Statistics for Maintenance Modelling: Repairable Systems

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# Fitting a NHPP Model and Estimating Parameters Using Lambda 1(t)¤

Let's look at some likelihood-based statistical methods for fitting a NHPP with (t) = 1(t) = or 2(t) = to some sets of repairable systems data.

## Estimating Parameters by Substituting Lambda 1(t) into the Log-Likelihood

Putting (t) = 1(t) = into the log-likelihood equation on the previous screen, the log-likelihood for a repairable system observed over the time interval (0, t0] with failure times at t1, t2, ... tn is:

ln L = ln (ti) - (t) dt

= ln () - dt

= n 0 + 1 ti - where n is the nth failure.

To obtain the maximum likelihood estimates of 0 and 1, one can maximise this equation using any optimisation software such as Excel Solver or Mathcad. Alternatively, one can differentiate it with respect to 0 and 1 and then set the derivatives to zero. We shall look at this next.