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Optimal Confidence Bands for Shape-restricted Regression in Multidimensions

Ashley, Datta, Somabha Mukherjee, Bodhisattva Sen

Abstract

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 $d Y(\mathbf{t}) = n^{1/2} f(\mathbf{t}) \,d\mathbf{t} + d W(\mathbf{t})$, where $Y$ is the observed stochastic process on $[0,1]^d$ ($d\ge 1$), $W$ 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).

Optimal Confidence Bands for Shape-restricted Regression in Multidimensions

Abstract

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 , where is the observed stochastic process on (), is the standard Brownian sheet on , and 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 and for every member of the underlying function class. Under monotonicity/convexity constraints on , the proposed confidence bands automatically adapt (in terms of width) to the global and local (Hölder) smoothness and intrinsic dimensionality of the unknown ; the bands are also shown to be optimal in a certain sense. These bands have (almost) parametric () widths when the underlying function has ``low-complexity'' (e.g., piecewise constant/affine).
Paper Structure (21 sections, 9 theorems, 197 equations, 4 figures, 1 table)

This paper contains 21 sections, 9 theorems, 197 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Confidence bands for multivariate shape-restricted functions
  3. Preliminaries
  4. Proposed confidence band
  5. Choice of kernels for function classes $\mathcal{F}_1$ and $\mathcal{F}_2$
  6. Adaptivity of the confidence band
  7. Adaptivity with respect to global smoothness
  8. Adaptivity with respect to local smoothness
  9. Adaptivity with respect to intrinsic dimension
  10. Optimality of our confidence band
  11. Simulation studies
  12. Proofs of the main results
  13. Proof of Proposition \ref{['exactcov']}
  14. Proof of Theorem \ref{['spadth']}
  15. Proof of Theorem \ref{['thm2']}
  16. ...and 6 more sections

Key Result

Theorem 1

For kernels $\psi^\ell$ and $\psi^u$ satisfying mainkercondition, we have: for all $f\in \mathcal{F}$ and for all $n\ge 1$.

Figures (4)

  • Figure 1: Confidence band for the function $f(x_1, x_2) = x_1 + x_2$ assuming $f$ is coordinate-wise isotonic and $n = 50^2$.
  • Figure 2: Confidence band for the function $f(x_1, x_2) = \mathbb{I}(x_1\geqslant0.5)$ assuming $f$ is coordinate-wise isotonic and $n = 50^2$.
  • Figure 3: Confidence band for the function $f(x_1, x_2) = |x_1-0.5|$ assuming $f$ is convex and $n = 40^2$.
  • Figure 4: Confidence band for the function $f(x_1, x_2) = 40\{(x_1-0.5)^2+(x_2-0.5)^2\}$ assuming $f$ is convex and $n = 40^2$.

Theorems & Definitions (22)

  • Definition 1: Brownian sheet
  • Theorem 1
  • Proposition 1
  • Theorem 2
  • remark 1: Confidence bands with (locally) parametric widths for "low-complexity" functions
  • Theorem 3
  • Definition 2: Hölder smoothness
  • Theorem 4
  • Theorem 5
  • Definition 3: Intrinsic dimensionality
  • ...and 12 more