Dirichlet–Kernel Methods for Geometric Conditional Quantiles: Bahadur Expansions and Boundary Adaptivity on the d-Simplex

This article develops a boundary-adaptive nonparametric methodology for estimating the geometric conditional quantiles of a multivariate response when the conditioning covariate is supported on the simplex—an important case, as it is the natural domain of compositional data. The statistical difficulty addressed here is twofold. First, geometric conditional quantiles for multivariate responses must be defined and estimated through a genuinely directional and convex framework rather than through any scalar ordering. Second, when the covariate is compositional or otherwise simplex-constrained, conventional symmetric kernel procedures suffer from intrinsic support mismatch and severe boundary distortion, thereby compromising both estimation accuracy and inferential validity near faces and edges of the simplex. The method proposed in this paper is designed precisely to overcome this combined obstacle. Our main innovation consists in embedding the spatial quantile formalism of Chaudhuri within a Dirichlet–Kernel smoothing scheme whose shape parameters depend deterministically on the evaluation point. This produces a convex M-estimator that respects the simplex geometry exactly, automatically adapts its local shape to the position of the target point, and removes the need for artificial boundary corrections. To the best of our knowledge, this is the first contribution to provide a complete asymptotic treatment of geometric conditional quantile estimation under simplex-supported covariates with location-adaptive asymmetric kernels. We establish a Bahadur-type linear representation with an explicit negligible remainder, from which we derive refined asymptotic bias and variance expansions. The variance analysis reveals a distinctive geometric phenomenon: each coordinate direction approaching the simplex boundary induces an additional b−1/2 inflation factor, so that the variance at a face of codimension |J| scales as n−1b−(s+|J|)/2. We further obtain the asymptotic mean squared error, an explicit optimal bandwidth rate, asymptotic normality under the nonstandard normalization n1/2b−s/4, and consistent plug-in covariance estimators yielding valid confidence ellipsoids. Numerical experiments and a real-data illustration based on the GEMAS data confirm the practical merit of the approach, especially in boundary regions where classical methods are known to deteriorate.