2024/01/23 by Ashley, Datta Datta, Datta +4
Mathematics · Medicine · #Bone health and osteoporosis research #FOS: Computer and information sciences #FOS: Mathematics #Methodology (stat.ME) #Point processes and geometric inequalities #Statistical Methods and Inference #Statistics Theory (math.ST)
paper · pdf · doi:10.48550/arxiv.2401.12753
openalex publication_date 2024/01/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper, we propose and study construction of confidence bands for shape-constrained regression functions when the predictor is multivariate. In particular, we consider the continuous multidimensional white noise model given by d Y(t) = n1/2 f(t) dt + d W(t), where Y is the observed stochastic process on [0,1]d (d≥ 1), W is the standard Brownian sheet on [0,1]d, and f is the unknown function of interest assumed to belong to a (shape-constrained) function class, e.g., coordinate-wise monotone functions or convex functions. The constructed confidence bands are based on local kernel averaging with bandwidth chosen automatically via a multivariate multiscale statistic. The confidence bands have guaranteed coverage for every n and for every member of the underlying function class. Under monotonicity/convexity constraints on f, the proposed confidence bands automatically adapt (in terms of width) to the global and local (Hölder) smoothness and intrinsic dimensionality of the unknown f; the bands are also shown to be optimal in a certain sense. These bands have (almost) parametric (n-1/2) widths when the underlying function has ``low-complexity'' (e.g., piecewise constant/affine).