Estimation error and bootstrapping in the chain-ladder model of Mack
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Estimation error and bootstrapping in the chain‑ladder model of Mack Alois Gisler1 Received: 12 May 2020 / Revised: 24 June 2020 / Accepted: 25 June 2020 © EAJ Association 2020
Abstract In 2006 there was quite some discussion on how to estimate the conditional estimation error in the chain-ladder (CL) model of Mack. Buchwalder, Bühlmann, Merz and Wüthrich (BBMW) (ASTIN Bull 36(2):521–542, 2006) proposed another estimator than the one derived by Mack (ASTIN Bull 23(2):213–225, 1993). These two estimators are also found in a broader context by new authors in recent papers. In the present paper we examine the theoretical properties of the two estimators and come to the conclusion that the BBMW estimator has some major deficiencies compared with the Mack estimator. It takes much less information of the observed triangle into account, the averaging is done over inappropriate sets and it does not properly fit to the Mack CL-model. Keywords Claims reserving · Distribution free chain-ladder model · conditional mean square error of prediction · Mack’s formula · BBMW-formula
1 Introduction In 1993 Mack [9] presented the stochastic CL-model and estimators for the conditional mean-square error of prediction (msep). The msep is decomposed into the conditional process variance (conditional variance of the ultimate claim) plus the conditional estimation error (squared deviation of the CL-forecast of the ultimate claim from its conditional expected value). Mack derived estimators for each of these two components. The Mack-estimator for the process variance was never questioned in the actuarial literature. In 2006, however, there was quite some discussion on how to estimate the estimation error (see [1, 2, 4, 10]). In [1] BBMW proposed another estimator than the one of Mack [9]. Their estimator was found by bootstrapping arguments. They also showed that the Mack * Alois Gisler [email protected] 1
Department of Mathematics, RiskLab, ETH Zurich, Rämistrasse 101, 8092 Zurich, Switzerland
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estimator is a first-order Taylor approximation of their estimator. Mack, Quarg and Braun [10] proved that the squares of neighbouring CL factors are conditionally negatively correlated and concluded that therefore the BBMW would overestimate the estimation error. However this conclusion is questionable (see Appendix). In a Bayesian context the estimation error is a measurable function of the observations and clearly defined. For this reason the present author considered in [4] two Bayesian CL-models with non-informative priors. In the normal-normal model the Bayes estimator of the estimation error coincides with the BBMW formula and in the Gamma-uniform model it is close to the BBMW formula but does only exist if the observations in the claims triangle fulfil certain conditions. The discussion in 2006 finished with a rather general one-page note of BBMW [2] stating that the right form of resampling is a fundamental question to be answered for all methods of stochastic claims reserving. The differenc
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