scholarly journals Improved Analytical Bounds for Some Risk Quantities

1996 ◽  
Vol 26 (2) ◽  
pp. 185-199 ◽  
Author(s):  
Werner Hürlimann
Keyword(s):  
Size Distribution ◽  
Upper Bounds ◽  
Ruin Probabilities ◽  
Claim Size ◽  
Lower And Upper Bounds ◽  
Relative Variance ◽  
Claim Size Distribution ◽  
The Mean ◽  
Stop Loss ◽  
Mean Variance

AbstractSimple analytical lower and upper bounds are obtained for stop-loss premiums and ruin probabilities of compound Poisson risks in case the mean, variance and range of the claim size distribution are known. They are based on stop-loss extremal distributions and improve the bounds derived earlier from dangerous extremal distributions. The special bounds obtained in case the relative variance of the claim size is unknown, but its maximal value is known, are related to other actuarial results.

scholarly journals Excess of Loss Reinsurance with Reinstatements Revisited

2005 ◽  
Vol 35 (01) ◽  
pp. 211-238 ◽  
Author(s):  
Werner Hürlimann
Keyword(s):  
Incomplete Information ◽  
Size Distribution ◽  
Claim Size ◽  
Inverse Gaussian ◽  
Insurance Risk ◽  
Distribution Free ◽  
Analytical Approximations ◽  
Claim Size Distribution ◽  
Aggregate Claims ◽  
Stop Loss

The classical evaluation of pure premiums for excess of loss reinsurance with reinstatements requires the knowldege of the claim size distribution of the insurance risk. In the situation of incomplete information, where only a few characteristics of the aggregate claims to an excess of loss layer can be estimated, the method of stop-loss ordered bounds yields a simple analytical distribution-free approximation to pure premiums of excess of loss reinsurance with reinstatements. It is shown that the obtained approximation is enough accurate for practical purposes and improves the analytical approximations obtained using either a gamma, translated gamma, translated inverse Gaussian or a mixture of the last two distributions.

scholarly journals Excess of Loss Reinsurance with Reinstatements Revisited

2005 ◽  
Vol 35 (1) ◽  
pp. 211-238 ◽  
Author(s):  
Werner Hürlimann
Keyword(s):  
Incomplete Information ◽  
Size Distribution ◽  
Claim Size ◽  
Inverse Gaussian ◽  
Insurance Risk ◽  
Distribution Free ◽  
Analytical Approximations ◽  
Claim Size Distribution ◽  
Aggregate Claims ◽  
Stop Loss

The classical evaluation of pure premiums for excess of loss reinsurance with reinstatements requires the knowldege of the claim size distribution of the insurance risk. In the situation of incomplete information, where only a few characteristics of the aggregate claims to an excess of loss layer can be estimated, the method of stop-loss ordered bounds yields a simple analytical distribution-free approximation to pure premiums of excess of loss reinsurance with reinstatements. It is shown that the obtained approximation is enough accurate for practical purposes and improves the analytical approximations obtained using either a gamma, translated gamma, translated inverse Gaussian or a mixture of the last two distributions.

scholarly journals An Investigation of the Use of Weighted Averages in the Estimation of the Mean of a Long-Tailed Claim Size Distribution

1978 ◽  
Vol 10 (1) ◽  
pp. 87-98 ◽  
Author(s):  
G. C. Taylor
Keyword(s):  
Size Distribution ◽  
Arithmetic Mean ◽  
Claim Size ◽  
Approximately Unbiased ◽  
Sufficient Statistic ◽  
Weighted Averages ◽  
Claim Size Distribution ◽  
The Mean ◽  
Biased Estimators ◽  
Vector Valued

AbstractThe paper discusses the problem of estimating the mean of a long-tailed claim size distribution when the investigator's knowledge of the distribution is only vague.One method of dealing with this problem, the method developed by Johnson and Hey, is examined and found to produce strongly biased estimators.The situation in which a sufficient statistic (but nothing else) for the claim size distribution is known is examined, and an approximately unbiased estimator developed. This estimator is substantially more efficient than the arithmetic mean in some cases. It appears to be quite successful when the sufficient statistic is real-valued. It is of limited use when the sufficient statistic is vector-valued.

scholarly journals A class of risk processes with delayed claims: ruin probability estimates under heavy tail conditions

2006 ◽  
Vol 43 (4) ◽  
pp. 916-926 ◽  
Author(s):  
Ayalvadi Ganesh ◽  
Giovanni Luca Torrisi
Keyword(s):  
Size Distribution ◽  
Ruin Probability ◽  
Heavy Tail ◽  
Claim Size ◽  
Probability Estimates ◽  
Claim Size Distribution ◽  
Risk Processes

We consider a class of risk processes with delayed claims, and we provide ruin probability estimates under heavy tail conditions on the claim size distribution.

Tail of compound distributions and excess time

1996 ◽  
Vol 33 (01) ◽  
pp. 184-195 ◽  
Author(s):  
Xiaodong Lin
Keyword(s):  
Size Distribution ◽  
Lower Bounds ◽  
Upper And Lower Bounds ◽  
Claim Size ◽  
Compound Distributions ◽  
Fundamental Identity ◽  
Claim Size Distribution ◽  
Simple Bounds ◽  
New Better Than Used ◽  
Better Than

Bounds on the tail of compound distributions are considered. Using a generalization of Wald's fundamental identity, we derive upper and lower bounds for various compound distributions in terms of new worse than used (NWU) and new better than used (NBU) distributions respectively. Simple bounds are obtained when the claim size distribution is NWUC, NBUC, NWU, NBU, IMRL, DMRL, DFR and IFR. Examples on how to use these bounds are given.

scholarly journals A Heuristic Review of some Ruin Theory Results

1985 ◽  
Vol 15 (2) ◽  
pp. 73-88 ◽  
Author(s):  
G. C. Taylor
Keyword(s):  
Size Distribution ◽  
Stochastic Dominance ◽  
Ruin Probability ◽  
Recursion Formula ◽  
Claim Size ◽  
Infinite Time ◽  
Ruin Theory ◽  
Interesting Connection ◽  
Claim Size Distribution ◽  
Aggregate Claims

AbstractThe paper deals with the renewal equation governing the infinite-time ruin probability. It is emphasized as intended to be no more than a pleasant ramble through a few scattered results. An interesting connection between ruin probability and a recursion formula for computation of the aggregate claims distribution is noted and discussed. The relation between danger of the claim size distribution and ruin probability is reexamined in the light of some recent results on stochastic dominance. Finally, suggestions are made as to the way in which the formula for ruin probability leads easily to conclusions about the effect on that probability of the long-tailedness of the claim size distribution. Stable distributions, in particular, are examined.

Sign in / Sign up

Export Citation Format

Share Document