Percentile estimation based on the power-normal distribution

The authors propose several estimation methods for the confidence interval of the percentile of the power-normal distribution, which specifies the distribution of the observed values before Box–Cox transformation and assumes that transformed distribution is the truncated normal distribution. Their approaches are based on the delta method and the inverse transformation method. Then the finite sample performance of each estimation method is compared through a simulation, and it is shown that the performance of the inverse transformation method is superior to the delta method. A reparametrization method of the power-normal distribution, which is useful parameter setting tool in simulations, is suggested. The authors also investigate the asymptotic behavior of the coverage probability of the confidence interval of the percentile of the power-normal distribution in the case wherein the variance inflation of the MLE associated with the estimation of the transformation parameter of the power-normal distribution is ignored.