Non-Normal Distribution Fitting - Notes and Technical Information

Parameter Estimation for Non-Normal Distributions

As described in the Overview, Statistica computes maximum likelihood parameter estimates for a wide selection of theoretical distributions.

For some distributions, this requires Statistica to compute solutions to complex simultaneous equations (such as, Weibull, gamma). For those distributions, Statistica first computes either least-squares estimates or matching-moments estimates, and then refines those solutions (to yield maximum likelihood estimates) in an iterative procedure. For details concerning the parameter estimation for those distributions, refer to Evans, Hastings, and Peacock (1993), Johnson and Kotz (1970), or Hahn and Shapiro (1967).

Johnson vs. Pearson Curves

In practice, the estimates (such as percentile values, probability densities) derived by fitting Pearson curves are very similar to those computed by fitting Johnson curves.

Moreover, as Hahn and Shapiro (1967) point out, the non-normal distribution fitted in this manner should only be regarded as an approximation, given the fact that the four moments (mean, standard deviation, skewness, and kurtosis) on which the fit is based are subject to (possibly substantial) sampling fluctuation.

Non-Normal Distribution Fitting in the Process Analysis Module

The Process Analysis module of Statistica will fit a non-normal Johnson curve to the data, and show the fitted curve along with the observed distribution. It is highly recommended, as a first step, always to visually examine the non-normal distribution fit to the observed data by clicking Summary Histogram button on the Advanced, non-normal tab of the Process Capability Analysis - Normal and General Non-Normal Distribution dialog.

Note: the Non-normal frequency distribution button on the same dialog computes the expected non-normal frequencies given the fitted curve. The frequency table spreadsheet will also report differences and relative differences between the observed data and the fitted distribution.

Capability Indices

For the computation of the standard capability indices (based on 6 times sigma as the process width), both percentile values derived from the Johnson distribution fit and the Pearson distribution fit are used.

As explained earlier (see the Overview), in most cases the capability indices computed from the two types of distributions will be very similar.

Only in extreme cases (such as a distribution with skewness greater than 1.7, and kurtosis equal to 10) may small differences in the second digit of the Cpk index occur. For such extreme distributions, the Johnson distributions' fit will give more conservative (slightly smaller) estimates of the process capability.

For distributions with extreme skewness only Johnson curves estimates are reported.

Non-standard process width

You can compute process capability indices based on the Johnson distributions' fit to the non-normal data for a process width other than 3 times sigma.

This option provides great flexibility when one wants to review a range of more or less conservative capability indices for the non-normal data.