DS-Optimal Designs. Rita SahaRay Theoretical Statistics and Mathematics Unit, Indian Statistical Institute Kolkata, India

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1 Design Workshop Lecture Notes ISI, Kolkata, Noember, 25-29, 2002, pp DS-Optimal Designs Rita SahaRay Theoretical Statistics and Mathematics Unit, Indian Statistical Institute Kolkata, India 1 Introduction The problem of characterization and construction of optimal designs under both discrete and continuous set up using the well known A-, D- and E- and uniersal optimality criteria has been extensiely studied in the literature. See for example, Shah and Sinha(1989), Pukelsheim(1993). Howeer, the study of the distance optimality criterion put forward by Sinha(1970) in certain treatment designs settings has receied relatiely less attention. Recently there has been a growing interest in this direction (cf. Liski, Luoma, Mandal and Sinha (1998); Liski, Luoma and Zaigrae (1999); Mandal, Shah and Sinha(2000), SahaRay and Bhandari(2001), Liski, Mandal, Shah, Sinha(2002)). Our present discussion is based on the papers cited aboe. In the process we first touch upon the definition of DS optimality criterion and then discuss its properties and finally characterize DS optimal designs in arious settings. We also mention about the connection of this optimality criterion with the well known D- and E- criteria. We start with a classical linear model Y N(Xβ, σ 2 I N ) (1.1) where the N 1 response ector Y = (Y 1,..., Y N ) follows a multiariate normal distribution, X = (X 1,..., X N ) is the N m design matrix, and β = (β 1,..., β m ) is the m 1 parameter ector. E(Y ) = Xβ, and D(Y ) = σ 2 I N are respectiely the expectation ector and the dispersion matrix of Y. Let ˆβ d be the least square estimator(lse) of β in (1.1) using the design d C where C denotes the class of competing designs. We are interested in characterizing an experimental design d which maximises the probability P r [ ˆβ d β < ɛ ] ɛ > 0 (1.2) oer the class C of all competing designs d, where ˆβ d β = [(ˆβ d β) (ˆβ d β)] 1/2, the Euclidean norm of ˆβ d β. As this criterion aims at minimizing the distance between the true parameter alue and its estimate in a stochastic sense, it is abbreiated as the DS (Distance Stochastic)- optimality criterion in the literature. Definition 1.1 A design d C is said to be DS(ɛ) optimal for the LSE of β if for a gien ɛ > 0, it maximizes the probability P r [ ˆβ d β < ɛ competing designs d. ] oer the class C of all

2 34 Rita SahaRay [Noember Definition 1.2 A design d C is said to be DS optimal for the LSE of β if d is DS(ɛ) optimal for all ɛ > 0, i.e. P r [ ˆβ d β < ɛ ] P r [ ˆβ d β < ɛ ] ɛ > 0 (1.3) and for any competing design d C. Note that the DS optimality criterion is defined ia peakedness of the distributions of ˆβ d and ˆβ d. According to a definition proposed by Birnbaum (1948), a random ariable Y 1 is more peaked about µ 1 than is a random ariable Y 2 about µ 2 if P r [ Y 1 µ 1 ɛ ] P r [ Y 2 µ 2 ɛ ] ɛ > 0 (1.4) When µ 1 = µ 2 = 0, we simply say that Y 1 is more peaked than Y 2. Sherman(1955) generalised this definition to the multiariate case. There is also a similarity between the DS-optimality criterion and Pitman nearness. Howeer, Pitman nearness is a stochastic criterion for comparing estimators while DS optimality criterion is for comparison of designs. Let us first consider a line fit model through the origin where Y ij = βx i + ɛ ij, (1.5) E(ɛ ij ) = 0, V (ɛ ij ) = σ 2 for i = 1, 2,..., n and j = 1, 2,..., n i. The responses Y ij s are uncorrelated normal random ariables. Then ( ˆβ β) 2 = χ 2 1 follows the central χ 2 distribution with 1 d.f and V ( ˆβ) P r [ ˆβ β ɛ ] = P r [ ( ˆβ β) 2 ɛ 2 ) = P (χ 2 1 ɛ2 σ 2 n n i x 2 i ) (1.6) Let χ = [a, b] be the regression range and let d p = {a, b; p} with 0 p 1 denote a design that assigns the weights p and 1 p to the regression alues b and a respectiely. If b > a, then the unique maximum of the probability (1.6) is P (χ 2 1 ɛ2 nb 2 σ ). Thus 2 d 1 = {a, b; 1} is the unique DS optimal design. Similarly, if b < a, then d 0 = {a, b; 0} is the unique DS optimal design. Finally, if χ 2 = [ a, a]; then eery design d p = { a, a; p} with any 0 p 1 is DS optimal. In fact under model (1.5) with normally distributed errors, the statements (i) P r [ ˆβ d β ɛ ] P r [ ˆβ d β ɛ ] for all ɛ > 0 (ii) P r [ ˆβ d β ɛ ] P r [ ˆβ d β ɛ ] for some ɛ > 0 (iii) V ( ˆβ d ) V ( ˆβ d ) are equialent (cf. Stepniak 1989). We now consider a more general solution where β may not be estimable. Let η k 1 = L k m β m 1 be the ector of the linear parametric functions of interest to us. We confine only to the class C of the designs d (i.e. the so called design matrix X d ) under which all the components of η are estimable. Let the Best Linear Unbiased Estimator (BLUE ) of η using the design d be denoted by ˆη d, where ˆη d = Lˆβ d,

