Lognormal Distribution - Maximum Likelihood Estimation

Techniques

Cumulative Distribution Function

Cumulative Hazard Function

Weibull Distribution

Normal Distribution

Lognormal Distribution

Exponential Distribution

Exam

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The following section describes maximum likelihood estimation for the lognormal 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 provides confidence limits for all parameters as well as for reliability and percentiles.  To estimate the parameters of the lognormal distribution using maximum likelihood estimation, follow these steps:

1. Enter the data using one of the data entry grids, or connect to a database.
2. Select the "Parameter Estimation"
3. Select "Lognormal"
4. Select "Maximum Likelihood (MLE)"

The estimated parameters are given along with 90% confidence limits; an example using the data set "Demo2.dat" is shown below.

 The default confidence level is 90%. The confidence level can be changed using the spin buttons, or by typing over the existing value. Changing the confidence level erases the confidence limits for the parameters. To re-calculate the confidence limits, click the "Compute Confidence Limits" button.

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

Maximum likelihood estimation for the lognormal distribution is accomplished by transforming the lognormal data to normal by taking the logarithm. After the transformation, the maximum likelihood procedure is the same as the procedure used for the normal distribution shown here.