A note on point estimation of the hazard ratio in exponential distributions

The maximum likelihood estimator (MLE) of the ratio of the hazard rates in two exponential distributions is biased. This bias can be important when sample sizes are small or the ratio of these two hazard rates is large. When there is either no censoring or type II censoring, we propose using the uniformly minimum variance unbiased estimator (UMVUE). We show that using the UMVUE instead of the MLE reduces the mean-squared error (MSE). We have found that the UMVUE always has MSEs smaller than the MLE. We have also found that the UMVUE leads to important reductions in the MSE when the sample size used to calculate the hazard rate in the numerator of the hazard ratio is small (say less than or equal to 20) regardless of the sample size in the denominator. In the presence of type I censoring, the proposed estimator and the MLE are both biased. On the basis of a Monte Carlo study, however, we obtain similar reductions in the MSE using the UMVUE, as for no censoring or type II censoring.