REVSTAT Statistical Journal Volume 3, Number 2, June 205, 9 29 NOTES ON THE REGULAR E-OPTIMAL SPRING BALANCE WEIGHING DESIGNS WITH CORRE- LATED ERRORS Authors: Bronis law Ceranka Department of Mathematical and Statistical Methods, Poznań Uniersity of Life Sciences, Wojska Polskiego 28, 60-637 Poznań, Poland bronicer@up.poznan.pl Ma lgorzata Graczyk Department of Mathematical and Statistical Methods, Poznań Uniersity of Life Sciences, Wojska Polskiego 28, 60-637 Poznań, Poland magra@up.poznan.pl Receied: May 203 Reised: September 203 Accepted: Noember 203 Abstract: The paper deals with the estimation problem of indiidual weights of objects in E- optimal spring balance weighing design. It is assumed that errors are equal correlated. The topic is focused on the determining the maximal eigenalue of the inerse of information matrix of estimators. The constructing methods of the E-optimal spring balance weighing design based on the incidence matrices of balanced and partially balanced incomplete block designs are gien. Key-Words: balanced incomplete block design; E-optimal design; partially incomplete block design; spring balance weighing design. AMS Subject Classification: 62K05, 62K5.
20 Bronis law Ceranka and Ma lgorzata Graczyk
Notes on Regular E-Optimal Designs 2. INTRODUCTION Let us suppose we want to estimate the weights of objects by weighing them b times using a spring balance, b. Suppose, that the results of this experiment can be written as. y = Xw + e, where y is an b random ector of the obserations, X Φ b 0,, where Φ b 0, denotes the class of b matrices X = x ij of known elements x ij = or 0 according as in the ith weighing operation the jth object is placed on the pan or not. Next, w is a ector of unknown measurements of objects and e is a b random ector of errors. We assume, that Ee = 0 b and Vare = σ 2 G, where 0 b denotes the b ector with zero elements eerywhere, G is the known b b diagonal positie definite matrix of the form.2 G = g ρi + ρ, g > 0, b < ρ <. It should be noticed that the conditions on the alues of g and ρ are equialent to the matrix G being positie definite. From now on, we will consider G on the form.2 only. Moreoer, let note, G = ρ g ρ I b +ρb b b. For the estimation of w we use the normal equations Mw = X G y, where M = X G X is called the information matrix of ŵ. A spring balance weighing design is singular or nonsingular, depending on whether the matrix M is singular or nonsingular, respectiely. From the assumption that G is positie definite it follows that matrix M is nonsingular if and only if matrix X is full column rank. If matrix M is nonsingular, then the generalized least squares estimator of w is gien by formula ŵ = M X G y and Varŵ = σ 2 M. Some considerations apply to determining the optimal weighing designs are shown in many books. Some problems related to optimality of the designs are presented in seeral papers 2 for G = I n, whereas in Katulska and Rychlińska 9 for the diagonal matrix G. In this paper, we emphasize a special interest of the existence conditions for E-optimal design, i.e. minimizing the maximum eigenalue of the inerse of the information matrix. The statistical interpretation of E-optimality is the following: the E-optimal design minimizes the maximum ariance of the component estimates of the parameters. It can be described in terms of the maximum eigenalue of the matrix M as λ max M or equialently as λ min M. Hence, for the gien ariance matrix of errors σ 2 G, any design X Φ b 0, is E-optimal if λ max M is minimal. Moreoer, if λ max M attains the lowest bound, then X Φ b 0, is called regular E-optimal. Notice that if the design See, Raghaarao 3, Banerjee, Shah and Sinha 5, Pukelsheim 2. 2 Jacroux and Notz 8, Neubauer and Watkins.
