Statistics for Maintenance Modelling: Proportional Hazards and Accelerated Life Models

# Estimation of the Survival and Hazard Functions¤

There are two ways to estimate the baseline hazard function. You can estimate = (1, 2, ... n) and the parametric form of h0(t) together using the maximum likelihood method provided you have sufficient life data. This is done by writing down the log-likelihood function based upon r distinct lifetimes and n - r censored times, that is:

Ln L () = Constant + +

= Constant + + (since f(ti; y) = h(ti; y) S(ti; y))

= Constant + + + (since S(ti; y) = )

= Constant + - (since ln(ex) = x, and combining the two sets)

where:

- D is the set of complete life data
- C is the set of censored data

- f(ti; y) is the probability density function of time to failure given y
- S(ti; y) is the survival (or reliability) function given y
- h(ti; y) is the hazard function given y.

Maximising the log-likelihood function given on the previous screen in terms of unknown parameters of , one should have the estimated parameter values. If the baseline hazard function depends on parameters , these can be estimated in this way.