3 2002] DS-Optimal Designs 35 Let ˆη d1 and ˆη d2 be the LSE s of ˆη in (1.1) under the designs d 1 and d 2 respectiely. If for a gien ɛ > 0, P r [ ˆη d1 η < ɛ ] P r [ ˆη d2 η < ɛ ], (1.7) then the design d 1 is at least as good as d 2 with respect to the DS(ɛ) criterion. A design d is said to be DS(ɛ) optimal for the LSE of η if for a gien ɛ > 0, it maximizes the probability P r [ ˆη d η < ɛ ]. When d is DS(ɛ) optimal for all ɛ > 0, we say that d is DS-optimal. In the particular case k = 1, the DS criterion coincides with the DS(ɛ) criterion for a gien ɛ > 0. Note that according to the usual definition of Stochastic ordering for random ariables (see Marshall & Olkin 1979, p. 481) ˆη d1 η is stochastically less than ˆη d2 η if (1.7) holds for all ɛ > 0. For the time being, we assume η to be nonsingularly estimable i.e rank(l) = k. Let σ 2 Σ d be he dispersion matrix and I(d) = 1 σ Σ 1 2 d be the information matrix of ˆη d under the gien model. Let T Λ d T be the spectral decomposition of I(d) where T is an orthogonal k k matrix and Λ d = Diag(λ d1,..., λ d,k ) is the diagonal matrix of the eigenalues of I(d), arranged in decreasing order. Define Z = 1 σ Λ1/2 d T (ˆη d η), so that Then Z N k (0, I k ). P r [ ˆη d η < ɛ ] = P r [(ˆη d η) (ˆη d η) < ɛ 2 ] (1.8) = P r[ Z Λ 1 Z ɛ 2 /σ 2 ] = P r[ Z i 2 λ di δ 2 ] (1.9) for δ = ɛ/σ. Thus δ 2 > 0, the DS(ɛ) optimality criterion depend on I(d) only through its eigenalues λ = (λ d1,..., λ dk ). We define the criterion function ψ ɛ or equialently ψ δ as ψ ɛ (I(d)) = P r [ ˆη d η 2 < ɛ 2 ] and ψ δ (λ d ) = P r[ Z i 2 λ di δ 2 ] (1.10) It is clear that ψ ɛ (I(d)) = ψ δ (λ d ) for δ = ɛ σ > 0. As a function of δ 2 the DS(ɛ) optimality criterion ψ δ (λ d ) is the cumulatie distribution function of Z i 2 λ di for eery λ d IR k +. 2 Properties of the DS optimality criterion In the below we sometimes skip the suffix d to aoid notational complexity. information matrix of the design d i be denote by I i. It directly follows from (1.10) that Let the ψ ɛ (ai) = ψ aɛ(i) and ψ δ (aλ d ) = ψ aδ(λ)

4 36 Rita SahaRay [Noember for all a > 0. It is clear that the optimality criterion ψ ɛ for gien ɛ > 0 induces an ordering among the designs and among the corresponding information matrices of designs. A design d 1 is said to be at least as good as d 2 relatie to the criterion ψ ɛ if ψ ɛ (I 1 ) ψ ɛ (I 2 ) which in other words mean that the informaton matrix I 1 is at least as good as I 2 with respect to ψ ɛ. 2.1 Isotonicity and Admissibility The function ψ ɛ preseres the matrix ordering. Theorem 2.1 The DS criterion is isotonic relatie to Loewner ordering i.e. Proof : Hence the eent E 1 : {Z : 2 Z i λ 2i consequently ψ ɛ (I 2 ) =Pr ( 2 Z i I 1 I 2 > 0 ψ ɛ (I 1 ) ψ ɛ (I 2 ) for all ɛ > 0. I 1 I 2 λ 1i λ 2i i = 1,..., k. δ 2 } implies the eent E 2 : {Z : 2 Z i 2 λ 1i δ 2 } and λ 2i δ 2 ) Pr( Z i λ 1i ] δ 2 ) = ψ ɛ (I 1 ) ɛ > 0. A reasonable weakest requirement for an information matrix I is that there be no competing information matrix Ĩ which is better than I in the Loewner ordering sense. We say that an information matrix I is admissible when eery competing moment matrix Ĩ with Ĩ I is actually equal to I (cf. Pukelsheim 1993, chapter 10 ). A design d is rated admissible when its information matrix I(d) is admissible. The admissible designs form a competing class (Pukelsheim 1993, Lemma 10.3). Thus eery inadmisible information matrix may be improed. If I is inadmissible, then there exists an admissible information matrix Ĩ I such that Ĩ I. Since ψ ɛ is isotonic relatie to Loewner ordering, DS(ɛ) optimal designs as well as DS- optimal ones can be found in the set of admissible designs. 2.2 Schur Concaity The notion of majorization proes useful in the study of the function ψ δ (λ d ). Majorization concerns the diersity of the components of a ector. (cf. Marshall and Olkin 1979, p.7). Let a = (a 1,..., a k ), and b = (b 1,..., b k ) be two k 1 ectors and a (1)... a (k), b (1)... b (k) be the ordered components. Definition 2.2 : For a, b IR k, a is said to majorize b, written a b if p a (i) p b } (i) p = 1,..., k 1, k a (i) = k b (i). (2.1) Definition 2.3 : For a, b IR k, a is said to weakly supermajorize b, written a w b if p a (i) p b (i) p = 1,..., k. (2.2) Majorization proides a partial ordering on IR k. The order a b implies that the elements of a are more dierse than the elements of b. Then for example, a ā = (ā,..., ā) for all a, IR k, where ā = 1 k k a i. Functions which reerse the ordering of majorization are said to be Schur Concae (cf. Marshall and Olkin 1979, p. 54). Definition 2.4 : A function f(x) : IR k IR is said to be a Schur Concae function if for x, y IR k the relation x y implies f(x) f(y). Thus the alue of f(x) becomes greater when the components of x become less dierse.

