Compound Geometric Distribution of Order k

2016 ◽  
Vol 19 (2) ◽  
pp. 377-393 ◽  
Author(s):  
Markos V. Koutras ◽  
Serkan Eryilmaz
Keyword(s):  
Geometric Distribution ◽  
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.

Reliability Analysis of Cold Standby Parallel System Possessing Failure And Repair Rate Under Geometric Distribution

2019 ◽  
Vol 13 ◽  
Author(s):  
Jasdev Bhatti ◽  
Mohit Kumar Kakkar
Keyword(s):  
Failure Rate ◽  
Geometric Distribution ◽  
Discrete Distribution ◽  
Maintenance Cost ◽  
Discrete Problem ◽  
New Techniques ◽  
Testing Strategies ◽  
Cold Standby ◽  
Mean Time ◽  
Iteration Technique

Background and Aim: With an increase in demands about reliability of industrial machines following continuous or discrete distribution, the important thing to be noticed is that in all previous researches where systems are having more than one failure no iteration technique has been studied to separate the failed unit on basis of its failure. Therefore, aim of our paper is to analyze the real industrial discrete problem following cold standby units arranged in parallel manner with newly concept of inspection procedure for failed units to inspect the exact failure and being communicator to the repairman for repairing exact failed part of unit for saving time and maintenance cost. Methods: The geometric distribution and regenerative techniques had been applied for calculating different reliability measures like mean time to system failure, availability of a system, inspection, repair and failed time of unit. Results: Graphical and analytical study had also been done to analyze the increasing/decreasing behavior of profit function w.r.t repair and failure rate. The system responded properly in fulfilling his basic needs. Conclusion: The calculated value of all reliability parameter is helpful for studying any other models following same concept under different environmental conditions. Thus, it concluded that, reliability increases/decreases with increase in repair/failure rate. Also, the evaluated results by this paper provides the better reliability testing strategies that helps to develop new techniques which leads to increase the effectiveness of system.

Use of Basu-Dhar bivariate geometric distribution in the analysis of the reliability of component series systems

2019 ◽  
Vol 36 (4) ◽  
pp. 569-586
Author(s):  
Ricardo Puziol Oliveira ◽  
Jorge Alberto Achcar
Keyword(s):  
Bayesian Approach ◽  
Geometric Distribution ◽  
Bivariate Distribution ◽  
Dependence Structure ◽  
Series System ◽  
Engineering Applications ◽  
Data Set ◽  
Content Type ◽  
Bivariate Geometric Distribution ◽  
Series Systems

Purpose The purpose of this paper is to provide a new method to estimate the reliability of series system by using a discrete bivariate distribution. This problem is of great interest in industrial and engineering applications. Design/methodology/approach The authors considered the Basu–Dhar bivariate geometric distribution and a Bayesian approach with application to a simulated data set and an engineering data set. Findings From the obtained results of this study, the authors observe that the discrete Basu–Dhar bivariate probability distribution could be a good alternative in the analysis of series system structures with accurate inference results for the reliability of the system under a Bayesian approach. Originality/value System reliability studies usually assume independent lifetimes for the components (series, parallel or complex system structures) in the estimation of the reliability of the system. This assumption in general is not reasonable in many engineering applications, since it is possible that the presence of some dependence structure between the lifetimes of the components could affect the evaluation of the reliability of the system.

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