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Fitting a Weibull Distribution Using the Kolmogorov-Simirnov GoF Test

We now develop an example of testing for the Weibull assumption, using the data in Table 6. These data consist of six measurements, drawn from the same Weibull (α = 10; β = 2) population. In our example, however, parameters are unknown and estimated.

Table 6. Data for Testing the Weibull Assumption
 11.7216 10.4286 8.0204 7.5778 1.4298 4.1154

We obtain the descriptive statistics (Table 7) and, using graphical methods in [11], also the point estimations of the assumed Weibull parameters: shape β = 1.3 and scale α = 8.7.

Table 7. Descriptive Statistics
Variable N Mean Median StDev Min Max Q1 Q2
DataSet 6 7.22 7.80 3.86 1.43 11.72 3.44 10.75

Using the KS statistic (1) and following the procedures already discussed in the previous section, we obtain the intermediate KS results shown in Table 8.

Table 8. Intermediate Values for the KS GoF Test for Weibull
Row DataSet F0 Fn Fn-1 D+=Fn-F0 D-=F0-Fn-1
1 1.430 0.091176 0.16667 0.000000 0.075491 0.091176
2 4.115 0.314692 0.33333 0.166667 0.018641 0.148025
3 7.578 0.566413 0.50000 0.333333 -0.066413 0.233079
4 8.020 0.593296 0.66667 0.500000 0.073371 0.093296
5 10.429 0.717949 0.83333 0.666667 0.115384 0.051282
6 11.722 0.770846 1.00000 0.833333 0.229154 -0.062488
Max: 0.229154 0.233079

The step-by-step procedure (Table 9) shows how the KS GoF test statistic (1) value (0.23) is smaller than the KS table critical values (0.52 or 0.411) for α = 0.05 and a sample of size n = 6. Based on these results, we do not reject the hypothesis that the population from where these data were obtained, is distributed Weibull (α = 8.7; β = 1.3). Again, recall that the KS test is theoretically used with the true (but usually unknown) distribution parameters and not their estimations. Therefore, when implementing KS with parameters estimated from the data, use an adaptive test procedure (e.g.,α ' = 4 times Error α), as we have done in the present case.

Table 9. Step-by-Step Summary of the KS GoF Test for the Weibull
 Establish the (Null Hypothesis) assumed distribution: Weibull (α; β). Estimate the Weibull parameters: α = 8.7; β = 1.3. Sort the data in ascending order (Col. 1, Table 8). Obtain the Theoretical distribution (Col. 2). Obtain the Empirical distributions (Cols. 3 and 4). Obtain D+ and D- (Cols. 5 and 6). Obtain the KS statistic: D = Maximum (D+, D-) = 0.2331. Obtain the KS table critical value(s) = 0.521 (or 0.411; Adaptive). Since D < CV do not reject the Weibull (α = 8.7; β = 1.3). Software for Weibull version of KS is not commonly available.

Finally, assume we want to assess the data for the Exponential (or any other continuous distribution) assumption. In such case, we obtain the Exponential mean (or any other appropriate distribution parameter) and apply the KS procedure, described in this section for the Weibull distribution. But now, we use instead the Exponential (or other pertinent probabilities) instead of the Weibull, in Column 2 of Table 8.