Least squares estimates are calculated by fitting a regression line to the points in a probability plot. The line is formed by regressing the data or log(data) on the transformed percent.
Maximum likelihood estimates are calculated by maximizing the likelihood function. The likelihood function describes for each set of distribution parameters the chance that the true distribution has these parameters based on the sample.
Note: For more information on when Minitab uses maximum likelihood versus least squares estimates, see Knowledgebase ID 1331.
Here are the major advantages of each method:
- The probability plot has a better graphical display because the line is fitted to the points.
- For small samples when doing a Reliability/Survival analysis, LSXY is more accurate than MLE. MLE tends to overestimate the shape parameter for a Weibull distribution and underestimate the scale parameter in other distributions. Therefore, MLE will tend to overestimate the low percentiles.
- For heavily censored samples when doing a Reliability/Survival analysis, MLE is more accurate than LSXY.
- Distribution parameter estimates are more precise than LSXY.
- When there are no failures when doing a Reliability/Survival analysis, MLE enables you to perform analyses. When there is only one failure and some right censored observations, maximum likelihood parameter estimates may exist.
- MLE has attractive mathematical qualities.
When possible, both methods should be tried. If the results are consistent, then there is more support for your conclusions. Otherwise, you may want to use the more conservative estimates or consider the advantages of both approaches and choose one based on your problem.
Robert Abernathy (1996). The New Weibull Handbook, 2nd Edition. Dr. Robert B. Abernethy.