Moderate- and large-deviation probabilities in actuarial risk theory

1989 ◽  
Vol 21 (04) ◽  
pp. 725-741 ◽  
Author(s):  
Eric Slud ◽  
Craig Hoesman
Keyword(s):  
Strong Approximation ◽  
Large Deviation ◽  
Age Distribution ◽  
Risk Theory ◽  
Renewal Processes ◽  
Collective Risk ◽  
Large Deviation Theorem ◽  
Large Deviation Probabilities ◽  
Actuarial Risk ◽  
Portfolio Size

A general model for the actuarial risk-reserve process as a superposition of compound delayed-renewal processes is introduced and related to previous models which have been used in collective risk theory. It is observed that non-stationarity of the portfolio ‘age-structure' within this model can have a significant impact upon probabilities of ruin. When the portfolio size is constant and the policy age-distribution is stationary, the moderate- and large-deviation probabilities of ruin are bounded and calculated using the strong approximation results of Csörg et al. (1987a, b) and a large-deviation theorem of Groeneboom et al. (1979). One consequence is that for non-Poisson claim-arrivals, the large-deviation probabilities of ruin are noticeably affected by the decision to model many parallel policy lines in place of one line with correspondingly faster claim-arrivals.

Moderate- and large-deviation probabilities in actuarial risk theory

1989 ◽  
Vol 21 (4) ◽  
pp. 725-741 ◽  
Author(s):  
Eric Slud ◽  
Craig Hoesman
Keyword(s):  
Strong Approximation ◽  
Large Deviation ◽  
Age Distribution ◽  
Risk Theory ◽  
Renewal Processes ◽  
Collective Risk ◽  
Large Deviation Theorem ◽  
Large Deviation Probabilities ◽  
Actuarial Risk ◽  
Portfolio Size

A general model for the actuarial risk-reserve process as a superposition of compound delayed-renewal processes is introduced and related to previous models which have been used in collective risk theory. It is observed that non-stationarity of the portfolio ‘age-structure' within this model can have a significant impact upon probabilities of ruin. When the portfolio size is constant and the policy age-distribution is stationary, the moderate- and large-deviation probabilities of ruin are bounded and calculated using the strong approximation results of Csörg et al. (1987a, b) and a large-deviation theorem of Groeneboom et al. (1979). One consequence is that for non-Poisson claim-arrivals, the large-deviation probabilities of ruin are noticeably affected by the decision to model many parallel policy lines in place of one line with correspondingly faster claim-arrivals.

scholarly journals Large deviation probabilities for sums of heavy-tailed dependent random vectors

1997 ◽  
Vol 13 (4) ◽  
pp. 647-660 ◽  
Author(s):  
Adam Jakubowski ◽  
Alexander. V. Nagaev ◽  
Zaigraev Alexander
Keyword(s):  
Large Deviation ◽  
Random Vectors ◽  
Large Deviation Probabilities ◽  
Heavy Tailed

On the limiting distribution of the failure time of fibrous materials

1983 ◽  
Vol 15 (02) ◽  
pp. 331-348
Author(s):  
Wagner De Souza Borges
Keyword(s):  
Materials Science ◽  
Failure Time ◽  
Large Deviation ◽  
Limiting Distribution ◽  
Statistical Characteristics ◽  
Fibrous Materials ◽  
Large Deviation Theorem ◽  
Science Area ◽  
The Individual ◽  
Failure Time Distributions

A large deviation theorem of the Cramér–Petrov type and a ranking limit theorem of Loève are used to derive an approximation for the statisticaldistribution of the failure time of fibrous materials. For that, fibrousmaterials are modeled as a series of independent and identical bundles of parallel filaments and the asymptotic distribution of their failure time is determined in terms of statistical characteristics of the individual filaments, as both the number of filaments in each bundle and the number of bundles in the chain grow large simultaneously. While keeping the numbernof filaments in each bundle fixed and increasing only the chain lengthkleads to a Weibull limiting distribution for the failure time, letting both increase in such a way that logk(n)= o(n), we show that the limit distribution isfor. Since fibrous materials which are both long and have many filaments prevail, the result is of importance in the materials science area since refined approximations to failure-time distributions can be achieved.

Sign in / Sign up

Export Citation Format

Share Document