Modelling a Bathtub-Shaped Failure Rate by a Coxian Distribution

We build a Coxian random variable model to describe characterizations of a bathtub-shaped failure rate and present a perturbation to the Coxian random variable. In the perturbation, the corresponding probability density function is a linear combination of exponentials, and weights may be negative. The clear mechanism of the perturbation reveals that numerical error of the computation includes cancellation error. By studying weights in a perturbation, we can control absolute errors of all intermediate results. As the involving probability density function is a summation, the total absolute error of the final result is not larger than the sum of absolute errors of intermediate results. Hence, we can suggest the level of numerical precision to control errors. With an accurate likelihood function, some algorithms of maximum likelihood estimates can be established. We evaluate proposed algorithms through simulation examples and empirical data.