5 2002] DS-Optimal Designs 37 Next we examine Schur Concaity of DS(ɛ) criterion. For that we quote below Proposition of Tong 1990, p.163. Proposition 1 For gien a = (a 1,..., a k ), a i > 0 consider an ellipsoid defined by A 2 (a) = {x : x IR k,, ( x i a i ) 2 λ}, λ > 0, fixed. If the density function f(x) is a Schur concae function of x then P [x A 2 (a)] is a Schur Concae function of a 2 = (a 2 1,..., a 2 n). Theorem 2.5 The DS(ɛ) criterion is Schur Concae for all ɛ > 0. Proof: The components of Z being i.i.d Normal with mean zero and ariance σ 2, the joint density function f(z) is Schur Concae (Tong 1990, Theorem 4.4.1). Then by the aboe Proposition ψ δ (λ d ) is a Schur Concae function of λ d = (λ d1,..., λ dk ) for all δ > 0. Corollary 2.6 ψ δ (λ d ) ψ δ ( λ d ) holds for all λ d IR k + and all δ > 0 where λ d = ( λ d,..., λ d ) where λ d = 1 k k λ di. Corollary 2.7 Let λ 1 and λ 2 denote the column ectors where components are the eigen alues of I 1 and I 2 respectiely arranged in decreasing order. If I is the information matrix with a ector of eigenalues (1 α)λ 1 + αλ 2 then, ψ α ((1 α)i 1 + αi 2 ) ψ ɛ (I) for all α [0, 1] and all ɛ > 0. Proof: Let λ[(1 α)i 1 + αi 2 ] denote the column ector of eigenalues of (1 α)i 1 + αi 2 arranged in decreasing order. Since by Theorem G.1 (Marshall & Olkin 1979, p. 241) (1 α)λ 1 + αλ 2 λ((1 α)i 1 + αi 2 ), the result follows. Theorem 2.8 ψ δ (λ) is Schur concae function of (log λ 1,..., log λ k ) for all δ > 0. Corollary 2.9 ψ δ (λ d ) ψ δ ( λ d ) holds for all λ d IR k + and all δ > 0 where λ d = ( λ,..., λ) and λ = k λ1/k i. Theorem 2.10 If λ d w λ d then ψ δ (λ d ) ψ δ ( λ d ). Proof: If λ d w λ d then there exists (cf. Marshall and Olkin p.11) a ector λ d0 such that λ d0 λ d and λ d0 λ d Thus P r( 2 Z i δ 2 ) P r( 2 Z i δ 2 ) P r( 2 Z i δ λ di λ 2 ) δ > 0. di λ d0i The first inequality follows from implication of eents as discussed earlier in the proof of Theorem 2.1. The last inequality now follows from Theorem 2.5. Theorem 2.8 is a ersion of Okamoto Lemma (1960). There are other generalisations of this useful result. We reproduce them below from Sinha(1970) and Liski et.al (1998). (We omit the proofs altoghether). To describe the results along similar fashion, Okamoto s Lemma is stated in a conentional form. Below Z i s are i.i.d N(0, 1). Note further that in Lemma 2.11 and 1 generalisations thereafter, λ di s are replaced by µ di s (omitting suffix d) and used in the numerator instead of the denominator.

6 38 Rita SahaRay [Noember Lemma 2.11 (Okamoto) P r( k µ i Z 2 i ɛ 2 ) P r(µχ 2 k ɛ 2 ) where χ k 2 refers to a central χ 2 ariate with k degrees of freedom and µ = ( µ i ) 1/k. Generalisation 1: where Generalisation 2: proided Generalisation 3: P r( k µ i Z 2 i ɛ 2 ) P r(µ χ 2 k + µ χ 2 k ɛ 2 ) k = k + k, µ = ( µ i ) 1/k and µ = ( k k i=k +1 P (µ 1 Z µ 2 Z 2 2 ɛ 2 ) P (γ 1 Z γ 2 Z 2 2 ɛ 2 ) µ i ) 1/k µ 1 µ 2 γ 1 γ 2 and max{µ 1, µ 2 } max{γ 1, γ 2 }. P (µ 1χ µ 2χ 2 2 ɛ 2 ) P (µ 1 χ µ 2 χ 2 2 ɛ 2 ) proided and (µ 1) 1 (µ 2) 2 > (µ 1 ) 1 (µ 2 ) 2 max{µ 1, µ 2} > max{µ 1, µ 2 } > min{µ 1, µ 2 } > min{µ 1, µ 2}. 2.3 Concaity Concaity is often regarded as a compelling property of an optimality criterion (cf. Pukelsheim 1993, p. 115), but DS(ɛ) optimality criterion is not, in general, concae. Theorem 2.12 The function ψ δ (λ) is concae on IR k + for eery fixed δ > 0 if and only if k 2. Remark 1 In the below, while characterising DS optimal Designs under arious settings, we use formulations (1.9) or the one stated in Lemma 2.11 depending on the tool employed to arrie at the optimal design. 3 Discrete DS optimal Designs In this section we derie DS optimal designs for the LSE of η for different choices of the parametric ector of interest.

