Optimal design measures under asymmetric errors, with application to binary design points

Bose, Mausumi; Mukerjee, Rahul

doi:10.1016/j.jspi.2014.10.006

Cited by 19 publications

(13 citation statements)

References 6 publications

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“…These results are similar to those in Bose and Mukerjee (2015), and the proofs are given in the Appendix. For practical purposes, we treat a design as optimal if its weight vectorŵ satisfies:…”

Section: Kiefer-wolfowitz Equivalence Theoremsupporting

confidence: 83%

Using SeDuMi to find various optimal designs for regression models

2017

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We introduce a powerful and yet seldom used numerical approach in statistics for solving a broad class of optimization problems where the search space is discretized. This optimization tool is widely used in engineering for solving semidefinite programming (SDP) problems and is called self-dual minimization (SeDuMi). We focus on optimal design problems and demonstrate how to formulate A-, A s-, c-, I-, and L-optimal design problems as SDP problems and show how they can be effectively solved by SeDuMi in MATLAB. We also show the numerical approach is flexible by applying it to further find optimal designs based on the weighted least squares estimator or when there are constraints on the weight distribution of the sought optimal design. For approximate designs, the optimality of the SDP-generated designs can be verified using the Kiefer-Wolfowitz equivalence theorem. SDP also finds optimal designs for nonlinear regression models commonly used in social and biomedical research. Several examples are presented for linear and nonlinear models.

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Section: Kiefer-wolfowitz Equivalence Theoremsupporting

confidence: 83%

Using SeDuMi to find various optimal designs for regression models

2017

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“…Using this result, Gao and Zhou (2014) proposed new optimality criteria under SLSE and obtained several results. Bose and Mukerjee (2015) and Gao and Zhou (2015) made further developments, including the convexity results for the criteria and numerical algorithms. Bose and Mukerjee (2015) applied the multiplicative algorithms in Zhang and Mukerjee (2013) for computing the optimal designs, while Gao and Zhou (2015) used the CVX program in MATLAB (Grant and Boyd (2013)).…”

Section: Introductionmentioning

confidence: 99%

Optimal designs for regression models using the second-order least squares estimator

Yin¹,

Zhou²

2018

STAT SINICA

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We investigate properties and numerical algorithms for A-and D-optimal regression designs based on the second-order least squares estimator (SLSE). Several results are derived, including a characterization of the A-optimality criterion. We can formulate the optimal design problems under SLSE as semidefinite programming or convex optimization problems and we show that the resulting algorithms can be faster than more conventional multiplicative algorithms, especially in nonlinear models. Our results also indicate that the optimal designs based on the SLSE are more efficient than those based on the ordinary least squares estimator, provided the error distribution is highly skewed.

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“…It is important that

{boldI}_{ξ_{N}} false(bold-italicθ false)

is linear in w , so

{scriptL}_{D} false(ξ_{N} false)

{scriptL}_{D}^{'} false(ξ_{N} false)

and

{scriptL}_{A C} false(ξ_{N} false)

are convex functions of w (Boyd & Vandenberghe, ; Bose & Mukerjee, ). We then apply CVX to find optimal designs after writing the convex optimization problem in a general form:

\begin{matrix} alignleft \\ align-1 \min_{w} L (ξ_{N}), subject to: w_{i} \geq 0, i = 1, …, N, {falsefalse}_{\sum}^{i = 1} w_{i} = 1, & align-2 \end{matrix}

where

scriptL false(ξ_{N} false)

can be

{scriptL}_{D} false(ξ_{N} false)

{scriptL}_{D}^{'} false(ξ_{N} false)

{scriptL}_{A C} false(ξ_{N} false)

, or other convex functions of w discussed later in Section 4.4.…”

Section: Cvx‐based Algorithmsmentioning

confidence: 99%

“…Similar to Bose & Mukerjee () and Wong, Yin & Zhou (), the optimality conditions are stated in Lemma . Let

ξ_{N}^{*}

be the optimal design with weight vector

\overset{boldw}{true}

, let

h_{T i} false(\overset{boldw}{true} false) = trace ((), {boldC}^{⊤} ((), {boldI}_{ξ_{N}^{*}}^{- 1} false(bold-italicθ false) {boldI}_{{boldx}_{i}} false(bold-italicθ false) {boldI}_{ξ_{N}^{*}}^{- 1} false(bold-italicθ false) - {boldI}_{ξ_{N}^{*}}^{- 1} false(bold-italicθ false)) boldC)

, and let

h_{D i} false(\overset{boldw}{true} false) = trace ((), {boldI}_{ξ_{N}^{*}}^{- 1} false(bold-italicθ false) {boldI}_{{boldx}_{i}} false(bold-italicθ false)) - q, i = 1, …, N .

The latter two functions represent the negative directional derivative of the criterion evaluated at

ξ_{N}^{*}

in the direction of the degenerate design at x i .…”

Section: Cvx‐based Algorithmsmentioning

confidence: 99%

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CVX‐based algorithms for constructing various optimal regression designs

Wong

Zhou

2019

Can J Statistics

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CVX‐based numerical algorithms are widely and freely available for solving convex optimization problems but their applications to solve optimal design problems are limited. Using the CVX programs in MATLAB, we demonstrate their utility and flexibility over traditional algorithms in statistics for finding different types of optimal approximate designs under a convex criterion for nonlinear models. They are generally fast and easy to implement for any model and any convex optimality criterion. We derive theoretical properties of the algorithms and use them to generate new A‐, c‐, D‐ and E‐optimal designs for various nonlinear models, including multi‐stage and multi‐objective optimal designs. We report properties of the optimal designs and provide sample CVX program codes for some of our examples that users can amend to find tailored optimal designs for their problems. The Canadian Journal of Statistics 47: 374–391; 2019 © 2019 Statistical Society of Canada

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Optimal design measures under asymmetric errors, with application to binary design points

Cited by 19 publications

References 6 publications

Using SeDuMi to find various optimal designs for regression models

Using SeDuMi to find various optimal designs for regression models

Optimal designs for regression models using the second-order least squares estimator

CVX‐based algorithms for constructing various optimal regression designs

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