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Minimum Variance Designs With Constrained Maximum Bias

Douglas P. Wiens

TL;DR

The paper develops a minimax robustness framework for regression design under model misspecification by decomposing the integrated mse into a variance term and a bias term. It introduces the tuning parameter $\nu$ and the objective $I_\nu(\xi) = (1-\nu) \mathrm{var}(\xi) + \nu \mathrm{maxbias}(\xi)$, proving that the minimax designs are exact solutions $\xi_\nu = \arg\min_\xi I_\nu(\xi)$ and that these designs simultaneously solve two constrained problems: minimize integrated predictor variance with a max-bias bound $b^2$ (rbb) and minimize max bias with a variance bound $s^2$ (rbv). Theoretical results show $\xi_0$ is $I$-optimal and $\xi_1$ is uniform, with $\nu$-indexed designs bridging the two extremes; conversely any minimax design arises from an appropriate bound pair. The paper also provides practical guidance for implementing continuous minimax designs in discrete settings via rounding, along with numerical examples for straight-line and quadratic regression that illustrate the tradeoffs and guide tunings via a coefficient of maximum bias $\mathrm{cmb}(\nu)$.

Abstract

Designs which are minimax in the presence of model misspecifications have been constructed so as to minimize the maximum, over classes of alternate response models, of the integrated mean squared error of the predicted values. This mean squared error decomposes into a term arising solely from variation, and a bias term arising from the model errors. Here we consider of designing so as to minimize the variance of the predictors, subject to a bound on the maximum (over model misspecifications) bias. We consider as well designing so as to minimize the maximum bias, subject to a bound on the variance. We show that solutions to both problems are given by the minimax designs, with appropriately chosen values of their tuning constants. Conversely, any minimax design solves each problem for an appropriate choice of the bound on the maximum bias or variance.

Minimum Variance Designs With Constrained Maximum Bias

TL;DR

The paper develops a minimax robustness framework for regression design under model misspecification by decomposing the integrated mse into a variance term and a bias term. It introduces the tuning parameter and the objective , proving that the minimax designs are exact solutions and that these designs simultaneously solve two constrained problems: minimize integrated predictor variance with a max-bias bound (rbb) and minimize max bias with a variance bound (rbv). Theoretical results show is -optimal and is uniform, with -indexed designs bridging the two extremes; conversely any minimax design arises from an appropriate bound pair. The paper also provides practical guidance for implementing continuous minimax designs in discrete settings via rounding, along with numerical examples for straight-line and quadratic regression that illustrate the tradeoffs and guide tunings via a coefficient of maximum bias .

Abstract

Designs which are minimax in the presence of model misspecifications have been constructed so as to minimize the maximum, over classes of alternate response models, of the integrated mean squared error of the predicted values. This mean squared error decomposes into a term arising solely from variation, and a bias term arising from the model errors. Here we consider of designing so as to minimize the variance of the predictors, subject to a bound on the maximum (over model misspecifications) bias. We consider as well designing so as to minimize the maximum bias, subject to a bound on the variance. We show that solutions to both problems are given by the minimax designs, with appropriately chosen values of their tuning constants. Conversely, any minimax design solves each problem for an appropriate choice of the bound on the maximum bias or variance.
Paper Structure (5 sections, 4 theorems, 23 equations, 3 figures)

This paper contains 5 sections, 4 theorems, 23 equations, 3 figures.

Table of Contents

  1. Introduction and summary
  2. Minimax robustness of design
  3. Robust Bounded Maximum Bias and Bounded Variance designs
  4. Proof of Theorem \ref{['thm: solution']}
  5. Examples

Key Result

Theorem 1

(a) Robust Bounded Bias designs with bias bound $b^{2}$ satisfying (b-range) are given by $\newline$(b) Robust Bounded Variance designs with variance bound $s^{2}$ satisfying (s-range) are given by

Figures (3)

  • Figure 1: (a) rbb($b^{2}(.28)$)/rbv($s^{2}(.28)$) designs for the values displayed (cmb = .33). (b) Implementation of the design in (a); design size $n=10$.
  • Figure 2: imse, var and maxbias vs. $\nu$ for the continuous optimal designs and their implementable approximations ($n=10$).
  • Figure 3: Continuous and implementable designs for quadratic regression; $n=14$.

Theorems & Definitions (4)

  • Theorem 1
  • Lemma 1
  • Lemma 2
  • Lemma 3