Meeker and Escobar (1998) give an example of the number of cycles to fatigue failure of specimens of a certain alloy. The first 67 specimens experienced failure, and the last five specimens had no failure at 300,000 cycles. The following statements create a SAS data set named Alloy that contains the number of cycles (in thousands) to failure or end of test for the specimens:

data Alloy;
   input kCycles@@;
   Cen = _n_ > 67;
   label kCycles = 'Fatigue Life in Thousands of Cycles';
   datalines;
94  96  99  99  104 108 112 114 117 117
118 121 121 123 129 131 133 135 136 139
139 140 141 141 143 144 149 149 152 153
159 159 159 159 162 168 168 169 170 170
171 172 173 176 177 180 180 184 187 188
189 190 196 197 203 205 211 213 224 226
227 256 257 269 271 274 291 300 300 300
300 300
;

The following SAS statements fit a three-parameter Weibull distribution to the specimen lifetimes, in thousands of cycles. The PROFILE option requests a profile likelihood plot for the threshold parameter. ODS Graphics must be enabled to create a profile likelihood plot with the PROFILE option.

proc Reliability data=Alloy;       
   distribution Weibull3;          
   Pplot kCycles*Cen(1) / Profile(noconf range=(50,100)) LifeUpper=500; 
run;                               

Figure 14.43 shows the maximum likelihood estimates of the Weibull threshold, shape and scale parameters, and the corresponding extreme value location and scale parameter estimates.

A probability plot of the failure lifetimes and the fitted three-parameter Weibull distribution is shown in Figure 14.44.

Figure 14.44 Three-Parameter Weibull Probability Plot

A profile likelihood plot for the threshold parameter is shown in Figure 14.45. The threshold value at the maximum log likelihood corresponds to the maximum likelihood estimate of the threshold parameter.

Figure 14.45 Profile Likelihood for Three-Parameter Weibull Threshold