Sequential nonparametric fixed-width confidence intervals for conditional quantiles

Let {(Xi, Yi): i ≥ 1} be a sequence of bivariate r.v.'s from a continuous distribution H with marginals F and G, respectively, and let Gx(·) denote the conditional distribution of Y1 given X1 = x,x ε Λ(F), the support of F. In this paper sequential fixed-width confidence interval procedures of length (at most) 2d for the conditional quantile qx(λ) = {y: Gx(y) ≥ λ}, 0 < λ < 1, are studied based on sample conditional quantiles: qnx(λ) = inf{y: Gnx*(y)≥ λ} where Gnx*(·) denotes a 'smoothed' or 'unsmoothed' rank nearest neighbor or Nadaraya-Watson-type kernel estimator of Gx(·). Asymptotics of these sequential confidence interval procedures including their consistency and (relative) efficiency properties are studied, as d→0, on the lines of Chow and Robbins (1965), Geertsema (1970), and Stute (1983). The relative efficiencies and deficiencies of these procedures with respect to each other along with some supportive Monte Carlo results are also presented. © Taylor & Francis Group, LLC.