This section describes the computations for the prediction intervals in the Reliability Forecast platform. Unlike plug-in intervals, prediction intervals take into account the parameter estimate variability. A simulation generates empirical distributions for the sequential and cumulative forecasts. These empirical distributions are then used to define the prediction intervals based on lower and upper quantiles. The following steps describe the simulation:
1. Simulate the maximum likelihood estimate parameter from the asymptotic normal distribution.
2. Use the simulated maximum likelihood estimates to compute the failure probabilities.
3. Generate the sample counts. The distribution that is used in this step depends on the setting of the Use Approximate Distribution option.
When the Use Approximate Distribution option is selected, the Poisson distribution is used to approximate the sequential and cumulative forecasts. This method can be used when the Poisson approximation to the binomial distribution is appropriate. This occurs when the number of items at risk is large and the probability of failure is small.
When the Use Approximate Distribution option is not selected, the multinomial distribution is used to estimate the sequential and cumulative forecasts. This method is more computationally intensive than the method that is used when the Use Approximate Distribution option is selected.
4. Compute the summations and generate the sample forecasts.
For more information about computing prediction intervals in a warranty forecasting system, see Liu and Wang (2013).