2. Lifetime Data Analysis

In this chapter we assume that populations are effectively unbounded; i.e., we are assuming the limiting population case (N ↑ ∞).2.1. Data analysis for the exponential model.In the previous chapter we derived the finite population exponential model for populations of size N. Only in the limit, as N ↑ ∞, did we obtain the usual exponential model, i.e., in the one-dimensional case where θ is the limiting average lifetimep(x|θ)=1θe−xθ2.1.1for x≥0 and θ>0 . We call (2.1.1) the “exponential model.”The first idea in Chapter 1 was that the derivation should be based on a judgment of indifference on submanifolds. The importance of this derivation was due to the fact that we focused on the average lifetime of the population. Were we to focus on other aspects of the lifetimes of the population, we would have derived a different conditional probability model.The second idea was that when we insert data in our conditional probability model, the corresponding function of θ, called the likelihood L(θ), provides the key tool in analyzing data. (In Chapter 1, θ was the population lifetime average.) Using the likelihood and a prior opinion concerning θ we can compute the posterior probability function for θ and from this answer any probability question we may have concerning θ or future lifetimes.The influence of failures on the posterior density. Suppose NO failures are observed but all items have survived for some time t. How can we analyze item lifetimes in this case? To make the situation clear we will consider a particular case. We will show that, although survivors tend to increase our estimate for the mean life θ, lack of failures also increases our uncertainty concerning θ. While the engineer naturally wants to see no failures, the statistician—analyst wants to see failures in order to decrease the uncertainty in the analysis.