By taking advantage of asymptotic results from the statistical literature, it is possible to develop marginal confidence intervals and joint confidence regions for hypocentral parameters derived from Jeffrey's uniform reduction earthquake location algorithm. The method currently used to compute confidence intervals is ad hoc and can result in substantial errors. Monte Carlo experiments have been used to indicate typical deviations from the asymptotic distribution for finite numbers of observations. Also, the statistical theory can be applied to estimating the efficiency of the uniform reduction algorithm when applied to various models of the travel-time residual probability density function. Further, the nonuniqueness of the uniform reduction algorithm can be understood and eliminated by appealing to well-known results from robust statistical theory. Finally, it is possible to derive a weighting function which is asymptotically efficient for any presumed residual model.