Research Article
Weibull Distribution and Approximation, by the Finite Volume Method, of the Ultim Ruin Probability Constructed from the Hawkes Variable Memory Process
Souleymane Badini*
,
Frédéric Bere
Issue:
Volume 14, Issue 4, August 2025
Pages:
118-125
Received:
31 May 2025
Accepted:
16 June 2025
Published:
4 July 2025
Abstract: It measures the risk that a system or company fails to maintain its elf over time. In this article, we provide an approximation of the probability of ruin at the infinite horizon whose inter-arrivals of claims follow the Hawks process and the amount of claims follows the Weibull distribution, with independence between these two processes. Using the Finite Volume Method is a numerical approach for solving partial differential equations. It consists of dividing the computational domain into discrete volumes and applying local approximations to obtain a global solution. This method can be used to estimate complex probabilities., a stochastic model with variable memory, it is possible to capture the temporal dependence of events. This allows us to analyze situations where the past directly influences the probability of occurrence of future events. This approximation is done using the finite volume method, which is a numerical approach for solving partial differential equations. It consists of dividing the computational domain into discrete volumes and applying local approximations to obtain a global solution. This method can be used to estimate complex probabilities. This is the case in our work; which consists of solving a second-order integro-differential equation, two cases of which are considered on the Weibull parameter η: if η=1, then the distribution of claim amounts is exponential. On the other hand, if η≥2, then the results lead us to a system of linear equations for which we use the finite volume method to obtain a numerical solution.
Abstract: It measures the risk that a system or company fails to maintain its elf over time. In this article, we provide an approximation of the probability of ruin at the infinite horizon whose inter-arrivals of claims follow the Hawks process and the amount of claims follows the Weibull distribution, with independence between these two processes. Using the...
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Research Article
Robust D-optimal Designs for the First-degree and the Second-degree Kronecker Model for Mixture Experiments
Mike Cherutich*,
Jopseph Arap Koske,
Mathew Kosgey
Issue:
Volume 14, Issue 4, August 2025
Pages:
126-137
Received:
24 May 2025
Accepted:
6 June 2025
Published:
14 July 2025
DOI:
10.11648/j.ajtas.20251404.12
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Views:
Abstract: Many practical problems are associated with the investigation of mixture of m ingredients, which are assumed to influence the response through the proportions in which they are blended together. Such problems lend their applicability to mixture experiments. Mixture experiments can be modeled using Scheffe’ or Kronecker models. For the first-, second-, and third-degree Kronecker models, D-optimal designs for the mixture experiments have been derived by various authors. This creates uncertainties to an experimenter, hence the need for robust designs. The objective of this study is to derive robust D-optimal designs for mixture experiments in the first- and the second-degree Kronecker model for mixture experiments. In order to achieve this, the D-optimal weights for the designs in the first-degree and those of the second degree Kronecker models are obtained. The model robust D-Optimality criterion is then used. The D-Optimal designs are obtained by maximizing this criterion which involves differentiating, equating to zero and solving for , r1 and r2 in order to obtain the optimal values. In conclusion the results of this study demonstrate the existence of model robust D-optimal Kronecker model mixture experiments for the first- and the second-degree Kronecker models.
Abstract: Many practical problems are associated with the investigation of mixture of m ingredients, which are assumed to influence the response through the proportions in which they are blended together. Such problems lend their applicability to mixture experiments. Mixture experiments can be modeled using Scheffe’ or Kronecker models. For the first-, secon...
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