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**Mixtures of tails in clustered automobile collision claims.**
*(English)*
Zbl 0864.62071

Summary: Knowledge of the tail shape of claim distributions provides important actuarial information. This paper dicusses how two techniques commonly used in assessing the most appropriate underlying distribution can be usefully combined. The maximum likelihood approach is theoretically appealing since it is preferable to many other estimators in the sense of best asymptotic normality. Likelihood based tests are, however, not always capable to discriminate among non-nested classes of distributions. Extremal value theory offers an attractive tool to overcome this problem. It shows that a much larger set of distributions is nested in their tails by the so-called tail parameter.

This paper shows that both estimation strategies can be usefully combined when the data generating process is characterized by strong clustering in time and size. We find that the extreme value theory is a useful starting point in detecting the appropriate distribution class. Once that has been achieved, the likelihood-based EM-algorithm is proposed to capture the clustering phenomena. Clustering is particularly pervasive in actuarial data. An empirical application to a four-year data set of Dutch automobile collision claims is therefore used to illustrate the approach.

This paper shows that both estimation strategies can be usefully combined when the data generating process is characterized by strong clustering in time and size. We find that the extreme value theory is a useful starting point in detecting the appropriate distribution class. Once that has been achieved, the likelihood-based EM-algorithm is proposed to capture the clustering phenomena. Clustering is particularly pervasive in actuarial data. An empirical application to a four-year data set of Dutch automobile collision claims is therefore used to illustrate the approach.

### MSC:

62P05 | Applications of statistics to actuarial sciences and financial mathematics |

60G70 | Extreme value theory; extremal stochastic processes |

### Keywords:

likelihood based tests; mixtures of distributions; tail shape of claim distributions; maximum likelihood approach; best asymptotic normality; tail parameter; EM-algorithm; automobile collision claims
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\textit{G. R. J. Kalb} et al., Insur. Math. Econ. 18, No. 2, 89--107 (1996; Zbl 0864.62071)

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