  The problem of constructing confidence regions for the median in the
nonparametric measurement error model (NMEM) is considered. This problem arises
in many settings, including inference about the median lifetime of a complex
system arising in engineering, reliability, biomedical, and public health
settings. Current methods of constructing CRs are discussed, including the
T-statistic based CR and the Wilcoxon signed-rank statistic based CR, arguably
the two default methods in applied work when a confidence interval about the
center of a distribution is desired. Optimal equivariant CRs are developed with
focus on subclasses of of the class of all distributions. Applications to a
real car mileage efficiency data set and Proschan's air-conditioning data set
are demonstrated. Simulation studies to compare the performances of the
different CR methods were undertaken. Results of these studies indicate that
the sign-statistic based CR and the optimal CR focused on symmetric
distributions satisfy the confidence level requirement, though they tended to
have higher contents; while two of the bootstrap-based CR procedures and one of
the developed adaptive CR tended to be a tad more liberal but with smaller
contents. A critical recommendation is that, under the NMEM, both the
T-statistic based and Wilcoxon signed-rank statistic based confidence regions
should not be used since they have degraded confidence levels and/or inflated
contents.
