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Sufficient conditions are obtained that ensure that a lifetime density has a bathtub-shaped failure rate. Analogous conditions handle increasing, decreasing, and upside-down bathtub-shaped failure rates. Application of these results to exponential families of densities is particularly straightforward and effective. Examples are furnished that introduce new bathtub models and illustrate the use of the general results for existing models. Examples involving mixtures are also considered.KEY WORDS : Increasing, decreasing, bathtub-shaped, and upside-down bathtub-shaped failure rates; Exponential family of densities : Mixture.

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[1] We describe a stochastic inversion method for mapping subsurface regions where the electrical resistivity is changing. The technique combines prior information, electrical resistance data, and forward models to produce subsurface resistivity models that are most consistent with all available data. Bayesian inference and a Metropolis simulation algorithm form the basis for this approach. Attractive features include its ability (1) to provide quantitative measures of the uncertainty of a generated estimate and (2) to allow alternative model estimates to be identified, compared, and ranked. Methods that monitor convergence and summarize important trends of the posterior distribution are introduced. Results from a physical model test and a field experiment were used to assess performance. The presented stochastic inversions provide useful estimates of the most probable location, shape, and volume of the changing region and the most likely resistivity change. The proposed method is computationally expensive, requiring the use of extensive computational resources to make its application practical.

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