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Cramér-Von Mises distance estimation for some positive infinitely divisible parametric families with actuarial applications. (English) Zbl 1401.62211

Summary: Cramér-von Mises (CVM) inference techniques are developed for some positive flexible infinitely divisible parametric families generalizing the compound Poisson family. These larger families appear to be useful for parametric inference for positive data. The methods are based on inverting the characteristic functions. They are numerically implementable whenever the characteristic function has a closed form. In general, likelihood methods based on density functions are more difficult to implement. CVM methods also lead to model testing, with test statistics asymptotically following a chi-square distribution. The methods are for continuous models, but they can also handle models with a discontinuity point at the origin such as the case of compound Poisson models. Simulation studies seem to suggest that CVM estimators are more efficient than moment estimators for the common range of the compound Poisson gamma family. Actuarial applications include estimation of the stop loss premium, and estimation of the present value of cash flows when interest rates are assumed to be driven by a corresponding Lévy process.

MSC:

62P05 Applications of statistics to actuarial sciences and financial mathematics
62F35 Robustness and adaptive procedures (parametric inference)
60E10 Characteristic functions; other transforms
91B30 Risk theory, insurance (MSC2010)
60G51 Processes with independent increments; Lévy processes

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