Statistics for Maintenance Modelling: Proportional Hazards and Accelerated Life Models

# Justifying Using the Likelihood Function¤

There are several alternative ways of justifying the use of the likelihood function on the previous screen. The simplest argument is due to Cox (1972). Consider the set of Rj of items at risk immediately prior to t(j). The conditional probability that the item failed at t(j) given items in Rj that could fail at t(j) is:

= (since P(fail at t(j)) hazard t if t is small)

= (since h(t(j); y(j)) = h0(t(j)) )

Treating the function on the previous screen as an ordinary likelihood, the general methods of maximum likelihood estimation discussed in the Model Fitting session may be used to estimate .