Abstract
Among Mixed Poisson processes, counting processes having geometrically distributed increments can be obtained when the mixing random intensity is exponentially distributed. Dealing with shock models and compound counting models whose shocks and claims occur according to such counting processes, we provide various comparison results and aging properties concerning total claim amounts and random lifetimes. Furthermore, the main characteristic distributions and properties of these processes are recalled and proved through a direct approach, as an alternative to those available in the literature. We also provide closed-form expressions for the first-crossing-time problem through monotone nonincreasing boundaries, and numerical estimates of first-crossing-time densities through other suitable boundaries. Finally, we present several applications in seismology, software reliability and other fields.
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Acknowledgements
We thank two anonymous referees for their useful comments that improved the paper. This research is partially supported by the groups GNAMPA and GNCS of INdAM.
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This paper is dedicated to the cherished memory of Moshe Shaked, to whom we are very grateful for much inspiration and advice on our studies in stochastic orderings and stochastic processes.
Appendix: Proof of Proposition 2.2
Appendix: Proof of Proposition 2.2
Fix \(\textbf {t}=(t_{1},t_{2},\ldots ,t_{m}) \in \mathcal {T}^{+}_{m}\). From Eq. 7 we have
Recall now that, for \(\textbf {k}=(k_{1},k_{2}, \ldots ,k_{m}) \in \mathcal {A}\),
and observe that it holds
if and only if the term t1t2⋯tm appears in expansion of the product \(t_{1}^{k_{1}} (t_{2}-t_{1})^{k_{2}} \cdots (t_{m}-t_{m-1})^{k_{m}}\). With a straightforward computation of such product, and considering the constraints in Eq. 8, it is easy to see that the condition
is fulfilled if and only if k1 + k2 + … + km = m − 1. Recalling that it should be k1 = 0, and again considering the constrains (8), we have that Eq. 40 holds if, and only if, k1 = 0 and ki = 1 for all i = 2, 3,…,m. In this case we have
In conclusion, from Eqs. 38, 39 and 41, for 0 < t1 < t2 < … < tm we have
while the density is zero whenever \(\textbf {t} \not \in \mathcal {T}^{+}_{m}\). Finally, this gives (9).
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Di Crescenzo, A., Pellerey, F. Some Results and Applications of Geometric Counting Processes. Methodol Comput Appl Probab 21, 203–233 (2019). https://doi.org/10.1007/s11009-018-9649-9
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DOI: https://doi.org/10.1007/s11009-018-9649-9
Keywords
- Counting processes
- Multivariate geometric distribution
- First-crossing time
- Shock models
- Stochastic orders
- Aging