Suppose I am producing units and have tested the failure time of some proportion of them (say 10%), possibly with some right censoring if they didn't fail within the testing period.

I'm able to fit a Weibull distribution that models the failure time of my units fairly accurately, using MLE to get estimates for the shape and scale parameters. This lets me say, for example, that 8% of my units will fail during the second month of operation.

But I'd like to put a prediction interval around that prediction, to be able to say that 90% of the time, between 3% and 12% of the units will fail during that time. Is there a formula or method for computing these?

(What I've found before asking: I've found formulas for computing confidence intervals. These are much too tight. Far more than 10% of the data falls outside the confidence interval in practice. I've also found papers that discuss prediction intervals around time-to-failure, but what I'm looking for is a pointwise confidence interval for reliability at each possible time interval.)

  • 2
    $\begingroup$ You could use Bayesian framework and MCMC. The predictive distribution of any random variable depending on the two Weibull parameters can be evaluated, for instance the number of units that will fail. You can compute a prediction interval corresponding a given probability $1 - \alpha$. Such an interval accounts for the uncertainty on the parameters as well as for the hazard on the predicted units. $\endgroup$
    – Yves
    Commented May 2, 2019 at 17:42

1 Answer 1


If I understand your question correctly, what you want is an interval containing $$ \theta=P(t_1 < T \le t_2)=e^{-(t_1/a)^b}-e^{-(t_2/a)^b}, $$ with a nominal probability $1-\alpha$ for given values of $t_1$ and $t_2$. Since $\theta$ is a parameter and not a random variable, this interval will be a confidence and not a prediction interval.

One approach would be to fit the model using e.g. MASS::fitdistr, compute the approximate asymptotic standard error of $\theta$ using the delta method, and then construct an approximate confidence interval for $\theta$ based on asymptotic normality of $\hat\theta$.

An approach that usually works better is be to construct the confidence interval based on the profile log likelihood. After reparameterising the model in terms of $\theta$ and perhaps $a$, the profile log likelihood for $\theta$ would be given by $$ l_p(\theta) = l(\theta,\hat a(\theta)), $$ where $l$ is the log likelihood and $\hat a(\theta)$ is the MLE of $a$ for a given value of $\theta$. I think you'll need to compute $\hat a(\theta)$ numerically. The approximate $1-\alpha$ confidence limits are then equal to the values of $\theta$ at which $l_p(\theta)$ falls $\frac{\chi_{1,\alpha}^2}2$ below the maximum log likelihood.


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