IDEAL Quantile Inference via Interpolated Duals of Exact Analytic L-statistics

The literature has two types of fractional order statistics: an `ideal' (unobserved) type based on a beta distribution, and an observable type linearly interpolated between consecutive order statistics. We show convergence in distribution of the two types at an O(n-1) rate, which we also show holds for joint vectors and linear combinations of fractional order statistics. This connection justifies use of the linearly interpolated type in practice when sampling theory is based on the `ideal' type. For example, the coverage probability error (CPE) has the same O(n-1) magnitude for one- sample nonparametric joint confidence intervals over multiple quantiles. For a single quantile, our new analytic calibration reduces the CPE to nearly O(n-3/2), and our new inference method on linear combinations of quantiles has O(n-2/3) CPE. With additional theoretical work, we propose a new method for two-sample quantile treatment effect inference, which has two-sided CPE of order O(n-2/3), or O(n-1) under exchangeability, and one-sided CPE of order O(n-1/2). In an application of our method to data from a recent paper on "gift exchange," we reveal interesting heterogeneity in the treatment effect of "gift wages." In simulations, our quantile treatment effect hypothesis test compares favorably with existing methods in both size and power properties. Along the way, we provide highorder approximations of the PDF and PDF derivative of a Dirichlet distribution in terms of the normal.