It is proportional to the probability that the experiment turned out the way it did.

So some Weibull model parameters are more likely than others because they explain the
observed failures better than other values. We choose the "best" parameter values,
*i.e.* those that maximize the likelihood,
which are called, not surprisingly, "*maximum likelihood parameter estimates*."

The *most likely* parameter values, given the data, for the
Weibull slope, ,
and location,
, are the "**+**"
in Figure 1, but other values are also plausible, although less likely.
That (, ) pair produces the best fit of the data, shown as the black
line in Figure 2.

The dataset below is severely right-censored: 58 of the 70 components
were removed from service at various times *before* they failed,
and are indicated by red tic-marks along the top of figure 2. 12
failed, plotted as points. The data are from Meeker and
Escobar (1998), **Statistical Methods for
Reliability Data**, Wiley, Table C-1, p630.

Notice that even after transformation the confidence "ellipse" isn't very elliptical. It is very asymmetric along its major and minor axes, especially the major axis. That is why we choose not to use methods that employ an ellipse to approximate the loglikeihood surface (like inverting the Fisher Information Matrix) and use the true loglikelihood instead.

Each point along the 95% confidence contour of the loglikelihood surface (Figure 1) produces a Weibull line (Figure 2).

The *most likely* parameter values, given the data, for the
Weibull slope, ,
and location,
, are the "**+**" but other values are also plausible, although less likely.
That (, ) pair produces the best fit of the data, shown as the black
line in Figure 2.

Note: If the figures get out of sync, please reload the page.
To see an example with *uncensored* data, click
here.

The animated figures illustrate how the model parameters (left
figure) control the behavior of the Weibull model (right figure).
The "**+**" plots the maximum likelihood estimates
for (, ),
and is represented by the black Weibull line. But values *near* the
"**+**" are not unreasonable and we can use the
*chi-square*
criterion to gauge "reasonableness." (More on this later.)

The pseudo-ellipse on the left translates to the confidence bounds on the
right, which are constructed one (, )
pair at a time. For each *point* on the left there is a
corresponding *line* on the right. Watch to see how the
critical isocline on the loglikelihood surface is traversed. Each
point represents a F(X) probability on the right where the Weibull fit
is as far away from the MLEs as possible while still remaining
"reasonable." The locus of all such "not unreasonable" lines forms
the loglikelihood ratio confidence bounds on the Weibull model.

Because the log of the ratio of likelihoods tends toward a chi-square
distribution *asymptotically* (*i.e.* as the sample size
increases), we have a criterion, chi-square, for constructing confidence
bounds on the Weibull model.