On the DFR property of the compound geometric distribution with applications in risk theory

2010 ◽  
Vol 47 (3) ◽  
pp. 428-433 ◽  
Author(s):  
Georgios Psarrakos
Keyword(s):  
Geometric Distribution ◽  
Risk Theory ◽  
Compound Geometric Distribution

scholarly journals Stein's Method for Compound Geometric Approximation

2010 ◽  
Vol 47 (1) ◽  
pp. 146-156 ◽  
Author(s):  
Fraser Daly
Keyword(s):  
Generating Functions ◽  
Analytical Approach ◽  
Geometric Distribution ◽  
Stein’S Method ◽  
Hitting Times ◽  
Stein's Method ◽  
Geometric Approximation ◽  
Compound Geometric Distribution ◽  
Probabilistic Approximation ◽  
Sequence Patterns

We apply Stein's method for probabilistic approximation by a compound geometric distribution, with applications to Markov chain hitting times and sequence patterns. Bounds on our Stein operator are found using a complex analytical approach based on generating functions and Cauchy's formula.

scholarly journals Three Methods to Calculate the Probability of Ruin

1989 ◽  
Vol 19 (1) ◽  
pp. 71-90 ◽  
Author(s):  
François Dufresne ◽  
Hans U. Gerber
Keyword(s):  
Stationary Distribution ◽  
Geometric Distribution ◽  
Random Variable ◽  
Claim Amount ◽  
Adjustment Coefficient ◽  
Exponential Distributions ◽  
Probability Of Ruin ◽  
The Third ◽  
Compound Geometric Distribution ◽  
Aggregate Loss

AbstractThe first method, essentially due to GOOVAERTS and DE VYLDER, uses the connection between the probability of ruin and the maximal aggregate loss random variable, and the fact that the latter has a compound geometric distribution. For the second method, the claim amount distribution is supposed to be a combination of exponential or translated exponential distributions. Then the probability of ruin can be calculated in a transparent fashion; the main problem is to determine the nontrivial roots of the equation that defines the adjustment coefficient. For the third method one observes that the probability, of ruin is related to the stationary distribution of a certain associated process. Thus it can be determined by a single simulation of the latter. For the second and third methods the assumption of only proper (positive) claims is not needed.

scholarly journals Nonexponential asymptotics for the solutions of renewal equations, with applications

2006 ◽  
Vol 43 (03) ◽  
pp. 815-824 ◽  
Author(s):  
Chuancun Yin ◽  
Junsheng Zhao
Keyword(s):  
Distribution Function ◽  
Birth Rate ◽  
Geometric Distribution ◽  
Renewal Processes ◽  
Renewal Function ◽  
Renewal Equations ◽  
Put Option ◽  
Special Cases ◽  
Compound Geometric Distribution ◽  
Defective Renewal Equations

Nonexponential asymptotics for solutions of two specific defective renewal equations are obtained. These include the special cases of asymptotics for a compound geometric distribution and the convolution of a compound geometric distribution with a distribution function. As applications of these results, we study the asymptotic behavior of the demographic birth rate of females, the perpetual put option in mathematics of finance, and the renewal function for terminating renewal processes.

scholarly journals Nonexponential asymptotics for the solutions of renewal equations, with applications

2006 ◽  
Vol 43 (3) ◽  
pp. 815-824 ◽  
Author(s):  
Chuancun Yin ◽  
Junsheng Zhao
Keyword(s):  
Distribution Function ◽  
Birth Rate ◽  
Geometric Distribution ◽  
Renewal Processes ◽  
Renewal Function ◽  
Renewal Equations ◽  
Put Option ◽  
Special Cases ◽  
Compound Geometric Distribution ◽  
Defective Renewal Equations

Nonexponential asymptotics for solutions of two specific defective renewal equations are obtained. These include the special cases of asymptotics for a compound geometric distribution and the convolution of a compound geometric distribution with a distribution function. As applications of these results, we study the asymptotic behavior of the demographic birth rate of females, the perpetual put option in mathematics of finance, and the renewal function for terminating renewal processes.

scholarly journals Stein's Method for Compound Geometric Approximation

2010 ◽  
Vol 47 (01) ◽  
pp. 146-156 ◽  
Author(s):  
Fraser Daly
Keyword(s):  
Generating Functions ◽  
Analytical Approach ◽  
Geometric Distribution ◽  
Stein’S Method ◽  
Hitting Times ◽  
Stein's Method ◽  
Geometric Approximation ◽  
Compound Geometric Distribution ◽  
Probabilistic Approximation ◽  
Sequence Patterns

We apply Stein's method for probabilistic approximation by a compound geometric distribution, with applications to Markov chain hitting times and sequence patterns. Bounds on our Stein operator are found using a complex analytical approach based on generating functions and Cauchy's formula.

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