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HISTOGRAM Statement |

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
and the scale parameter are unknown (unless
specified) and that the threshold is known
(it is either specified with the THETA=
option or assumed to be zero).

If it is necessary to estimate in addition to and , 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 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 , as described
by Cohen (1951).
This estimate is then
used to compute maximum likelihood estimates for and .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.

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