Due to their heterogeneity, insurance risks can be properly described as a mixture of different fixed models, where the weights assigned to each model may be estimated empirically from a sample of available data. If a risk measure is evaluated on the estimated mixture instead of the (unknown) true one, then it is important to investigate the committed error. In this paper we study the asymptotic behaviour of estimated risk measures, as the data sample size tends to infinity, in the fashion of large deviations. We obtain large deviation results by applying the contraction principle, and the rate functions are given by a suitable variational formula; explicit expressions are available for mixtures of two models. Finally, our results are applied to the most common risk measures, namely the quantiles, the Expected Shortfall and the shortfall risk measure

Bignozzi, V., Macci, C., Petrella, L. (2018). Large deviations for risk measures in finite mixture models. INSURANCE MATHEMATICS & ECONOMICS, 80, 84-92 [10.1016/j.insmatheco.2018.03.005].

### Large deviations for risk measures in finite mixture models

#### Abstract

Due to their heterogeneity, insurance risks can be properly described as a mixture of different fixed models, where the weights assigned to each model may be estimated empirically from a sample of available data. If a risk measure is evaluated on the estimated mixture instead of the (unknown) true one, then it is important to investigate the committed error. In this paper we study the asymptotic behaviour of estimated risk measures, as the data sample size tends to infinity, in the fashion of large deviations. We obtain large deviation results by applying the contraction principle, and the rate functions are given by a suitable variational formula; explicit expressions are available for mixtures of two models. Finally, our results are applied to the most common risk measures, namely the quantiles, the Expected Shortfall and the shortfall risk measure
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Articolo in rivista - Articolo scientifico
Contraction principle, Lagrange multipliers, Quantile, Entropic risk measure, Relative entropy
English
2018
80
84
92
reserved
Bignozzi, V., Macci, C., Petrella, L. (2018). Large deviations for risk measures in finite mixture models. INSURANCE MATHEMATICS & ECONOMICS, 80, 84-92 [10.1016/j.insmatheco.2018.03.005].
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Utilizza questo identificativo per citare o creare un link a questo documento: `https://hdl.handle.net/10281/198785`