7 2002] DS-Optimal Designs Mean ector and the set of all elementary contrasts in a CRD Model Sinha(1970) introduced DS-optimality criterion for optimal allocation of obserations with a gien total in a CRD model: Y ij = µ + τ i + ɛ ij, i = 1,..., ; j = 1,..., n i, (3.1) where τ = (τ 1,..., τ ) is the ector of treatment effects. The parametric ector of interest is the mean ector η = (µ + τ 1,..., µ + τ ). For a design d C, let the ith treatment be allocated n di times, n di 1, 1 i, n di = n. Referring to (1.9) we note that λ di = n di for each i. When n is diisible by it turns out that a symmetrical allocation with n di = n/ i, is uniquely DS- optimal. The general case, when n is not diisible by is quite inoled and Sinha (1978) established the partial result when is een and n = 2 (mod). Very recently this problem is resoled by Liski et. al.(1998). A most symmetrical allocation d with n d i n d j 1, i j, turns out to be DS- optimal as is expected. We outline the proof below. For any design d( d ) C there exists at least a pair of treatments (i, j) such that n di n dj > 1. It is easy to check that n d = (n d1,..., n d(i 1), n di, n d(i+1),..., n d(j 1), n dj, n d(j+1)..., n d ) n d0 = (n d1,..., n d(i 1), n di 1, n d(i+1),..., n d(j 1), n dj + 1, n d(j+1)..., n d ). Then using Theorem 2.5 P r( 2 Z i δ 2 ) P r( 2 Z i δ 2 ) (3.2) n di By repeated application of (3.2) it can be shown that wheneer allocation numbers for a pair of treatments differ by more than 1, successiely reducing thier difference by 2, but keeping the total fixed, a better design can be obtained and finally, a most symmetrical allocation with n di n dj 1, i j, turns out to be DS- optimal. Recently, SahaRay and Bhandari(2001) examined the nature of DS -optimal designs in a CRD set up for inference on the set of all elementary contrasts of the form τ i τ j, i < j iz. η = (τ 1 τ 2,..., τ 1 τ,..., τ 1 τ ). Writing η = Lτ, we note that R(L) = 1. Thus it corresponds to a singularly estimable full rank problem. Here weak majorization plays an important role to establish DSoptimality of symmetrical or most symmetrical allocations depending on the diisibility of the total number of obserations by the number of treatments or not. We sketch the proof below. Let P be a ( 1) submatrix of an orthogonal matrix such that n d0i P 1 P 1 = I 1, P P = (I J/), (3.3) D = Diag(1/n d1,..., 1/n d ), and D 1/2 = Diag(1/ n d1,..., 1/ n d ). Writing ˆη d = Lˆτ d, where ˆτ d = (ȳ 1.,..., ȳ. ), we hae from (1.7) (3.4) P ɛ = P r[(ˆη d η) (ˆη d η) ɛ 2 ]

8 40 Rita SahaRay [Noember = P r[(ˆτ d τ) L L(ˆτ d τ) ɛ 2 ] = P r[(ˆτ d τ) (I J/)(ˆτ d τ) ɛ 2 /] = P r[(ˆτ d τ) P P (ˆτ d τ) ɛ 2 /] = P r[(ξ d ξ d ) ɛ 2 /], (3.5) where ξ d = P (ˆτ d τ) N 1 (0, σ 2 Σ d ) with Σ d = P DP. For this specific problem, σ 2 does not play any role in the determination of DS optimal designs. So we assume σ 2 = 1 in particular. It is not hard to erify that for any d C, Σ d = P DP is nonsingular. Let µ d = (µ d1,..., µ d,( 1) ) denote the ector of ordered eigenalues of P DP where µ d1 µ d2 µ d,( 1). Let T Λ T = Σ d be the spectral decomposition of Σ d, the dispersion matrix of ξ, where T is an orthogonal ( 1) ( 1) matrix and Λ = Diag(µ d1,..., µ d,( 1) ) is the diagonal matrix of the eigenalues of the dispersion matrix Σ d, in other words of P DP. Then using cannonical reduction as discussed in (1.9), we get P r [ ˆη d η < ɛ ] = P r[ µ di Z i 2 δ 2 ] (3.6) for δ 2 = ɛ 2 /. (Note that here µ di s are eigenalues of the corresponding dispersion matrix whereas in (1.9) λ di s are the eigenalues of the corresponding information matrix.) Thus δ 2 > 0, the DS(ɛ) optimality criterion P r[ ηˆ d η < ɛ ] depend on the design d with allocation numbers n di, i = 1,..., only through the eigenalues µ d1,..., µ d,( 1) of the matrix (P DP ). It is worthwhile to note that µ di s are ery nontriial functions of n di s, unlike the problem of estimation of mean ector. Furthermore, µ di s do not depend on the choice of the P matrix where P P = I J/ and P P = I 1 as is clear, by defining A = P D 1/2 and B = A and noting that the positie eigenalues of AB = P DP and BA = D 1/2 P P D 1/2 = D 1/2 (I J/)D 1/2 are equal. Remark 2 Instead of η if we had considered the set of all contrasts of the form τ i τ j, i j, the problem would hae remained the same except that δ 2 in (3.6) would change to a scalar multiple of it, iz δ 2 /2. Wheneer n is diisible by, using Okamoto s Lemma and well known inequality between the Arithmetic Mean(A.M) and the Geometric Mean(G.M) of a set of positie quantities, DS optimality of a symmetric design d with n d i = n/ can be shown through the following steps. First we note that, in the present context of estimation of the set of all elementary contrasts, So P DP = 1 n µ di = (3.7) n di P r ( 1 µ di Z 2 i δ 2 ) P r [(n/( n di )) 1/ 1 χ 2 1 δ 2 ] P r [(n/( n d i)) 1/ 1 χ 2 1 δ 2 ] (follows by implication of eents)

