On Distribution Function Estimation Using Log-Odds Interpolation

This paper considers the problem of distribution function estimation and quantile estimation of a studentized statistic that admits an asymptotic normal distribution. We propose an interpolation method that is built upon a linear approximation to the distribution function on the log-odds scale. The proposal combines a jackknife-type estimator obtained at a subsample size and the limiting normal distribution. We show that with the optimal subsample size the convergence rate of the proposal is at least \(O(n^{-5/6})\), making it more preferable than the delete-d jackknife and at least as accurate as the corrected jackknife method in Booth and Hall (in Ann Stat 21(3):1476–1485, 1993). Extensive simulation studies suggest that the proposal generally yields better performance than existing methods in achieving a smaller mean squared error in the context of quantile estimation across different distributions and tail percentiles.