Non‐parametric inference on (conditional) quantile differences and interquantile ranges, using L‐statistics

Abstract

We provide novel, high-order accurate methods for non-parametric inference on quantile differences between two populations in both unconditional and conditional settings. These quantile differences correspond to (conditional) quantile treatment effects under (conditional) independence of a binary treatment and potential outcomes. Our methods use the probability integral transform and a Dirichlet (rather than Gaussian) reference distribution to pick appropriate L-statistics as confidence interval endpoints, achieving highorder accuracy. Using a similar approach, we also propose confidence intervals/sets for vectors of quantiles, interquantile ranges and differences of linear combinations of quantiles. In the conditional setting, when smoothing over continuous covariates, optimal bandwidth and coverage probability rates are derived for all methods. Simulations show that the new confidence intervals have a favourable combination of robust accuracy and short length compared with existing approaches. Detailed steps for confidence interval construction are provided in online Appendix E as supporting information, and code for all methods, simulations and empirical examples is provided.

Fractional order statistic approximation for nonparametric conditional quantile inference

“…The theoretical results we develop contribute to the fractional order statistic literature and provide the basis for inference on other objects of interest explored in Goldman and Kaplan (2016b) and Kaplan (2014). In particular, Theorem 2 tightly links the distributions of L-statistics from the observed and 'ideal' (unobserved) fractional order statistic processes.…”

“…Hutson (1999) had proposed CIs for quantiles using L-statistics (interpolating between order statistics) as endpoints and found they performed well, but formal proofs were lacking. Using the analytic n −1 term we derive in the CPE, we provide a new calibration to achieve O n −3/2 [log(n)] 3 CPE, analogous to the Ho and Lee (2005a) analytic calibration of the CIs in Beran and Hall (1993).The theoretical results we develop contribute to the fractional order statistic literature and provide the basis for inference on other objects of interest explored in Goldman and Kaplan (2016b) and Kaplan (2014). In particular, Theorem 2 tightly links the distributions of L-statistics from the observed and 'ideal' (unobserved) fractional order statistic processes.Additionally, Lemma 7 provides Dirichlet PDF and PDF derivative approximations.High-order accuracy is important for small samples (e.g., for experiments) as well as nonparametric analysis with small local sample sizes.…”

“…Additionally, our framework can be extended to CIs for interquantile ranges and two-sample quantile differences (Goldman and Kaplan, 2016b), which has not been done in the Rényi representation framework.…”

Three essays on crime policy and the Bayesian bootstrap

“…Various other methods have been used to construct such confidence intervals including Kaplan (2015), who uses Edgeworth expansions, and Beran and Hall (1993), who also use L-statistics but with an interpolation weight based on the binomial distribution. Goldman and Kaplan (2018) extend the results of Goldman and Kaplan (2017) to interquartile ranges, among other things.…”

Quantile regression approach for measuring production inefficiency with empirical application to the primary production sector for the Xinjiang Production and Construction Corps in China

“…For example, the regression lines for some quantiles cross each other in some local regions, as also noted by Hao and Naiman ( 2007 ). To overcome this problem, Kuosmanen and Zhou ( 2021 ) include shape constraints, while Cai and Xiao ( 2012 ), Goldman and Kaplan ( 2018 ), and Yu and Jones ( 1998 ) propose use of a semiparametric or local linear model instead of simple linear regression. However, in this paper, we do not employ any of these complex approaches.…”