, are estimated
from:The method used by the application in estimating the different types of confidence bounds for lognormally distributed data is presented in this section. Note that there are closed-form solutions for both the normal and lognormal reliability that can be obtained without the use of the Fisher information matrix. However, these closed-form solutions only apply to complete data. To achieve consistent application across all possible data types, Weibull++ always uses the Fisher matrix in computing confidence intervals. The complete derivations were presented in detail for a general function in the Confidence Bounds chapter. For a discussion on exact confidence bounds for the normal and lognormal, see the Normal Distribution chapter.
This section includes the following subsections:
Fisher Matrix Confidence Bounds for the Lognormal Distribution
Likelihood Ratio Confidence Bounds for the Lognormal Distribution
The lower and upper bounds on the mean, , are estimated
from:
For the standard deviation, 
, ln() is treated
as normally distributed, and
the bounds are estimated from:
where Kα is defined by:
If δ
is the confidence level, then 
for the two-sided bounds
and α
= 1 - δ
for the one-sided bounds.
The variances and covariances of 
and are
estimated as follows:
where Λ is the log-likelihood function of the lognormal distribution.
The reliability of the lognormal distribution is:
Let 
then 
For t
= 
, 
, and for t = 
, 
= . The above equation then becomes:
The bounds on z are estimated from:
where:
or:
The upper and lower bounds on reliability are:
The bounds around time for a given lognormal percentile, or unreliability, are estimated by first solving the reliability equation with respect to time, as follows:
where:
and:
The next step is to calculate the variance of (,):
The upper and lower bounds are then found by:
Solving for TU and TL we get:
Using the data of Example 2 and assuming a lognormal distribution, estimate the parameters using the MLE method.
In this example we have only complete data. Thus, the partials reduce to:
Substituting the values of Ti and solving the above system simultaneously, we get:
Using Eqns. (2) and (3) we get:
and:
The variance/covariance matrix is given by:
See the discussion regarding bias with the normal distribution in the Normal Distribution chapter for information regarding parameter bias in the lognormal distribution.
As covered in the Confidence Bounds chapter, the likelihood confidence bounds are calculated by finding values for θ1 and θ2 that satisfy:
(15)
This equation can be rewritten as:
(16)
For complete data, the likelihood formula for the normal distribution is given by:
(17)
where the xi values represent the
original time-to-failure data. (Note: Note that the procedures outlined
here are for complete data only. For data with suspensions or intervals,
one merely needs to expand the likelihood function to included these other
data types. See the Data & Data
Types chapter for more information.) For a given value of α,
values for and 
can be found which represent
the maximum and minimum values that satisfy Eqn. (16). These represent
the confidence bounds for the parameters at a confidence level δ,
where α
= δ
for two-sided bounds and α
= 2δ - 1
for one-sided.
Five units are put on a reliability test and experience failures at
45, 60, 75, 90, and 115 hours. Assuming a lognormal
distribution, the MLE parameter estimates are calculated to be = 4.2926 and = 0.32361. Calculate the two-sided
75% confidence bounds on these parameters using the likelihood ratio method.
The first step is to calculate the likelihood function for the parameter estimates:
(18)
(19)
where xi are the original time-to-failure data points. We can now rearrange Eqn. (16) to the form:
Since our specified confidence level, δ, is
75%, we can calculate the value of the chi-squared statistic, 
We can now substitute this information into the equation:
It now remains to find the values of and which satisfy this equation. This is an iterative process
that requires setting the value of and finding the appropriate
values of , and vice versa.
The following table gives the values of based on given
values of .
These points are represented graphically in the following contour plot:
(Note that this plot is generated with degrees of freedom k
= 1,
as we are only determining bounds on one parameter. The contour plots
generated in Weibull++ are done with degrees of freedom
k
= 2,
for use in comparing both parameters simultaneously.) As can be determined
from the table the lowest calculated value for is 4.1145,
while the highest is 4.4708. These represent the two-sided 75% confidence
limits on this parameter. Since solutions for the equation do not exist
for values of below 0.24 or above 0.48, these can be considered
the two-sided 75% confidence limits for this parameter. In order to obtain
more accurate values for the confidence limits on , we can
perform the same procedure as before, but finding the two values of σ
that correspond with a given value of . Using this method,
we find that the 75% confidence limits on are 0.23405 and
0.48936, which are close to the initial estimates of 0.24 and 0.48.
