Abstract
Although both over-dispersed Poisson and log-normal chain-ladder models are popular in claim reserving, it is not obvious when to choose which model. Yet, the two models are obviously different. While the over-dispersed Poisson model imposes the variance to mean ratio to be common across the array, the log-normal model assumes the same for the standard deviation to mean ratio. Leveraging this insight, we propose a test that has the power to distinguish between the two models. The theory is asymptotic, but it does not build on a large size of the array and, instead, makes use of information accumulating within the cells. The test has a non-standard asymptotic distribution; however, saddle point approximations are available. We show in a simulation study that these approximations are accurate and that the test performs well in finite samples and has high power.
Highlights
Which is the better chain-ladder model for claim reserving: over-dispersed Poisson or log-normal?While the expert may have a go-to model, the answer should be informed by the data
We develop a test that can distinguish between over-dispersed Poisson and log-normal data generating processes, both of which have a long history in claim reserving
Building on the encompassing literature, we find the distribution of the over-dispersed Poisson model estimators under a generalized log-normal data generating process and vice versa
Summary
Which is the better chain-ladder model for claim reserving: over-dispersed Poisson or log-normal?. Harnau and Nielsen (2017) developed a theory that gives the over-dispersed Poisson model a rigorous statistical footing They propose an asymptotic framework based on infinitely-divisible distributions that keeps the dimension of the data array fixed and instead builds on large cell means. Kuang and Nielsen (2018) proposed a theory that includes closed-form distribution forecasts for cell sums, such as the reserve, in the log-normal model, remedying one of its drawbacks. In this application, dropping a clearly needed calendar effect turns the results of the encompassing tests upside down Taking these insights into account, we implement a testing procedure that makes use of a whole range of recent results: deciding between the over-dispersed Poisson and generalized log-normal model, evaluating misspecification and testing for the need for a calendar effect. These include further misspecification tests, a theory for the bootstrap and empirical studies assessing the usefulness of the recent theoretical developments in applications
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