Example 4.6: Fitting a Three-Parameter Lognormal Curve

HISTOGRAM Statement

Example 4.6: Fitting a Three-Parameter Lognormal Curve

See CAPL3A in the SAS/QC Sample Library

If you request a lognormal fit with the LOGNORMAL option, a two-parameter lognormal distribution is assumed. This means that the shape parameter $\sigma$ and the scale parameter $\zeta$ are unknown (unless specified) and that the threshold $\theta$ is known (it is either specified with the THETA= option or assumed to be zero).

If it is necessary to estimate $\theta$ in addition to $\zeta$ and $\sigma$ , the distribution is referred to as a three-parameter lognormal distribution. The equation for this distribution is the same as the equation given at "Lognormal Distribution" , but the method of maximum likelihood must be modified. This example shows how you can request a three-parameter lognormal distribution.

A manufacturing process (assumed to be in statistical control) produces a plastic laminate whose strength must exceed a minimum of 25 psi. Samples are tested, and a lognormal distribution is observed for the strengths. It is important to estimate $\theta$ to determine whether the process is capable of meeting the strength requirement. The strengths for 49 samples are saved in the following data set:

   data plastic;
      label strength='Strength in psi';
      input strength @@;
      datalines;
   30.26  31.23  71.96  47.39  33.93  76.15  42.21
   81.37  78.48  72.65  61.63  34.90  24.83  68.93
   43.27  41.76  57.24  23.80  34.03  33.38  21.87
   31.29  32.48  51.54  44.06  42.66  47.98  33.73
   25.80  29.95  60.89  55.33  39.44  34.50  73.51
   43.41  54.67  99.43  50.76  48.81  31.86  33.88
   35.57  60.41  54.92  35.66  59.30  41.96  45.32
   ;

The following statements use the LOGNORMAL option in the HISTOGRAM statement to display the fitted three-parameter lognormal curve shown in Output 4.6.1:

   title 'Three-Parameter Lognormal Fit';
   proc capability data=plastic noprint;
      spec lsl=25 cleft=orange clsl=black;
      histogram strength / lognormal(fill 
                                     color = paoy
                                     theta = est)
                           cfill  = paoy
                           cframe = ligr
                           nolegend;
      inset lsl='LSL' lslpct / cfill = blank pos=nw;
      inset lognormal        / format=6.2 pos=ne cfill = blank;
   run;

Specifying THETA=EST requests a local maximum likelihood estimate (LMLE) for $\theta$ , as described by Cohen (1951). This estimate is then used to compute maximum likelihood estimates for $\sigma$ and $\zeta$ .The sample program CAPL3A illustrates a similar computational method implemented as a SAS/IML program.

See CAPW3A in the SAS/QC Sample Library

Note that you can specify THETA=EST as a Weibull-option to fit a three-parameter Weibull distribution.

Output 4.6.1: Three-Parameter Lognormal Fit

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