In order to calculate the bounds on a time estimate for a given reliability, or on a reliability estimate for a given time, the likelihood function needs to be rewritten in terms of one parameter and time/reliability, so that the maximum and minimum values of the time can be observed as the parameter is varied. This can be accomplished by substituting a form of the normal reliability equation into the likelihood function. The normal reliability equation can be written as:
This can be rearranged to the form:
where Φ-1 is the inverse
standard normal. This equation can now be substituted into Eqn. (17) to
produce a likelihood equation in terms of , t
and R:
(20)
The "unknown" variable t/R depends on what type of bounds are being determined. If one is trying to determine the bounds on time for a given reliability, then R is a known constant and t is the unknown variable. Conversely, if one is trying to determine the bounds on reliability for a given time, then t is a known constant and R is the unknown variable. Either way, Eqn. (20) can be used to solve Eqn. (16) for the values of interest.
For the data given in Example 5, determine the two-sided 75% confidence bounds on the time estimate for a reliability of 80%. The ML estimate for the time at R(t) = 80% is 55.718.
In this example, we are trying to determine the two-sided 75% confidence
bounds on the time estimate of 55.718. This is accomplished by substituting
R
= 0.80
and α = 0.75
into Eqn. (20), and varying until the maximum and minimum
values of t are found.
The following table gives the values of t based on
given values of .
This data set is represented graphically in the following contour plot:
As can be determined from the table, the lowest calculated value for t is 43.634, while the highest is 66.085. These represent the two-sided 75% confidence limits on the time at which reliability is equal to 80%.
For the data given in Example 5, determine the two-sided 75% confidence bounds on the reliability estimate for t = 65. The ML estimate for the reliability at t = 65 is 64.261%.
In this example, we are trying to determine the two-sided 75% confidence
bounds on the reliability estimate of 64.261%. This is accomplished by
substituting t = 65 and α = 0.75
into Eqn. (20), and varying until the maximum and minimum
values of R
are found. The following table gives the values of R based on
given values of .
This data set is represented graphically in the following contour plot:
As can be determined from the table, the lowest calculated value for R is 43.444%, while the highest is 81.508%. These represent the two-sided 75% confidence limits on the reliability at t = 65.
From the Confidence Bounds chapter,
we know that the marginal distribution of parameter is:
where:
φ() is 
, non-informative
prior of .
φ()
is an uniform distribution from - to +,
non-informative prior of .
With the above prior distributions, f(|Data) can be rewritten as:
The one-sided upper bound of is:
The one-sided lower bound of is:
The two-sided bounds of is:
The same method can be used to obtained the bounds of .
The reliable life of the lognormal distribution is:
The one-sided upper on time bound is given by:
(21)
Eqn. (21) can be rewritten in terms of as:
From the posterior distribution of get:
(22)
Eqn. (22) is solved w.r.t. TU. The same method can be applied for one-sided lower bounds and two-sided bounds on Time.
The one-sided upper bound on reliability is given by:
From the posterior distribution of is:
(23)
Eqn. (23) is solved w.r.t. RU. The same method is used to calculate the one-sided lower bounds and two-sided bounds on Reliability.
Determine the two-sided 90% Bayesian confidence bounds on the lognormal parameter estimates for the data given next:
|
Data Point Index |
State End Time |
|
1 |
2 |
|
2 |
5 |
|
3 |
11 |
|
4 |
23 |
|
5 |
29 |
|
6 |
37 |
|
7 |
43 |
|
8 |
59 |
The data is entered into a Times-to-failure data sheet. The lognormal distribution is selected under Distributions. The Bayesian confidence bounds method only applies for the MLE analysis method, therefore, Maximum Likelihood (MLE) is selected under Analysis Method and Use Bayesian is selected under the Confidence Bounds Method on the Analysis tab.
The two-sided 90% Bayesian confidence bounds on the lognormal parameter are obtained using the QCP and clicking the Calculate Bounds button on the Parameter Bounds tab as follows:
See Also:
The Lognormal Distribution
Go to weibull.com
Go to ReliaSoft.com
©1996-2006. ReliaSoft Corporation. ALL RIGHTS RESERVED.