Performance of confidence interval tests for the ratio of two lognormal means applied to Weibull and gamma distribution data

Abstract

Five estimation approaches have been developed to compute the confidence interval (CI) for the ratio of two lognormal means: (1) T, the CI based on the t-test procedure; (2) ML, a traditional maximum likelihood-based approach; (3) BT, a bootstrap approach; (4) R, the signed log-likelihood ratio statistic; and (5) R*, the modified signed log-likelihood ratio statistic. The purpose of this study was to assess the performance of these five approaches when applied to distributions other than lognormal distribution, for which they were derived. Performance was assessed in terms of average length and coverage probability of the CIs for each estimation approaches (i.e., T, ML, BT, R, and R*) when data followed a Weibull or gamma distribution. Four models were discussed in this study. In Model 1, the sample sizes and variances were equal within the two groups. In Model 2, the sample sizes were equal but variances were different within the two groups. In Model 3, the variances were different within the two groups and the larger variance was paired with the larger sample size. In Model 4, the variances were different within the two groups and the larger variance was paired with the smaller sample size. The results showed that when the variances of the two groups were equal, the t-test performed well, no matter what the underlying distribution was and how large the variances of the two groups were. The BT approach performed better than the others when the underlying distribution was not lognormal distribution, although it was inaccurate when the variances were large. The R* test did not perform well when the underlying distribution was Weibull or gamma distributed data, but it performed best when the data followed a lognormal distribution.

Keywords:

• skewed data
• lognormal
• gamma
• Weibull
• confidence interval