Identifying locally optimal designs for nonlinear models: A simple extension with profound consequences

Abstract

We extend the approach in [Ann. Statist. 38 (2010Statist. 38 ( ) 2499Statist. 38 ( -2524 for identifying locally optimal designs for nonlinear models. Conceptually the extension is relatively simple, but the consequences in terms of applications are profound. As we will demonstrate, we can obtain results for locally optimal designs under many optimality criteria and for a larger class of models than has been done hitherto. In many cases the results lead to optimal designs with the minimal number of support points.

Optimal experimental designs for accelerated failure time with Type I and random censoring

“…Checking whether Q(τ ) satisfies their conditions may be used to compute easily D-optimal and c-optimal designs. Moreover, the complete class of results given by Yang (2010), Yang and Stufken (2012), Dette and Melas (2011), Dette and Schorning (2013) can be applied to prove that the optimal designs have two support points, one of which is a boundary point for given models with two parameters. Otherwise the equivalence theorem will be checked for each particular case.…”

Optimal Group Testing Designs for Estimating Prevalence with Uncertain Testing Errors

“…Hence, the group testing model satisfies the conditions for Theorem 2(a) in Yang & Stufken (2012), and therefore the designs having at most two points together with the designs having three points including x L (which coincides with a = 1) form an essentially complete class. In addition, according to the proof of Theorem 2 in Yang & Stufken (2012), it can be shown that there exists s 0 , …, s 5 = ±1 such that $${false\{s_i{\mathrm{\Psi }}_{i}false\}}_{i=0}^5\text{ is a chebyshev system on }false[r,1false].$$ This property will be used in the proof of Theorem 3. …”

“…Let $${𝒞}_0=true\{{false\{false(x_i,w_{i}false)false\}}_{i=1}^3:x_L=x_1<x_2<x_3\le x_U,\sum _{i=1}^3{w}_i=1,w_i\ge 0true\}.$$ Note that by allowing w i = 0 at some points, this class contains all one- and two-point designs, and all three-point designs with a support point at x L . By applying Theorem 2(a) in Yang & Stufken (2012), we have the following theorem which allows us to search for a D - and a D s -optimal group testing design in 𝒞 0 …”

“…We use Theorem 2(a) in Yang & Stufken (2012) to show that 𝒞 0 is essentially complete. First we rewrite the information matrix (3) as $$Mfalse(\xi false)=\frac{1}{{false(p_1+p_2-1false)}^2{false(1-p_{0}false)}^{2x_L}}false(P_1·P_{2}false)true(\sum _{i=1}^k{w}_{i}Cfalse(a_{i}false)true){false(P_1·P_{2}false)}^{\mathrm{T}}.$$ Let f i 1 ( a ) = d Ψ i ( a )/ da for i = 1, …, 5 and let $$f_{i,j}false(afalse)=\frac{d}{da}true(\frac{f_{i,j-1}false(afalse)}{f_{j-1,j-1}false(afalse)}true),\text{ for }j=2,\dots ,5\text{ and }i=j,\dots ,5.$$ Then for a ∈ [ r , 1] ⊂ (0, 1], we have $$f_{1,1}false(afalse)=-\frac{c+\delta c}{{false(\delta c+afalse)}^2}<0;f_{2,2}false(afalse)=-\frac{2false(c+\delta cfalse)false(\delta c+afalse)}{{false(c-afalse)}^3}<0;f_{3,3}false(afalse)=-\frac{{false(c-afalse)}^{2}false(2a+cfalse)}{2a^2{false(c+\delta cfalse)}^2}<0;f_{4,4}false(afalse)=-\frac{2false(c+\delta cfalse)}{{false(2a+cfalse)}^2}<\mathrm{0...}$$…”

Design Admissibility, Invariance and Optimality in Multiresponse Linear Models

“…One may then restrict oneself to this subclass Ξ com . Along this line, a series of remarkable papers by Yang and Stufken (2009), Yang (2010), Dette and Melas (2011), Yang and Stufken (2012) and Dette and Schorning (2013) derived several complete classes of designs for single response models with respect to the Loewner ordering of the information matrices, based on considerations of admissibility and invariance.…”