9 2002] DS-Optimal Designs = P r [ ( µ d i) ( 1) χ 2 1 δ 2 ] = P r ( 1 µ d iz 2 i δ 2 ) ( as µ d i s are all equal). Thus wheneer n is diisible by, the design d with symmetrical allocation turns out to be DS- optimal. The case when n is not diisible by is dealt below. From now onwards we assume that n = k + t, t 1, where k = [n/] denotes the greatest integer less than or equal to n/. Let d C be a design with n d = (k,..., k, k + 1,..., k + 1), k occuring t times and corresponding D = Diag(1/k,..., 1/k, 1/(k + 1),..., 1/(k + 1) ). }{{}}{{} t times t times In order to establish d to be DS-optimal, in iew of (3.6) and Theorem 2.10, it suffices to show that µ d 1 w µ d 1 where µ d 1 and µ d 1 denote respectiely the ectors of eigenalues of (P DP ) 1 and (P D P ) 1, the information matrices. Step I: We will establish that µ(d 1 (A + ɛi) µ(d 1 (A + ɛi)), ɛ IR, ɛ 0 and 1. We first note that, for any ɛ IR, ɛ 0 and 1, A + ɛ I is a nonsingular matrix and the eigenalues of D 1/2 (A + ɛ I)D 1/2 and D 1 (A + ɛ I) are identical. It is clear that for any design d( d ) C, there exists at least one pair of treatment symbols i and j such that (n di n dj ) 2 and n di = n. We permute i and j treatment symbols, keeping others fixed and obtain d C as n di = n di i i, j n di = n dj n dj = n di and hence D 1 = QD 1 Q, where Q represents the corresponding permutation matrix. In iew of the relation Q Q = QQ = I, and Q(A + ɛ I)Q = A + ɛ I, Hence D 1 (A + ɛi) = QD 1 Q (A + ɛi) = QD 1 Q Q(A + ɛi)q = QD 1 (A + ɛi)q. (3.8) µ( D 1 (A + ɛ I)) = µ(d 1 (A + ɛ I)) Now it is easy to see that for some 0 < α < 1, (n di 1, n dj + 1) can be represented as a conex combination of (n di, n dj ) and (n dj, n di ). Choosing this α, it follows that µ(d 1 (A + ɛi)) = αµ(d 1 (A + ɛi)) + (1 α)µ(qd 1 (A + ɛi)q ) = αµ(d 1 (A + ɛi)) + (1 α)µ( D 1 (A + ɛi))

10 42 Rita SahaRay [Noember µ(αd 1 (A + ɛi) + (1 α) D 1 (A + ɛi)) (3.9) (cf. proof of Corollary 2.7) = µ((αd 1 + (1 α) D 1 )(A + ɛi)) = µ(d 0 1 (A + ɛi)), (3.10) where D 1 0 = αd 1 + (1 α) D 1 = Diag(n d1,..., n d(i 1), n di 1, n d(i +1),..., n d(j 1), n dj + 1, n d(j +1),..., n d ). Thus when the pair of allocation numbers (n di, n dj ) is transferred to a pair (n di 1, n dj + 1), reducing their mutual difference by two, but keeping the total fixed, we get µ(d 1 (A + ɛ I)) µ(d 0 1 (A + ɛ I)), ɛ IR, ɛ 0and 1. Note that starting from D 1 successie aeraging by taking conex combination of any two co-ordinates of n d = (n d1,... n d ) in the aboe sense, while keeping the rest of the co-ordinates fixed, we will eentually get D 1 = diag(k,..., k, k + 1,..., k + 1) and similar successie steps of majorization will yield Step II: We will now establish that µ(d 1 (A + ɛi)) µ(d 0 1 (A + ɛi)) µ(d 1 (A + ɛi)). (3.11) µ d 1 w µ d 1 where µ d 1 and µ d 1 denote respectiely the ectors of eigenalues of (P DP ) 1 and (P D P ) 1. In Step I, (3.10) can be rewritten alternatiely as As A = I J/, Call 1+ɛ ɛ µ 1 (D(A + ɛ I) 1 ) µ 1 (D (A + ɛ I) 1 ). (3.12) (A + ɛi) 1 = ɛ (A ɛ J/), ɛ IR, ɛ 0 and 1. (3.13) ɛ = θ. Thus for any θ > 0, (3.4) can be rewritten as Now use the result that and µ 1 (D(A + θj/)) µ 1 (D (A + θj/)). (3.14) lim 1(D(A + θj/)) θ 0 = 0, lim µ i (D(A + θj/)) θ 0 = λ i (DA), i 1. and note that µ i (DA) = µ i 1 (P DP ), i = 2,...,. Referring to the Def 2.2, it can be seen that (3.13) will yield inequalities of which the first ( 1) after imposing limit will yield µ d 1 = (1/µ d1,..., 1/µ d( 1) ) w (1/µ d 1,..., 1/µ d ( 1)) = µ d 1 (3.15) and hence the result. It can be easily seen that d as well as any permutation of d is uniquely DS optimal.

