Three-Parameter Weibull

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

Figure 16.45: Three-Parameter Weibull Parameter Estimates

The RELIABILITY Procedure

Three-Parameter Weibull Parameter Estimates
Parameter Estimate Standard Error Asymptotic Normal
95% Confidence Limits
Lower Upper
EV Location 4.5354 0.1009 4.3377 4.7332
EV Scale 0.7575 0.0898 0.6005 0.9556
Weibull Scale 93.2642 9.4082 76.5329 113.6531
Weibull Shape 1.3202 0.1565 1.0465 1.6654
Weibull Threshold 92.9928 1.9516 89.1676 96.8179

A probability plot of the failure lifetimes and the fitted three-parameter Weibull distribution is shown in Figure 16.46.

A profile likelihood plot for the threshold parameter is shown in Figure 16.47. The threshold value at the maximum log likelihood corresponds to the maximum likelihood estimate of the threshold parameter.