Analytical best upper bounds on stop-loss premiums

Devylder, F.; Goovaerts, Marc

doi:10.1016/0167-6687(82)90007-5

Cited by 34 publications

(11 citation statements)

References 4 publications

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“…Especially for the stop-loss treaty a vast body 12 E. Kremer of mathematical literature has appeared during the last few years (see e.g. De Vylder & Goovaerts, 1982;Gerber, 1982;Panjer, 1981). However, for the moment these results seem to be mainly of theoretical interest, since in practice one usually does not have sufficient statistical data for giving reliable estimates for the unknown claims distribution and for the unknown model parameters respectively.…”

Section: Introductionmentioning

confidence: 99%

An asymptotic formula for the net premium of some reinsurance treaties

Kremer¹

1984

Scandinavian Actuarial Journal

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In the following a rather general class of reinsurance treaties is defined, including the excess-of-Ioss, largest claims and ECOMOR reinsurance treaties. Based on asymptotic considerations a simple premium formula is developed for the general treaty. The result is applied to two special types of reinsurance treaties. As a byproduct the paper indicates that there are intimate connections between reinsurance and the field of robust statistics.

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Section: Introductionmentioning

confidence: 99%

An asymptotic formula for the net premium of some reinsurance treaties

Kremer¹

1984

Scandinavian Actuarial Journal

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“…This kind of problem has been treated in a rather fundamental way by , De Vylder and Goovaerts (1982) and Goovaerts et al (1982). De Vylder transforms the basic problem into the "associated dual problem".…”

Section: Methodsmentioning

confidence: 99%

Distribution-free option pricing

Schepper

Heijnen

2007

Insurance: Mathematics and Economics

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“…More specifically, the stop-loss function can be maximised, but the corresponding class of distributions is not closed and maximising distribution will not have a finite variance. We first state the following results which can be found in [12,23,22]. E Q (W − t) + , has the following solutions for different values of t, where Q * is a diatomic distribution with the given atoms,…”

Section: Stratificationmentioning

confidence: 99%

“…Extremal properties. In this section we investigate upper and lower bounds on the performance of pseudo-marginal algorithms by establishing a, perhaps surprising, link to the actuarial science literature in terms of extremal moments and stop-loss functions [12,23,22]. More specifically we consider unit expectation distributions Q * and Q * which are minimal and maximal in the convex orders, subject to some constraints.…”

Section: Stratificationmentioning

confidence: 99%

Establishing some order amongst exact approximations of MCMCs

Andrieu¹,

Vihola²

2016

Ann. Appl. Probab.

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Abstract. Exact approximations of Markov chain Monte Carlo (MCMC) algorithms are a general emerging class of sampling algorithms. One of the main ideas behind exact approximations consists of replacing intractable quantities required to run standard MCMC algorithms, such as the target probability density in a Metropolis-Hastings algorithm, with estimators. Perhaps surprisingly, such approximations lead to powerful algorithms which are exact in the sense that they are guaranteed to have correct limiting distributions. In this paper we discover a general framework which allows one to compare, or order, performance measures of two implementations of such algorithms. In particular, we establish an order with respect to the mean acceptance probability, the first autocorrelation coefficient, the asymptotic variance and the right spectral gap. The key notion to guarantee the ordering is that of the convex order between estimators used to implement the algorithms. We believe that our convex order condition is close to optimal, and this is supported by a counter-example which shows that a weaker variance order is not sufficient. The convex order plays a central role by allowing us to construct a martingale coupling which enables the comparison of performance measures of Markov chain with differing invariant distributions, contrary to existing results. We detail applications of our result by identifying extremal distributions within given classes of approximations, by showing that averaging replicas improves performance in a monotonic fashion and that stratification is guaranteed to improve performance for the standard implementation of the Approximate Bayesian Computation (ABC) MCMC method.

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Analytical best upper bounds on stop-loss premiums

Cited by 34 publications

References 4 publications

An asymptotic formula for the net premium of some reinsurance treaties

An asymptotic formula for the net premium of some reinsurance treaties

Distribution-free option pricing

Establishing some order amongst exact approximations of MCMCs

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