Relating Distribution Parameters to Real-World Applications
For managers and engineers who
are new to reliability, it can be difficult to relate the parameters of a
statistical distribution to their real-life reliability
requirements. Consider, for example, the fact that the eta parameter of
the Weibull distribution is the time by which about
63.2% of the units in a population are expected to fail.
In practice, this value may not be particularly useful.
When looking at warranties, for instance, the maximum percentage of units that can
fail during a warranty period is often 10%
or less. For this reason, it can be easier to
understand and explain reliability concepts in terms of
metrics other than those that statisticians typically
use. This article presents an example of
constructing a contour plot for the Weibull distribution
using the parameter beta and the time at which 10% of the
units in the population are expected to fail (i.e., the B10
Joe the reliability engineer was approached by his
boss with a new problem. For years, the company
management had been focused on one reliability
requirement—the B10 life of their product. Joe’s
boss really liked the contour plots available in
which show the possible combinations of two
parameters of a distribution that could fit a particular
data set. The standard contour plot for a Weibull
distribution, for example, shows the possible values
of beta for each possible value of eta at
one or more specified confidence levels (e.g., at a 90%
confidence level, there is a 90% chance that the actual
parameters of the distribution fall somewhere within or
on the boundary of the contour area). Joe's boss asked if, instead of looking at a
plot of the parameters for a Weibull distribution, he
could see a contour plot of the shape parameter, beta, and
the B10 life. Eager to please his boss, Joe read the
Reliability HotWire article "Individual and Joint Parameter Bounds"
(July 2009) to help him get started.
Joe knew that the first thing he needed to do was to
recast the likelihood function in terms of BX life
instead of the scale parameter, eta. He started with the
equation for the unreliability for a Weibull
- Q is unreliability.
- t is time.
- beta and eta are
the Weibull shape and scale parameters, respectively.
He solved the equation for eta:
Since the BX life is the time by which X percent of
the units in a population are expected to fail, Joe
substituted X/100 for Q and BX for t. This resulted in the
following expression for eta in terms of beta, X and BX:
Since the data set that Joe’s boss was the most
interested in consisted of specimens that had all been
run to failure, Joe substituted the above expression for
eta into the likelihood function for a Weibull
distribution with exact times to failure. After a
few lines of algebra, he obtained the following
expression for the likelihood function for any
percentile, X, in terms of beta, BX and time:
where N is the number of exact times to failure in
the data set and the ti values represent those
times to failure. As a quick
check, Joe noted that when X=1-1/e≈63.2% and BX=eta, the
above equation reduces to the familiar equation for the
likelihood function for a complete data set in terms of
beta, eta and time. Since his boss specifically asked
about a contour plot using B10 life, Joe substituted in
X=10 to get the likelihood function in terms of beta, B10
Referring again to the previous Reliability
HotWire article on individual and joint
parameter bounds, Joe noted that since he was creating a
contour plot, he was constructing joint parameter
bounds. Therefore, he used the Chi-Squared
statistic with 2 degrees of freedom:
- L is the likelihood function given above in
beta, B10 and time
is the maximum likelihood
Chi-Squared statistic with a
significance level of α and 2 degrees of freedom.
Rearranging this equation, Joe obtained the expression
for the likelihood value corresponding to the beta vs.
B10 contour as:
From this equation, the beta vs. B10 contour can be
obtained by choosing specified values of beta in the
likelihood function and iteratively solving for the
corresponding two values of B10, or vice versa.
As a test of his method, Joe chose a simple data set
with the following times to failure: 10, 20, 30, 40, 50.
He entered this data set into Weibull++ and calculated
parameters using MLE. He found that the LK value
was -20.1840, which he converted to a maximum likelihood
value of 1.715E-9 using a General Spreadsheet.
(Note: For small data sets with exact times to failure,
RRX is the preferred analysis method, but in this case
Joe needed to use MLE in order to obtain the maximum
likelihood value to generate his special contour plot.)
He also viewed the resulting beta vs. eta contour plot shown
in Figure 1.
Figure 1 - Traditional
contour plot (beta vs. eta) at
a 90% confidence level
Next, Joe chose a 90% confidence level for his
special contour plot, which is equivalent to a 10%
significance level. This information allowed him to
compute the likelihood value corresponding to the beta vs. B10
Joe substituted this value into the likelihood
function in terms of B10 life to obtain the following
equation, which he used to find two solutions to B10
life for different values of beta.
Using the fact that beta on the beta vs. eta contour plot
ranged from 0.89 to 4.55, Joe constructed the following
Table 1 - Solutions to the likelihood unction for
various values of beta
Then, he put this information into a general
spreadsheet in Weibull++ and made a scatter plot to
obtain the beta vs. B10 contour plot shown in Figure 2.
For more information on generating scatter plots in
Weibull++, please see the Reliability HotWire
the Chart Wizard to Create a Custom Plot" (February 2010).
Figure 2 - Beta vs. B10
contour plot at a 90%
This contour plot allowed Joe and his boss to see how
the beta parameter for the data set relates to the B10
life, which is easier to understand in "real-world" terms
than eta. Now Joe can create his special contour plots
to compare data sets and perform hypothesis tests in
terms that his boss and his other colleagues can relate
to the company reliability goals.