11 2002] DS-Optimal Designs A full set of orthonormal contrasts in the block design model Now we consider a block design model. We explain the connection between DS optimal designs and uniersally optimal designs in the sense of Kiefer(1975). For the inferential problem of a full set of orthonormal treatment contrasts i.e η d = P τ d the DS-optimality criterion boils down to maximize P r [ ˆη d η < ɛ ] = P r[ξ d ξ d ɛ2 ] where ξ d = P (ˆτ d τ) N 1 (0, Σ d ) with Σ d = (P C d P ) 1 and C d is the usual C matrix under the design d. Now application of Corollary 2.6 and arguments used through implication of eents yield that if there exists a design d for which the C d is completely symmetric and tr(c d ) = max d C tr(c d ) then the design d is DS optimal. This coers BIBDs, BBDs, Yoden square designs, in particular. (ide Kiefer 1975). 3.3 Control s. test treatment comparisons in a CRD model Optimality studies in the context of treatment s.control comparisons has receied considerable attention of the researchers in the recent years. See Majumder (1996)and references therein. Dealing with a CRD model and block design set up, Mandal, Shah Sinha (2000) characterizes DS optimal designs for inference on the ector of parametric contrasts inoling a set of treatments and one control, i.e η = (τ 0 τ 1, τ 0 τ 2,..., τ 0 τ ) = Lτ (3.16) where L = (1 I), τ = (τ 0, τ 1,..., τ ) and τ 0 refers to the effect of the control treatment and τ 1,..., τ refer to the test treatments. Let under a CRD d in the competing class C, n d0,, n d1,..., n d denote allocation numbers subject to a total of n obserations. Setting p di = n di n, i = 0, 1, 2,..., so that i=0 p di = 1, d C, the problem is to seek optimal alues of p di s so as to maximize P ɛ = P r [ ˆη d η < ɛ ] ɛ > 0. It is not hard to erfy that ariance- coariance matrix of ˆη d turns out to be D(ˆη d ) = σ 2 [D + J n d0 ] where D = diag( 1 n d1,..., 1 n d ) and J = ((1)), the matrix of all ones. Let µ d1,..., µ d denote the eigenalues of σ 2 D(ˆη d ). Then Further µ di = D + J n d0 = 1 n ( n ) ( di n d0 ) n (A.M G.M) (3.17) n n d0 n d0 µ max 1 J (D + n d0 ) n n d0 n d0 n = n d0 (n n d0 ) = 1 n di + 2 n d0 (usinga.m H.M) (3.18)

12 44 Rita SahaRay [Noember In (3.16) and (3.17) = holds wheneer n d1 = n d2 = = n d = (n n d0 )/ for eery n d0. Referring to the probability inequality in Generalisation 3, we set Note that 1 = 1, 2 = 1, µ 1 = µ max, µ 2 = ( µ n 1 = n d0 (n n d0 ) and µ 2 = µdi µ max ) 1/( 1) (n n d0 ) µ 1 = max(µ 1, µ 2) max(µ 1, µ 2 ) = µ 1 (3.19) Now we deal with the two cases µ 2 µ 2 and µ 2 < µ 2 separately. Suppose first that µ 2 µ 2. Then an application of Generalisation 1 gies P r( k µ i Z 2 i ɛ 2 ) P r(µ max χ µ 2χ 2 1 ɛ 2 ) P r(µ 1 χ µ 2 χ 1 2 ɛ 2 ) (3.20) The last equation follows by implication of eents. If on the other hand, µ 2 < µ 2 conditions reqiured in Generalisation 3 are satisfied and hence the aboe result follows from application of Generalisation 1 followed by that of Generalisation 3. Thus, for eery fixed n d0 = n 0 say, the allocation (n 0, n n0,..., n n0 ) is uniformly beter than the allocation (n 0, n d1,..., n d ) for all choices of n di s subject to n di = n n 0. Thus for this problem, P r( k µ i Z 2 i ɛ 2 n ) P r( n 0 (n n 0 ) χ2 1 + χ 2 1 ɛ 2 ) n n 0 = P r(χ p 0 χ 1 2 p 0 (1 p 0 )ɛ 2 ) = P ɛ (say) According to the approximate design theory, p 0 is chosen optimally to maximize P ɛ for eery ɛ > 0. For seeral choices of ɛ and using 10,000 simulation Mandal and Shah and Sinha (2000) obsere that no single p 0 is optimal for all alues of ɛ. Further the DS optimal design d are almost the same as A- optimal allocations. Since the optimal designs are ery much ɛ dependent it would be reasonable to choose a alue of p 0 which maximizes P ɛ aeraged with respect to an appropriate weight function for ɛ. Assuming the p.d.f of ɛ 2 as that of χ 2 2 Mandal et al. (2000) set a alue for the weighted coerage probability P and search for aalue of p 0 which will maximise /n and thereby minimize the alue of n required to attain P. Results are tabulated in the paper for choices of P =.9 and Control-treatment comparisons in treatment- connected designs Referring to (3.15) for a treatment connected design d with the usual C- matrix denoted by C d D(ˆη d ) = (LC + d L )σ 2 = σ 2 Σ d

