Simultaneous Confidence Interval for the Weibull Parameters

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This Demonstration shows the construction of the simultaneous confidence interval (CI) for the Weibull parameters (shape) and (scale) maximum likelihood estimators (MLE). In reliability analysis, the Weibull parameters MLE derive from failure and suspension time data of independently operating systems.

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Use the "seed" slider to generate a pseudorandom test scenario for 20 independently operating systems. Also, you may use the "suspend" slider to randomly suspend up to seven of these test times (right-censored), implying that the corresponding systems have not failed during their operating period, whereas the rest of the systems did fail at the indicated test times. A table on the right of the graph shows the failure (red "f") and suspension (black "s") times that correspond to the specific test scenario.

You can choose to display:

• the event matrix plot showing the failure (boxes) and suspension (arrows) events across the test period.

• the survival plot showing the selected % pointwise bands (brown dashed lines) about the Kaplan-Meier survival function (blue line), and the Weibull fitted model (red curve).

• the contour plot for the likelihood function, depending on the parameters and . The contours correspond to the simultaneous likelihood CI percentiles. The frame ticks indicate the parameters MLE within the selected CI limits.

• the 3D illustration of the contour plot.

• the CI that corresponds to each parameter (blue area).

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Contributed by: Michail Bozoudis (January 2016)
Suggested by: Michail Boutsikas
Open content licensed under CC BY-NC-SA


Snapshots


Details

Assuming that failure times follow a two-parameter Weibull distribution, the maximum likelihood method is applied to estimate the Weibull parameters and .

Given a time dataset that refers to failures and suspensions, the likelihood function for the Weibull parameters is:

,

Reference

[1] R. B. Abernethy, The New Weibull Handbook, 5th ed., North Palm Beach: R. B. Abernethy, 2006.



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