ABSTRACT
The framework for the results in the preceding chapters has been random sample of n observations that are either complete or censored in one way or the other. We have re-
garded these observations as realizations of independent and identically distributed random
variables with cumulative distribution function F (x |θ) and density function f(x |θ) = dF (x |θ)/dx, where θ is a labeling vector of parameters. We also assumed that these parameters are constants with unknown values. However, we agreed upon the parame-
ter space, i.e., the set of all possible values of the parameters, which we denoted by Ω. Mostly we have assumed that Ω is the natural parameter space; i.e., Ω contains those values of θ for which the density function f(x |θ) is well defined, i.e., f(x |θ) ≥ 0 and∫∞ −∞ f(x |θ) dx = 1. But there are cases in which one can assume a little more about the unknown parameters.
For example, we could assume that θ is itself a realization of a random vector, denoted by Θ, with density function g(θ). In the WEIBULL model the scale parameter b may be regarded as varying from batch to batch over time, and this variation is represented by a
probability distribution over Ω. Thus, the set-up is now as described in Sect. 14.1.
