Bayesian analysis of masked series system lifetime data from a family of lifetime distributions

Abstract

We consider the Bayesian analysis of lifetime data obtained from multi-component series systems when lifetime of each component of the system follows a ‘family of lifetime distributions’. We obtain Bayes estimates of parameters included in this family using incomplete system lifetime data under competing risk model. In order to show the applicability of results to the real life problems, we present analysis of a real data set of electric appliances in which failures occur due to several causes. We also give an ad-hoc technique to generate sample observations from a complicated distribution.

Appendix: An ad-hoc numerical technique for generating sample observations

Here we propose a technique for generating sample observations from a very complicated or improper probability density function. This is a parallel attempt to the traditional inverse transformation technique. The technique may be useful while using Gibbs sampler since in most of the cases the full conditionals used appear in the proportionality form. The only requirement for the technique here is that the density under consideration should be log-concave. We propose the technique as follows.

Let f(y) is a log-concave (density) function of y, then the integral from −∞ to c (say) may be finite, where c is a sufficiently large value of y. The choice of c is done such that the integral −∞ to c takes a constant value ϕ and we do not observe any further increment in ϕ on increasing the c, that is

$$\int\limits_{ - \infty }^{c} {f(y)} dy = \Upphi$$

(1)

Now we generate a random number U in the range [0, ϕ] and use it as a support for sample generation as follows.

$$\int\limits_{ - \infty }^{z} {f(y)dy} = U$$

(2)

Now, we solve the integral (2) numerically to get the value of z. For the numerical solution, we divide the range of variable in very small intervals (depends on accuracy required) and find the ordinate for which calculated integral value is nearest to U. This ordinate is a required sample observation from the considered density function. For generating another sample value we generate another U and solve (2) again. Proceeding this way a sample of required length may be obtained from the density f(y).

The algorithm for simulating sample using this technique is as follows.

• Step 1 Calculate the area ϕ under the curve f(y) for sufficiently large value of variable y such that it (area) becomes constant with respect to further increment in y.

• Step 2 Generate a uniform random variate U in the range [0, ϕ].

• Step 3 Find a sample observation z by solving (2) numerically.

• Step 4 Go to Step 2 and follow steps 2 and 3 to generate more observations.