Adjusted profile likelihood inference for the scale parameter of the Gumbel distribution

We consider inference about the scale parameter of the Gumbel distribution. The maximum likelihood estimator of the scale parameter based on the profile likelihood is biased. This is because the profile likelihood function is not a genuine likelihood. To address this problem, adjustments of the profile likelihood function of the scale parameter were obtained with the purpose of obtaining improved estimators. The adjustments aim at approximating a conditional likelihood using the so called $p^{\ast }$ formula for the conditional distribution of the maximum likelihood estimator. The resulting point estimators are investigated and compared with the maximum likelihood estimator in terms of their biases and mean squared errors using simulation techniques. We derived confidence intervals based on the profile likelihood and its adjustments. The confidence intervals are investigated in terms of the attainment of lower, upper and total error rates in addition to their simulated expected lengths. Our investigations show that the adjustments improve the performance of point and interval estimators. The methods were applied to real data examples.