devoted to the applications, two examples of actuarial loss function estimation are presented. They analyse automobile claim data. Finally, section five describes the results of a large simulation study that shows the performance of the different kernel methods when estimating loss function curves and their tail index, i.e. when describing their behaviour in the tail.