22 Bronis law Ceranka and Ma lgorzata Graczyk X Φ b 0, with the ariance matrix of errors σ 2 G is regular E-optimal then is also E-optimal. But the inerse implication may not be true. Moreoer, the E-optimal design in the set of all design matrices Φ b 0, exists but the regular E-optimal design may not exist. The problem presented in this paper is focused to determining such matrix that λ max M takes the minimal alue oer all possible matrices in Φ b 0, for gien matrix G. 2. REGULAR E-OPTIMAL SPRING BALANCE WEIGHING DESIGN In this section we gie some new results concerning the lower bound for λ max M depending on ρ and number of objects is een or odd. Additionally, let Π be the set of all permutation matrices. We shall denote by M the aerage of M oer all elements of Π, i.e. M =! P Π P MP. It is not difficult to see that M = trm M I + M trm, moreoer, trm = tr M and M = M. The matrix M has two eigenalues µ = trm M multiplicity. Let 2. M = M = g ρ From aboe, eigenalues of M are with the multiplicity and µ 2 = M X ρ X g ρ + ρb X b b X. For X Φ b 0, and G we hae trm = k k µ = µ 2 = ρ +ρb g ρ k k g ρ with the g ρ ρ r +ρb r r and b k 2, where X = k, X b = r, r = b k. b k k k + ρ b + ρb k 2 ρ + ρb. b k2 r r, Thus the matrix M has also two eigenalues λ = µ and λ 2 = µ 2. Next, comparing these two eigenalues we become following lemma. Lemma 2.. For any nonsingular spring balance weighing design X Φ b 0, with the ariance matrix of errors σ 2 G, the matrix M has two eigenalues λ and λ 2 and moreoer
Notes on Regular E-Optimal Designs 23 i λ > λ 2 if and only if ρ < ii λ = λ 2 if and only if ρ = iii λ < λ 2 if and only if ρ > k k b k b b k k k+ b k2 r r, k k b k b b k k k+ b k2 r r, k k b k b b k k k+ b k2 r r. Proof: We hae µ µ 2 = g ρ b k k k + ρ +ρb b k2 r r = g ρ+ρb b k k k g ρ k k + ρb ρ +ρb b k2 + ρ b k2 r r. Because g ρ + ρb > 0 then µ µ 2 > 0 if and only if ρ b b k k k + b k2 r r > k k bk and we obtain the Lemma. Lemma 2. imply that in order to determine E-optimal design we hae to delimit the lowest bound of eigenalues of M according to the alue of ρ. Theorem 2.. Let be een. In any nonsingular spring balance weighing design X Φ b 0, with the ariance matrix of errors σ 2 G i if ρ 2 b+ 2 then λ max M 4g ρ and the equality is fulfilled if and only if X = 2 b, ii if ρ 2 b+ 2, then λ max M > g+ρb. Proof: In order to determine regular E-optimal spring balance weighing design we hae to gie the lowest bound of the maximal eigenalue of the matrix M. Let M denote the aerage of M oer all elements of Π for the design X Φ b 0, with the ariance matrix of errors σ 2 G. From the monotonicity theorem gien by Rao and Rao 2004 it follows λ max M λ max M. The proof falls naturally in two parts according to the alue ρ in.2. If ρ 2 b+ 2 then λ max M g ρ+ρb =. As we want to minimize λ max M, we should find the maximum alue +ρb b k+ρbk k+ρ b k2 ρr r for A = + ρb b k + ρb k k + ρ b k2 ρ r r.
24 Bronis law Ceranka and Ma lgorzata Graczyk If p is een 2.2 A + ρb b k + ρb bk 2 + ρ b k2 ρ 2 r 2 = 2.3 +ρb bk +ρb bk 2 +ρbk 2 ρ 2 r 2 4 2 +ρb. Hence λ max M 4g ρ. The equality in inequality 2.2 holds if and only if k = k 2 = = k b = k and r = r 2 = = r = r, whereas the equality in 2.3 is fulfilled if and only if k = 2. If ρ 2 b+ 2, then λ max M = g ρ+ρb k k+ρbk k ρ k2. So, we obtain λ maxm > g+ρb. Thus the result. b Theorem 2.2. Let be odd. In any nonsingular spring balance weighing design X Φ b 0, with the ariance matrix of errors σ 2 G i if ρ 3 ++ 3 then λ max M 4g ρ + and the equality is satisfied if and only if X = 2 b or X = + 2 b, ii if ρ 3 ++ 3, b+ then λ max M 4g ρ + and the equality is satisfied if and only if X = + 2 b, iii if ρ b+, then λ max M > g+ρb. Proof: The proof is similar to gien in Theorem 2. one. Now, we can formulate the definition of the regular E-optimal spring balance weighing design. So, we hae Definition 2.. Any nonsingular spring balance weighing design X Φ b 0, with the ariance matrix of errors σ 2 G is regular E-optimal if the eigenalues of the information matrix attains the bounds of Theorems 2. and 2.2, i.e. i is een and ρ 2 if λ max M = 4g ρ, ii is odd and ρ b+ 2 b+ if λ max M = 4g ρ +. A direct consequence of aboe considerations is Theorem 2.3. Any nonsingular spring balance weighing design X Φ b 0, with the ariance matrix of errors σ 2 G is regular E-optimal design if and only if is een and ρ X G X = g ρ X = 2 b, 2 b+ 2 4 I + 2 4 ρb 2 4+ρb and
Notes on Regular E-Optimal Designs 25 2 p is odd and 2. ρ 3 ++ 3 2.. X G X = g ρ and X = 2 b, or 2..2 X G X = g ρ and X = + 2 b, 2.2 ρ 3 ++ 3, X G X = g ρ and X = + 2 b. + 4 I + 3 4 ρb2 2 4 2 +ρb + 4 I + + 4 ρb2 + 2 4 2 +ρb b+ + 4 I + + 4 ρb2 + 2 4 2 +ρb 2 b+ 2 Proof: Since the proofs for een and odd are similar, we gie the proof only for the case of een and ρ. Notice, that λ max M attains the lowest bound in Theorem 2.i if equalities λ max M = λ max M and = k 2 hold. It follows easily that trm = 2+ρb 2 4g ρ+ρb and M = 2 4g+ρb. We apply formulas on µ and µ 2 to gie the form of M. Thus the Theorem. It can be noted that g =, ρ = 0 and G = I b, we become the equalities gien by Jacroux and Notz 8. Theorem 2.4. In any nonsingular spring balance weighing design X Φ b 0, with the ariance matrix of errors σ 2 G, if i is een and ρ 2 b+ 2, or ii is odd and ρ b+,, then regular E-optimal spring balance weighing design does not exist. Proof: Since the proofs for een and odd are similar, we shall only gie the proof for the case odd. If ρ b+, then the lowest bound of the maximal eigenalue of the design matrix X is gien in Theorem 2.2iii. The lowest bound is attained if and only if k =. It means X = b. Such matrix is singular one. Thus the regular E-optimal design does not exist. Theorem 2.5. Any nonsingular spring balance weighing design X Φ b 0, with the ariance matrix of errors σ 2 pk G for ρ = np k+pk, k =, 2,..., + 2 for een or k =, 2,..., 2 for odd is regular E-optimal design if and only if X nk nkk nkk X = I + p pp pp.
26 Bronis law Ceranka and Ma lgorzata Graczyk Proof: Let us take into consideration the case gien in Lemma 2.ii. k b k λ = λ 2 = λ = k µ if and only if ρ =. Now, we hae to b b k k k+ b k2 r r consider two cases is een and is odd. For both the proofs are similar, so we gie the case for een, only. According to the proof of Theorem 2., the equality in 2.2 holds if and only if k = k 2 = = k b = k and r = r 2 = = r = r, for k =, 2,..., 2. Hence ρ = pk np k+pk. From the aboe results M = µi. Thus µi = g ρ X ρn X 2 k 2 2 +ρb and X X = nkp k nkk pp I+ pp. Putting ρ and µ = nkp k g ρpp we obtain the form of the matrix X X. If k > 2, then from Theorem 2.4 regular E-optimal spring balance weighing design does not exist. 3. CONSTRUCTION OF THE REGULAR E-OPTIMAL DESIGN For the construction of the regular E-optimal spring balance weighing design, from all possible block designs, we choose the incidence matrices of the balanced incomplete block designs and group diisible designs. The definitions of these block designs are gien in Raghaarao and Padgett 4. Theorem 3.. Let N be the incidence matrix of the balanced incomplete block design with the parameters i = 2t, b = 22t, r = 2t, k = t, λ = t or ii = 2t, b = 2t t, r = 2t t, k = t, λ = 2t t 2, t = 2, 3,... Then, any X = N Φ b 0, with the ariance matrix of errors σ 2 G for ρ 2 b+ 2 is the regular E-optimal spring balance weighing design. Proof: An easy computation shows that the matrix X = N satisfies of Theorem 2.3. Now, let 3. X = N N, 2 where N u is the incidence matrix of the group diisible design with the same association scheme with the parameters, b u, r u, k = 2, λ u, λ 2u, u =, 2. Furthermore, let the condition 3.2 λ + λ 2 = λ 2 + λ 22 = λ
