Confidence Intervals for Threshold Parameters

The calculation of a confidence interval for an unknown threshold is sometimes a non-standard problem. We use a modification of a score statistic initially suggested by Smith for hypothesis testing. The limiting distribution follows a stable law. We use the Mellin transform to obtain explicit formulas for the parameters defining this distribution. We show that if the distribution is to remain non-degenerate over the range of possible parameters values then a special standardization has to be used. We also so show how the score statistic can be modified so that it converges to just a single distribution. This simplifies the tabulation required for practical confidence interval calculations.