13 2002] DS-Optimal Designs 45 where C + d denotes the Moore-Penrose inerse of C d. We note that µ max (Σ d ) x (LC + d L )x x for any x 0 x In particular, taking x = 1 we get It is not difficult to erify that so that µ max (Σ d ) µ max (Σ d ) C + ( + 1) 2 d00 C + d00 C d00 ( + 1 )2 ( + 1)2 1 ( + 1 )2 = c d00 Let d be a design for which C d has the structure a -a/ a/ -a/ b/ -c/... -c/ C d = a/ -c/ -c/... b/ C d00 (3.21) (3.22) where a = C d00 and b = tr(c d ) C d00. It is easy to erify that the eigenalues of Σ d are ( 1) a with multiplicity 1 and b a with multiplicity ( 1). Wheneer b a, µ max (Σ d ) = a, and in iew of (3.20) µ max(σ d ) µ max (Σ d ). Mandal et.al.(2002) arguing exactly as in the case of CRD indicated in preious subsection claim that d improes oer d uniformly in ɛ > 0 in terms of increasing the coerage probability wheneer the condition tr(c d ) 2C d00 (3.23) is satisfied. Thus, if we restrict our attention to the class of designs of the type d, the problem reduces to that of choosing C d00 so as to maximize P r( a χ ( 1) b a χ 1 2 ɛ 2 ) where a = C d00 and tr(c d ) = a + b, in situations where b a. The solution is ery much ɛ 2 dependent and hence appropriate weighted aerage probability by using a suitable weight distribution for ɛ 2 can be taken up. Let C R0,b,,k denote the class of binary connected block designs under consideration for gien alues of b, and k(< ) where R 0 is the replication number for the control treatment. As in the case of CRD assuming χ 2 2 distribution for ɛ 2 Mandal et.al (2000) present optimal alues of n and R 0 for fixed alues of wighted coerage probability P =.9 and.95 and conclude that a Balanced treatment Incomplete Block (BTIB) design defined by Bechhofer and Tamhane(1981) with aboe alues of n and R 0 is optimal for the gien coerage probability.

14 46 Rita SahaRay [Noember 4 DS-optimal Regression Designs In this section we derie DS optimal designs for the LSE of β for a m-factor first degree model on asymmetric experimental domain and briefly summarize certain results on symmetric polynomial designs. 4.1 First degree polynomial fit model We consider a m- factor first degree polynomial fit model: Y ij = β 0 + β 1 x i1 + + β m x im + ɛ ij (4.1) with m regression ariables and experimental conditions x i = (x i1,..., x im ) i = 1,..., n. The n point design d n = {x 1,..., x n : p 1,..., p n } has n i replications of the leel x i with relatie weight p i = ni n, i = 1,..., n We assume tha the experimental domain is an m dimensional Euclidean ball of radius m that is T m = {x IR m : x m}. If the ectors x i T m fulfill the conditions 1 + x i x i = m + 1, 1 + x i x j = 0 (4.2) for all 1 i j m + 1, then the ectors span a conex body in IR m called a regular simplex (cf. Pukelsheim 1993, p. 391). The ertices x i belong to the boundary sphere of the ball T m. A design which places weights p i, i = 1,..., m + 1 on the ertices of a regular simplex in IR m is called a simplex design. A design with equal weights p 1 = p 2 = = 1 m+1 is called a uniform simplex design. For m = 1, the support points are x 1 = 1 and x 2 = 1 so that (1, 1) and (1, 1) satisfy the conditions (4.2). For m = 2, the support points x 1, x 2, x 3 satisfying the conditions (4.2) belong to the boundary of T 2 and span an equilateral triangle on the sphere T 2. For example, the support points (1, 1), 1 2 (1 + 3, 1 3), 1 2 (1 3, 1 + 3) and eery rotation of them span an equilateral triangle. For all m 1 it is always possible to choose ectors x 1, x 2,..., x m+1 such that they fulfill the condition (4.2). A regular simplex design always exists. Note that any rotation of a regular simplex design is also a regular simplex design. The smallest possible support size of a feasible desgn for the LSE of β in a m factor first degree model is n = m + 1 because β has m + 1 components. Since the DS optimality criterion is isotonic with respect to the Lowener ordering and it depends on the information matrix only through the eigen alues, there exists an (m + 1) point simplex design on the ball T m+1 that dominates any other design with respect to DS optimality criterion. Therefore the search of DS optimal design for the LSE of β = (β 0, β 1,..., β m ) can be restricted to the calss of (m + 1)point simplex designs. Theorem 4.1 Let d be an (m + 1)point design for the LSE of β in (4.1) oer the ball T m+1. Then d is DS optimal if and only if it is an (m + 1) uniform simplex design. Proof: Note that the rows of the design matrix X under the (m + 1) point design d are the support etors (1, x 1), (1, x 2),..., (1, x m+1) with information matrix I(d) = X DX = p i (1, x i ) (1, x i )

