The UNIVARIATE Procedure

Example 4.24 Fitting a Three-Parameter Lognormal Curve

If you request a lognormal fit with the LOGNORMAL primary 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. This example shows how you can request a three-parameter lognormal distribution.

A manufacturing process produces a plastic laminate whose strength must exceed a minimum of 25 pounds per square inch (PSI). Samples are tested, and a lognormal distribution is observed for the strengths. It is important to estimate $\theta$ to determine whether the process meets the strength requirement. The following statements save the strengths for 49 samples in the data set Plastic:

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 primary option in the HISTOGRAM statement to display the fitted three-parameter lognormal curve shown in Output 4.24.1:

title 'Three-Parameter Lognormal Fit';
ods graphics on;
proc univariate data=Plastic noprint;
   histogram Strength / lognormal(fill theta = est noprint)
                        odstitle = title;
   inset lognormal    / format=6.2 pos=ne;
run;

The NOPRINT option suppresses the tables of statistical output produced by default. 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 $\zeta$ .

Note: You can also specify THETA=EST with the WEIBULL primary option to fit a three-parameter Weibull distribution.

A sample program for this example, uniex14.sas, is available in the SAS Sample Library for Base SAS software.

Output 4.24.1: Three-Parameter Lognormal Fit