Median Rank Regression

Finds the best-fit straight line by using least squares regression (curve fitting) to minimise the sum of the squared deviation (regressing X on Y). Median regression is considered the standard parameter estimation method because it provides the most accurate results on the majority of data sets. In addition to using the simplest method, the Weibull probability plots that this method generates are easily understood.

Cannot be used with a single failure.

Statisticians, who prefer MLE, claim that median regression is not rigorous enough.

Mean Regression

A regression method based on mean values (as originally proposed by Weibull) rather than median values.

Because of the non-symmetrical nature of life data, mean values are generally not as accurate as median values.

Mean Regression Special

A regression method based on mean values instead of median values, where the percentage of failure is the dependent variable and time is the independent variable (regressing Y on X).

Because of the non-symmetrical nature of life data, mean values are generally not as accurate as the median values.

Regressing Y (component age) on X (time to failure) is generally not as accurate as regressing X on Y. This is because the times to failure are much more scattered and have more error than the component ages.

Hazen Regression

A regression method based on midpoint values instead of median values.

Cannot be used with a single failure.

Hazen Regression Special

A method that uses the midpoint to calculate rank regression, where the percentage of failure is the dependent variable and time is the independent variable.

Regressing Y (component age) on X (time to failure) is generally not as accurate as regressing X on Y. This is because the times to failure are much more scattered and have more error than the component ages.

Binomial Regression

An exact method that uses binomial distribution to find the median rank values. This is generally the default parameter estimation method in Weibull software.

Calculations are intensive.

Binomial Regression Special

An exact method that uses binomial distribution to find the rank values, where the percentage of failure is the dependent variable and time is the independent variable.

Calculations are intensive.

Regressing Y (component age) on X (time to failure) is generally not as accurate as regressing X on Y because the times to failure are much more scattered and have more error than the component ages.

Benard Regression

An approximation method to binomial regression that requires less computational time to determine median rank values.

Approximations are used.

Benard Regression Special

A simplified approximation method to binomial regression, where the percentage of failure is the dependent variable and time is the independent variable.

Approximations are used.

Regressing Y (component age) on X (time to failure) is generally not as accurate as regressing X on Y because the times to failure are much more scattered and have more error than the component ages.

Maximum Likelihood Estimation (MLE)

Finds the β and η values that maximise the probability or “likelihood” of obtaining the observed data. MLE is probably the best practice to use with 500 or more failures; however, if the right suspensions exist, MLE can be used with a single failure. If inspection intervals are not the same with all units, MLE should be used.

Calculations are complex and iterative, and convergence does not always occur.

Generally requires more than 500 failures for accurate results. Smaller samples are likely to be biased and yield results that are overly optimistic.

Lacks a good method for plotting the data to produce the graphic data displays important to engineers.

Modified Maximum Likelihood Estimation (MMLE)

To reduce the bias of the estimation, uses the square root of an unbiased estimate of variance, SQR(Var-U), rather than the MLE of the standard deviation from the normal distribution. MMLE is considered the best method if a large sample has many suspensions or dirty data.

All of the disadvantages for MLE apply to MMLE. Although the square root is less biased for small sample than the standard deviation of the normal distribution, small sample bias still exists.