IDEAL Inference on Conditional Quantiles 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. From the nonparametric perspective of local smoothing, we examine inference on conditional quantiles, as well as linear combinations of conditional quantiles and conditional quantile treatment effects. This paper develops a framework for translating the powerful, high-order accurate IDEAL results (Goldman and Kaplan, 2012) from their original unconditional context into a conditional context, via a uniform kernel. Under mild smoothness assumptions, our new conditional IDEAL method's two-sided pointwise coverage probability error is O(n-2/(2+d)), where d is the dimension of the conditioning vector and n is the total sample size. For d ≤ 2, this is better than the conventional inference based on asymptotic normality or a standard bootstrap. It is also better for other d depending on smoothness assumptions. For example, conditional IDEAL is more accurate for d = 3 unless 11 or more derivatives of the unknown function exist and a corresponding local polynomial of degree 11 is used (which has 364 terms since interactions are required). Even as d → ∞, conditional IDEAL is more accurate unless the number of derivatives is at least four, and the number of terms in the corresponding local polynomial goes to infinity as d → ∞. The tradeoff between the effective (local) sample size and bias determines the optimal bandwidth rate, and we propose a feasible plug-in bandwidth. Simulations show that IDEAL is more accurate than popular current methods, significantly reducing size distortion in some cases while substantially increasing power (while still controlling size) in others. Computationally, our new method runs much more quickly than existing methods for medium and large datasets (roughly n ≥ 1000). We also examine health outcomes in Indonesia for an empirical example.