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Individual and Joint Parameter Bounds
in Weibull++
Weibull++
can provide both individual and joint parameter bounds for a given data
set. This article briefly explains the difference between these
bounds with respect to their calculation and application. The
article also provides step-by-step guidance on how to obtain
these bounds using Weibull++.
Confidence Bounds in Weibull++
In
Weibull++, there are several choices to obtain confidence bounds
for a given data set based on the method used to calculate these
bounds. For bounds on individual parameters, Weibull++ provides
Fisher matrix bounds, likelihood ratio bounds, beta binomial
bounds and Bayesian confidence bounds. Joint parameter bounds in
Weibull++ are presented as contour plots.
Individual Parameter Bounds
Individual
parameter bounds are used to evaluate uncertainty in terms of
the expected (or mean) values of the parameters.
(A) Fisher Matrix Bounds
Fisher
Matrix
bounds are used widely in many statistical applications. These bounds are calculated using the Fisher information
matrix. The inverse of the Fisher information matrix yields the
variance-covariance matrix, which provides the variance of the
parameter. The bounds on the parameters are then calculated
using the following equations:
Lower bound =
Upper bound =
where:
 is
the estimate of mean value of the parameter θ.- Var() is the variance of the parameter.
- α = 1 - CL, where
CL is the confidence level.
- zα/2
is the standard
normal statistic.
Parameters that do not take negative
values are assumed to follow the lognormal distribution
and the following equations are used to obtain the
confidence bounds:
Lower bound =
Upper
bound =
Interested readers can find further
details on the procedure in [1].
(B) Likelihood Ratio Bounds
For data sets with very few data points, Fisher matrix
bounds are not sufficiently conservative. The likelihood
ratio method produces results that are more conservative
and consequently more suitable
in such cases. (For data sets with larger numbers of
data points, there is not a significant
difference in the results of these two methods.)
Likelihood ratio bounds are calculated using the
likelihood function as follows:
where:
- L(θ) is
the likelihood function for the unknown parameter θ.
- L() is
the likelihood function calculated at the estimated
parameter value .
- α = 1 - CL, where
CL is the confidence level.
-
is the Chi-Squared statistic with k degrees of
freedom, where k is the number of quantities
jointly estimated.
In the calculations of the likelihood ratio bounds on
individual parameters, only one degree of freedom (k =
1) is used in the
statistic. This is due to the fact
that these calculations provide results for a single
confidence region. For details, refer to [2].
(C) Beta Binomial Bounds
The beta-binomial
method of confidence bounds calculation is a
non-parametric approach to confidence bounds
calculations that involves the use of rank tables. In
Weibull++, these bounds are available only with
mixed-Weibull distributions. Details on the calculation
of these bounds are available in [3].
(D)
Bayesian Confidence Bounds
This method of
estimating confidence bounds is based on the Bayes
theorem. These confidence bounds rely on a different
school of thought in statistical analysis, where prior
information is combined with sample data in order to
make inferences about model parameters and their
functions. Details on the calculation of these bounds
are available in [4].
Obtaining Individual
Parameter Bounds in Weibull++
For individual
parameter bounds, the user can choose the confidence bound
calculation method in one of two ways:
- On the Main page of the Standard Folio Control
Panel, the Settings area contains shortcuts that allow
the user to cycle through available settings. The user
can click the Confidence Bounds Method setting, circled
in Figure 1, until the desired option is displayed.
- On the Analysis tab of the Standard Folio Control
Panel, the
user can select the desired method from all available
confidence bounds methods, as shown in Figure 2. Note
that any methods that
do not apply to the specific analysis are unavailable.
Figure 1: Selecting the Confidence Bounds Method
on the Main Page of the Control Panel
Figure 2:
Selecting the Confidence Bounds Method on the
Analysis Page of the Control Panel
Once the data set has been analyzed, confidence bounds for individual parameters can
be calculated using the Quick Calculation pad, accessed
by
clicking the QCP icon.
For demonstration purposes, we
have analyzed the times-to-failure data shown in
Figure 1 using the two-parameter Weibull distribution
and the likelihood ratio confidence bound
method. We then access the QCP. The first step before
calculating parameter bounds in the QCP is to choose the
desired confidence level and the type of confidence
bound on the Confidence Bounds page of the QCP, as shown in Figure 3. Here, we
have chosen two-sided 90%
confidence bounds.
Figure 3: Setting Confidence Level and Type of
Confidence Bounds in the QCP
We can then calculate the
parameter bounds by clicking the Calculate Bounds
button on
the Parameter Bounds page, as shown next.
Figure 4: The Parameter Bounds
Page in the QCP
Joint Parameter Bounds or Contour
Plots
Joint parameter bounds or contour
plots are used to compare two data sets and graphically
perform hypothesis testing at a specific confidence
level in order to evaluate whether the two data sets are
statistically different. These
plots are calculated by using the joint region for the
selected distribution’s parameters. In the case of the
Weibull distribution, the contour plots show the joint
region of the parameters Beta and Eta at a specific
confidence level. The equation used to calculate the
contour plots is the same as the one used to obtain
likelihood ratio bounds on individual parameters.
However, the calculations that go into the generation of
the contour plots use the
statistic with two degrees of
freedom (k = 2) in order to include the joint confidence
region for both the parameters.
Obtaining
Contour Plots in Weibull++
The user can
create contour plots by clicking the
Plot icon and then choosing
Contour Plot in the Plot Type
drop-down list on the Plot Sheet Control Panel.
In the
Contours Setup window that appears, we select the
90% confidence level, as shown in Figure 5. (Additional confidence levels can
be plotted simultaneously, if desired.)
Figure 5: Contours Setup Window
The resulting contour plot
is shown in Figure 6.
Figure 6: Contour Plot for Weibull Distribution
Parameters Beta and Eta Based on Data from Figure 1
This plot shows the joint region of the parameters
Beta and Eta at the 90% confidence level for this
particular data set. To compare data sets, we can create
a contour plot that includes both data sets in a
MultiPlot (added by choosing Project > Add
Additional Plot > Add MultiPlot and then
selecting the data sets for inclusion). We can then
evaluate the difference between the data sets based on
the overlap of the contours. For more detail on this,
see [5].
Conclusion
In reliability engineering, it is important to be able
to obtain bounds around parameters in order to get a
sense of the variability around the mean estimates of
the parameters. We call these bounds individual
parameter bounds. Note that the parameter bounds become
narrower with more test samples and more complete data
(failures rather than suspensions).
It is equally important to be able to compare two different
data sets and evaluate whether they are statistically
different at a specific confidence level. Contour plots
provide a graphical method that can help us visually
evaluate this hypothesis test. Contour plots give a
joint region for the parameters of the underlying
lifetime distribution.
While similar, individual and
joint parameter bounds yield different results because
of the difference in the degrees of freedom that are
examined in the two cases. They are both powerful tools
in the hands of the reliability engineer. Here we showed
how to obtain these bounds, provided a discussion of
their differences, and explained what answers each one
can provide.
References:
[1] ReliaSoft
Corporation, Life Data Analysis Reference, Tucson, AZ:
ReliaSoft Publishing, 2005.
[2]
https://www.weibull.com/hotwire/issue18/relbasics18.htm
[3]
https://www.weibull.com/hotwire/issue48/relbasics48.htm
[4]
https://www.weibull.com/hotwire/issue60/relbasics60.htm
[5]
https://www.weibull.com/hotwire/issue19/relbasics19.htm
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