Continuous Mixtures of Exponentials and IFR Gammas Having Bathtub-Shaped Failure Rates

It can be seen that a mixture of an exponential distribution and a gamma distribution with increasing failure rate for the right choice of parameters can yield a distribution with a bathtub-shaped failure rate. In this paper we consider a continuous mixture of exponentials and a continuous mixture of gammas with increasing failure rates and show that the resulting mixture has a bathtub-shaped failure rate.

Continuous Mixtures of Exponentials and IFR Gammas Having Bathtub-Shaped Failure Rates

It can be seen that a mixture of an exponential distribution and a gamma distribution with increasing failure rate for the right choice of parameters can yield a distribution with a bathtub-shaped failure rate. In this paper we consider a continuous mixture of exponentials and a continuous mixture of gammas with increasing failure rates and show that the resulting mixture has a bathtub-shaped failure rate.

Statistical analysis of mixtures underlying probability of ruin

If the hypothesis on exponentially distributed claims in a risk (or surplus) model is untenable then, in many cases, the assumption that they are mixtures of two (or more) exponentials is a suitable substitute. In the first part of the paper tests of homogeneity for exponentially distributed claims are discussed and their properties are stated. The statistical properties of parameter estimations for such claims are also mentioned. In the second part the classical Cramer-Lundberg ruin model is discussed when claims are distributed as mixtures of exponentials. Our attention is focussed primarily on assesment of accuracy of approximations obtained. Then our results are compared to those already known.

OPTIMAL MAINTENANCE DECISIONS FOR DIKES

To protect the Dutch polders against flooding, more than 2500 km of dikes have been constructed. Due to settlement, subsoil consolidation, and relative sea-level rise, these dikes slowly sink “away into the sea” and should therefore be heightened regularly (at present, every 50 years). In this respect, one is interested in safe and cost-optimal dike heightenings for which the sum of the initial cost of investment and the future (discounted) cost of maintenance is minimal.For optimization purposes, a maintenance model has been developed for dikes subject to uncertain crest-level decline. On the basis of engineering knowledge, crest-level decline has been modeled as a monotone stochastic process with expected decline being either linear or nonlinear (i.e., linear after transformation) in time. For both models and for a particular unit time, the increments are distributed according to mixtures of exponentials.In a case study, the maintenance decision model has been applied to the problem of heightening the Dutch “Oostmolendijk.”