Methods and formulas for estimation methods in Parametric Distribution Analysis (Arbitrary Censoring)

In This Topic

Maximum likelihood (MLE)
Least squares (LSE)

Maximum likelihood (MLE)

Maximum likelihood estimates of the parameters are calculated by maximizing the likelihood function with respect to the parameters. The likelihood function describes, for each set of distribution parameters, the chance that the true distribution has those parameters based on the sample data.

Minitab uses the Newton-Raphson algorithm¹ to calculate maximum likelihood estimates of the parameters that define the distribution. The Newton-Raphson algorithm is a recursive method for computing the maximum of a function. All resulting functions, such as the percentiles and survival probabilities, are calculated from that distribution.

Note

For some data, the likelihood function is unbounded and, therefore, yields inconsistent estimates for distributions with a threshold parameter (such as the 2-parameter exponential, 3-parameter Weibull, 3-parameter lognormal, and 3-parameter loglogistic distributions). In these cases, the usual maximum likelihood estimation method can break down. When this happens, Minitab assumes a fixed threshold parameter using a bias correction algorithm and finds the maximum likelihood estimates of the other two parameters. For more information, see references 2, 3, 4, and 5.

References

W. Murray, Ed. (1972). Numerical Methods for Unconstrained Optimization, Academic Press.
F. Giesbrecht and A.H. Kempthorne (1966). "Maximum Likelihood Estimation in the Three-parameter Lognormal Distribution", Journal of the Royal Statistical Society, B 38, 257-264.
H.L. Harter and A.H. Moore (1966). "Local Maximum Likelihood Estimation of the Parameters of the Three-parameter Lognormal Populations from Completed and Censored Samples", Journal of the American Statistical Association, 61, 842-851.
R.A. Lockhart and M.A. Stephens (1994). "Estimation and Tests of Fit for the Three-parameter Weibull Distribution", Journal of the Royal Statistical Society, 56, No. 3, 491-500.
R.L. Smith (1985). "Maximum Likelihood Estimation in a Class of Non-regular Cases", Biometrika, 72, 67-90.

Least squares (LSE)

Least squares estimates are calculated by fitting a regression line to the points in a probability plot from a data set that has the minimal sum of the deviations squared (least square error). The line is formed by regressing either the time to failure or the logarithm of the time to failure (X) to the transformed percent (Y).

Note

For information on how the assumption of common shape or scale parameters affects the LSE or MLE estimates, go to Least squares estimation method and maximum likelihood estimation method and click "Assume common shape or scale parameters for parametric distribution analysis".