Rational spline estimators of the quantile function

There exist several interpolation methods used in conjunction with the order statistics for estimating a quantile function: the simple step function estimator, linear interpolation, quasi-quantile estimators, the Hermitian quantile function estimator, kernel estimators and Bernstein polynomial estimators, to name a few. In this note we explore using monotone rational splines for developing quantile function estimators. Delbourgo and Gregory [1] give a representation of a piecewise rational cubic function that is a monotone interpolant. The piecewise cubic interpolant is a generalization of their earlier rational quadratic monotone interpolant defined and analyzed in Gregory and Delbourgo [2] and Delbourgo and Gregory [3], [4]. We construct and examine the estimators based on these interpolants in terms of their exact bootstrap expressions for the finite sample size bias and variance estimates given sample profiles from a standard normal population. Recommendations for the choice of quantile estimator are given.