From Dimitri Kececioglu, Reliability & Life Testing Handbook, Page 418 [20].
Sample of 10 units, all tested to failure. The times-to-failure were recorded at 16; 34; 53; 75; 93; 120; 150; 191; 240; and 339 hours.
Published Results (using Rank Regression on Y):
This same data set can be entered into Weibull++ 7 by selecting the Times to Failure type. Use RRY for the estimation method.
Weibull++ computed parameters for RRY are:
The small difference between the published results and the ones obtained from Weibull++ is due to the difference in the median rank values between the two (in the publication, median ranks are obtained from tables to 3 decimal places, whereas in Weibull++ they are calculated and carried out up to the 15th decimal point).
You will also notice that in the examples that follow, a small difference may exist between the published results and the ones obtained from Weibull++. This can be attributed to the difference between the computer numerical precision employed by Weibull++ and the lower number of significant digits used by the original authors. In most of these publications, no information was given as to the numerical precision used.
From Wayne Nelson, Applied Life Data Analysis, Page 415 [30]. One hundred and sixty-seven (167) identical parts were inspected for cracks. The following is a table of their last inspection times and times-to-failure:
Table 6.7- Nelson's Part Cracking Data [26]
|
Data point index |
Number in State |
Last Inspection |
State |
State End Time |
|
1 |
5 |
0 |
F |
6.12 |
|
2 |
16 |
6.12 |
F |
19.92 |
|
3 |
12 |
19.92 |
F |
29.64 |
|
4 |
18 |
29.64 |
F |
35.4 |
|
5 |
18 |
35.4 |
F |
39.72 |
|
6 |
2 |
39.72 |
F |
45.24 |
|
7 |
6 |
45.24 |
F |
52.32 |
|
8 |
17 |
52.32 |
F |
63.48 |
|
9 |
73 |
63.48 |
S |
63.48 |
Published results (using MLE):
Published 95% FM confidence limits on the parameters:
Published variance/covariance matrix:
This same data set can be entered into Weibull++ 7 by selecting the data sheet Times to Failure, with Right Censored Data (Suspensions), with Interval and Left Censored Data and with Grouped Observations options, and using MLE.
Weibull++ computed parameters for maximum likelihood are:
Weibull++ computed 95% FM confidence limits on the parameters:
Weibull++ computed/variance covariance matrix:
From Dallas R. Wingo, IEEE Transactions on Reliability Vol. R-22, No 2, June 1973, Pages 96-100 [37]:
Wingo uses the following times-to-failure: 37, 55, 64, 72, 74, 87, 88, 89, 91, 92, 94, 95, 97, 98, 100, 101, 102, 102, 105, 105, 107, 113, 117, 120, 120, 120, 122, 124, 126, 130, 135, 138, 182. In addition, the following suspensions are used: 4 at 70, 5 at 80, 4 at 99, 3 at 121 and 1 at 150.
Note that you must have the Use True 3-P MLE on Weibull option in the Weibull++ User Setup selected to replicate these results.
From Wayne Nelson, Fan Example, Applied Life Data Analysis, page 317 [30].
Seventy diesel engine fans accumulated 344,440 hours in service and twelve of them failed. A table of their life data is shown next (+ denotes non-failed units or suspensions, using Dr. Nelson's nomenclature). Evaluate the parameters with their two-sided 95% confidence bounds, using MLE for the two-parameter Weibull distribution.
Table
6.9 - Nelson's Fan Failure Data (hr), for Example 17 [30]
Weibull parameters (two-parameter Weibull, MLE):
Published 95% FM confidence limits on the parameters:
Published variance/covariance matrix:
Note that Nelson expresses the results as multiples of 1000 (or 
= 26.297,
etc.). The published results were adjusted by this factor to correlate
with Weibull++ results.
This same data set can be entered into Weibull++ 7 by selecting the data sheet Times to Failure, with Right Censored Data (Suspensions) and I want to enter data in groups (in order to group identical values) options, and using two-parameter Weibull and MLE to calculate the parameter estimates.
You can also enter the data as given in Table 6.9 without grouping them by opening a Data Sheet with Times to Failure and the with Right Censored Data (Suspensions) options. Then click the Group Data icon and chose Group exactly identical values.
The data will be automatically grouped and put into a new grouped data sheet.
Weibull++ computed parameters for maximum likelihood are:
Weibull++ computed 95% FM confidence limits on the parameters:
Weibull++ computed/variance covariance matrix:
The two-sided 95% bounds on the parameters can be determined from the QCP, in the Parameter Bounds tab.
From Dimitri Kececioglu, Reliability & Life Testing Handbook, Page 406 [20].
Estimate the parameters for three-parameter Weibull, for a sample of ten units all tested to failure. The times-to-failure were recorded at 200; 370; 500; 620; 730; 840; 950; 1,050; 1,160; and 1,400 hours.
Published results (using probability plotting):
Weibull++ computed parameters for rank regression on X are:
The small difference between the published results and the ones obtained from Weibull++ are due to the difference in the estimation method. In the publication the parameters were estimated using probability plotting (i.e. the fitted line was "eye-balled"). In Weibull++, the parameters were estimated using non-linear regression (a more accurate, mathematically fitted line). Note that γ in this example is negative. This means that the unadjusted for γ line is concave up, as shown next.
Figure 6-11: Probability Plot of data in Example 18
See Also:
The Weibull Distribution
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