Weibull Distribution - Maximum Likelihood Estimation
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Weibull Distribution - Maximum Likelihood Estimation |
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Maximum likelihood estimation is tedious and computer routines are often employed. Commercial software is available for these calculations, such as the Reliability & Maintenance Analyst. The following section describes maximum likelihood estimation for the Weibull distribution using the Reliability & Maintenance Analyst. The manual method is located here.
The maximum likelihood estimation routine is considered the most accurate of the parameter estimation methods, but does not provide a visual goodness-of-fit test. It is recommended to verify goodness-of-fit using probability plotting or hazard plotting, and then, if the fit is acceptable, use maximum likelihood estimation to determine the parameters. Maximum likelihood estimation supports the 2-parameter and 3-parameter Weibull distribution, and provides confidence limits for all parameters as well as for reliability and percentiles. To estimate the parameters of the Weibull distribution using maximum likelihood estimation, follow these steps:
The estimated parameters are given along with 90% confidence limits; an example using the data set "Demo2.dat" is shown below.
In the Location Parameter frame, the software will determine the best value for the location parameter if "Software Estimate" is selected. The user is also given the option of entering a location parameter in the text box below the "User Entered" option. Be cautious when using a non-zero location parameter. A positive location parameter indicates a zero probability of failure for time less than the value of the location parameter, and a negative location parameter means that the population had failures before testing began. Unless there is a sound engineering for one of these conditions, it is best to use a location parameter equal to zero. Clicking the "Plot" button gives a plot of expected reliability with upper and lower confidence limits at the level specified. A plot of percentiles (time as a function of reliability) is produced by selecting the "Percentiles" option in the Plot Type frame before clicking the "Plot" button. The title of the graph can be changed by editing the text in the Graph Title frame. To check the spelling of the title, click the "Spell Check" button. To predict reliability or time-to-fail using the estimated parameters use the Predicting Module. Manual Maximum Likelihood Estimation The maximum likelihood equations for the Weibull distribution arewhere: r is the number of failures, and Iterative techniques are required to solve these equations. The estimated parameters are asymptotically normal. The variances of the estimates can be found by inverting the local information matrix. The local information matrix is The second partial derivatives of the log-likelihood equation are where: The variances of the estimated parameters are Approximate (1–a )100% confidence intervals for the estimated parameters are where These confidence intervals are approximate, but approach exactness as the sample size increases. Confidence intervals for reliability can be found using the expressions Confidence intervals for percentiles can be found using the expressions
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represents summation over all failures,
and
represents summation over all censored points.
is the inverse of the standard normal
probability density function.