15 2002] DS-Optimal Designs 47 where D = diag(p 1,..., p m+1 ). As already noted aboe, a DS optimal design can always be found in the set of (m + 1) point simplex designs. If d is a simplex design, the design matrix X is square and the non-zero eigenalues of X DX and DXX are the same. Hence m+1 I(d) = X DX = DXX = (m + 1) Now the arithmetic mean -geometric mean unequality yields m+1 1 p i ( m + 1 )m+1 (4.3) p i Equality holds in (4.3) if and only if p 1 = p 2 = = p m+1 = 1 simplex design. By Corollary 2.9 m+1 i.e. d is a uniform ψ ɛ [I(d)] P [χ 2 m+1 δ 2 (m + 1) 1 m+1 ( m+1 p i ) 1 m+1 ] where the R.H.S is an increasing function of m+1 p i. Thus by 4.3 d is a DS optimal design if and only if it is a uniform simplex design.. If m = 1 then we hae the line fit model y ij = β 0 + β 1 x i + ɛ ij (4.4) wih experimental domail T = [ 1, 1] it follows from Theorem 4.1 that the design d = { 1, 1; 1/2} which assigns weight 1/2 to the points -1 and 1 is the unique DS optimal design for the LSE of β = (β 0, β 1 ) in 4.4. Corollary 4.2 Let D n be the set of designs with support size n m + 1 in th m- way first degree model (4.1) on the experimental domain T m+1. Then a design d D is DS optimal if I(d) = I m+1. We omit the proof as it is a simple modification of the proof of the theorem 6.1. By Theorem 6.1 I(d) = I m+1 holds for a uniform simplex design d. Hence a design d satisfying the condition I(d) = I m+1 exists for the minimal feasible support size n = m+1. Existence of a DS optimal design in general for any gien pair of positie integers m and n m + 1 seems to be an unsoled problem. Neertheless, A DS optimal design can be found for certain alues of m and n m + 1. For example, the complete factorial design 2 m is a DS optimal design with n = 2 m. The information matrix of the complete factorial design 2 m is I m+1 and consequently it is a DS optimal design for te LSE of β. 4.2 Symmetric Polynomial Designs We postulate now a polynomial fit model of degree k 1. Y ij = β 0 + β 1 x i + + β k x k i + ɛ ij (4.5) where E(ɛ ij ) = 0 and V (ɛ ij ) = σ 2 i = 1,..n and j = 1,.., n i The responses Y ij s are uncorrelated and the experimental conitions x 1,..., x n are assumed to lie in [ 1, 1]. Let d D be a design for the LSE of β = (β 0,..., β k ) on T = [ 1, 1]. We now consider the reflection operation. The reflected design d R is gien by d R =

16 48 Rita SahaRay [Noember { x 1,..., x n : p 1,..., p n }. The design d and d R hae the same een moments, while the odd moments of d R hae a reersed sign. If I(d R ) denotes the (k +1) (k +1) information matrix of d R, then I(d R ) = QI(d)Q where Q = diag(1, 1, 1, 1,..., ±1). The symmetrized design d = 1 2 (d + dr ) = {+x i ; assigns the weights pi 2 to x i and x i for each i. The information matrix of d is p i 2, p i i n} 2 I( d) = 1 [I(d) + QI(d)Q] 2 It is easy to see that I(d R ) and I(d) hae the same eigenalues. The DS optimality criterion for ψ ɛ is inariant with respect to the reflection operation and it follows from the Corollary 2.7 that the symmetrized design d is at least as good as d with respect to the DS optimality criterion. Therefore, in symmetric experimental domain it is sufficient to consider symmertrized designs only. Howeer, it turns out that in the polynomial regression model of degree k > 1 there exists no DS optimal design for the LSE of β. The illustration using a quadratic regression model is to be found in Liski et.al. (1999). 5 D-, E- and DS(ɛ) optimality Liski et.al (1999) studied the behaiour of the DS(ɛ) criterion when ɛ approaches 0 and respectiely. These limiting cases hae an interesting relationship wih the traditional D- and E- optimality criteria. We now refer to a theorem from Liski et.al.(1999) without proof. Theorem 5.1 Let λ and γ IR k denote ectors whose components are eigenalues of the information matrices I(d 1 ) and I(d 2 ) respectiely, arranged in decreasing order. Then the following statements hold: a) If ψ δ (λ) ψ δ (γ) for all sufficiently small δ > 0, then I(d 1 ) I(d 2 ) if I(d 1 ) I(d 2 ), then ψ δ (λ) > ψ δ (γ) for all sufficiently small δ > 0. b) If ψ δ (λ) ψ δ (γ) for all sufficiently large δ, then λ k γ k ; if λ k > γ k then ψ δ (λ) > ψ δ (γ) for all sufficiently large δ. This theorem shows that DS(ɛ) criterion is equialent to the D- criterion as ɛ 0 and to the E- criterion as ɛ. These conclusions are due to the fact that both D -and E optimal designs are unique. (cf. Hoel 1958, Pukelsheim and Studden(1993)). References Birnbaum, Z. W. (1948). On random ariables with comparable peakedness. Annals of Mathematical Statistics.19: Hoel, P. G. (1958). Efficiency problems in polynomial estimation. Annals of Mathematical Statistics. 29 :

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