Efficient Construction of Test-Inversion Confidence Intervals Using Quantile Regression, With Application To Population Genetics

Modern problems in statistics tend to include estimators of high computational complexity and with complicated distributions. Statistical inference on such estimators usually relies on asymptotic normality assumptions, however, such assumptions are often not applicable for available sample sizes, due to dependencies in the data and other causes. A common alternative is the use of re-sampling procedures, such as the bootstrap, but these may be computationally intensive to an extent that renders them impractical for modern problems. In this paper we develop a method for fast construction of test-inversion bootstrap confidence intervals. Our approach uses quantile regression to model the quantile of an estimator conditional on the true value of the parameter, and we apply it on the Watterson estimator of mutation rate in a standard coalescent model. We demonstrate an improved efficiency of up to 40% from using quantile regression compared to state of the art methods based on stochastic approximation, as measured by the number of simulations required to achieve comparable accuracy.