The evaluation of the p-value as an estimator for the null hypothesis in the exponential distribution

This paper is concerned with investigating the adequacy of using the p-value as an estimator for the set specified by the null hypothesis in the Exponential distribution. It is shown that the p-value is an admissible estimator in the one-sided test of the location parameter. When the one-sided test of the scale parameter is considered, the p-value is found to be a generalized Bayes estimator with infinite Bayes risk. However, it is very difficult to find an estimator that dominates it. When the parameter space is restricted, the modified p-value is an admissible estimator in the one-sided test of the scale parameter and performs better than the usual p-value. Although the usual p-value is generally inadmissible in the two-sided test, it can be useful as an estimator in this type of test for the scale parameter.