Notes on Regular E-Optimal Designs 27 be satisfied. For X in 3., b = b + b 2. The limitation on the t and s gien in next Theorem 3.2 follow from the restrictions: r, k 0 gien in Clatworthy 5. Theorem 3.2. Let N u, u =, 2, be the incidence matrix of the group diisible block design with the same association scheme and with the parameters = 4, k = 2 and. b = 23t+, r = 3t+, λ = t+, λ 2 = t, t =, 2, 3, and b 2 = 23s + 2, r 2 = 3s + 2, λ 2 = s, λ 22 = s +, s = 0,, 2,.2 b = 23t + 2, r = 3t + 2, λ = t + 2, λ 2 = t, t =, 2, and b 2 = 23s + 4, r 2 = 3s + 4, λ 2 = s, λ 22 = s + 2, s = 0,, 2,.3 b = 2t + 3, r = t + 3, λ = t +, λ 2 = and b 2 = 4t, r 2 = 2t, λ 2 = 0, λ 22 = t, t =, 2,...,5,.4 b = 6, r = 8, λ = 0, λ 2 = 4 and b 2 = 23s+4, r 2 = 3s+4, λ 2 = s + 4, λ 22 = s, s =, 2,.5 b = 8, r = 9, λ = 5, λ 2 = 2 and b 2 = 6s+2, r 2 = 3s+2, λ 2 = s, λ 22 = s + 3, s = 0,, 2 = 6, k = 3 and 2. b = 4t, r = 2t, λ = 0, λ 2 = t and b 2 = 6t, r 2 = 3t, λ 2 = 2t, λ 22 = t, t =, 2, 3, 2.2 b = 22t+5, r = 2t+5, λ = t+, λ 2 = t+2 and b 2 = 6t, r 2 = 3t, λ 2 = t +, λ 22 = t, t =, 2, 2.3 b = 2, r = 6, λ = 4, λ 2 = 2 and b 2 = 25s+4, r 2 = 5s+4, λ 2 = 2s, λ 22 = 2s +, s = 0,, 2.4 b = 6, r = 8, λ = 4, λ 2 = 3 and b 2 = 25s+2, r 2 = 5s+2, λ 2 = 2s, λ 22 = 2s +, s = 0,, 3 = 8, k = 4 and 3. b = 4t+, r = 2t+, λ = 0, λ 2 = t+ and b 2 = 46 t, r 2 = 26 t, λ 2 = 6, λ 22 = 5 t, t =, 2, 3, 3.2 b = 23t + 2, r = 3t + 2, λ = t + 2, λ 2 = t + and b 2 = 64 t, r 2 = 34 t, λ 2 = 4 t, λ 22 = 5 t, t =, 2, 4 = 0, k = 5 and b = 8t, r = 4t, λ = 0, λ 2 = 2t and b 2 = 0t, r 2 = 5t, λ 2 = 4t, λ 22 = 2t, t =, 2, 5 = 22t +, k = 2t + and b = 4t, r = 2t, λ = 0, λ 2 = t and b 2 = 22t +, r 2 = 2t +, λ 2 = 2t, λ 22 = t, t =, 2, 3, 4, 6 = 4t +, k = 2t + and b = 22t +, r = 2t +, λ = 2t +, λ 2 = t and b 2 = 4t+, r 2 = 2t+, λ 2 = 0, λ 22 = t+, t =,2,3,4. Then any X Φ b 0, in the form 3. with the ariance matrix of errors σ 2 G for ρ 2 b+ 2 is the regular E-optimal spring balance weighing design.
28 Bronis law Ceranka and Ma lgorzata Graczyk Proof: This is proed by checking that the matrix X in 3. satisfies of Theorem 2.3. Theorem 3.3. Let N be the incidence matrix of balanced incomplete block design with the parameters i = 2t +, b = 22t +, r = 2t +, k = t +, λ = t +, ii = b = 4t 2, r = k = 2t 2, λ = t 2, iii = b = 8t + 7, r = k = 4t +, λ = 2t +, i = b = 4t, r = k = 2t, λ = t, = 4t +, b = 24t +, r = 22t +, k = λ = 2t +, i = 2t +, b = 2t+ t+, r = 2t t, k = t +, λ = 2t t, t =, 2,... Then any X = N Φ b 0, with the ariance matrix of errors σ 2 G for ρ b+ is the regular E-optimal spring balance weighing design. Proof: For X = N Φ b 0, the equalities 2..2 and 2.2 of Theorem 2.3 are satisfied. Theorem 3.4. Let N be the incidence matrix of balanced incomplete block design with the parameters i = 2t +, b = 22t +, r = 2t, k = t, λ = t, t = 2, 3,... ii = b = 4t 2, r = k = 2t 2, λ = t 2, t = 2, 3,... iii = b = 8t + 7, r = k = 4t + 3, λ = 2t +, t =, 2,... i = b = 4t, r = k = 2t, λ = t, t = 2, 3,... = 4t +, b = 24t +, r = 4t, k = 2t, λ = 2t, t =, 2,... i = 2t +, b = 2t+ t+, r = 2t t, k = t, λ = 2t t 2, t = 2, 3,... Then any X = N Φ b 0, with the ariance matrix of errors σ 2 G for ρ is the regular E-optimal spring balance weighing design. 3 ++ 3 Proof: It is oious that for X = N Φ b 0, the equality 2.. gien in Theorem 2.3 is fulfilled.
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