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Maximum Likelihood and Estimation Efficiency of the Chain Ladder

Published online by Cambridge University Press:  09 August 2013

Greg Taylor
Greg Taylor*
Affiliation:
Level 8, 30 Clarence Street, Sydney NSW 2000, Australia Centre for Actuarial Studies, Faculty of Economics and Commerce, University of Melbourne, Parkville VIC 3052, Australia School of Actuarial Studies, Australian School of Business, University of New South Wales, Kensington NSW 2033, Australia, Phone: 61 2 9249 2901, Fax: 61 2 9249 2999, E-Mail: [email protected]
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Abstract

The chain ladder is considered in relation to certain recursive and non-recursive models of claim observations. The recursive models resemble the (distribution free) Mack model but are augmented with distributional assumptions. The non-recursive models are generalisations of Poisson cross-classified structure for which the chain ladder is known to be maximum likelihood. The error distributions considered are drawn from the exponential dispersion family.

Each of these models is examined with respect to sufficient statistics and completeness (Section 5), minimum variance estimators (Section 6) and maximum likelihood (Section 7). The chain ladder is found to provide maximum likelihood and minimum variance unbiased estimates of loss reserves under a wide range of recursive models. Similar results are obtained for a much more restricted range of non-recursive models.

These results lead to a full classification of this paper's chain ladder models with respect to the estimation properties (bias, minimum variance) of the chain ladder algorithm (Section 8).

Keywords

Chain laddercross-classified modelcompletenessexponential dispersion familyloss reserveMack modelmaximum likelihoodminimum variance unbiased estimatornon-recursive modelover-dispersed Poissonrecursive modelsufficient statisticTweedie family
Type
Research Article
Information
ASTIN Bulletin: The Journal of the IAA , Volume 41 , Issue 1 , May 2011 , pp. 131 - 155
Copyright
Copyright © International Actuarial Association 2011

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