Optimum designs for parameter estimation in mixture experiments with group synergism
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...The inverse of the information matrix of n0 is given by M 1ðn0Þ ¼ pþ 12 þ qþ 1 2 ðX1X10Þ 1 Or t O0 ðX2X20Þ 1 where X1 ¼ Ip B O I p 2 2 64 3 75, X2 ¼ Iq C O I q 2 2 64 3 75 (3.3) are pþ 1 2 pþ 1 2 and qþ 1 2 þ qþ 1 2 matrices respectively with ðX1X01Þ 1 ¼ Ip 4B 4B0 4I p 2 24 3 5, ðX2X02Þ 1 ¼ Iq 4C 4C0 4I q 2 2 64 3 75, Im is an identity matrix of order m m, O is a null matrix of order pþ 12 qþ 1 2 , and B and C are respectively p p 2 and q q 2 matrices given by b12 b13::: b1p 1 b1p b23 b24:::b2p 1 b2p:::bp 1, p B ¼ 1=2 1=2 ::: 1=2 1=2 0 0 ::: 0 0 ::: 0 1=2 0 ::: 0 0 1=2 1=2 ::: 1=2 1=2 ::: 0 ::: ::: ::: ::: ::: ::: ::: ::: 0 0 ::: 1=2 0 0 0 ::: 1=2 0 ::: 1=2 0 0 ::: 0 1=2 0 0 ::: 0 0 ::: 1=2 2 6666664 3 7777775 bpþ1, pþ2 bpþ1, pþ3::: bpþ1pþq 1 bpþ1pþq bpþ2, pþ3 :::bpþ2pþq 1 bpþ2, pþq:::bpþq 1, pþq C ¼ 1=2 1=2 ::: 1=2 1=2 0 ::: 0 0 ::: 0 1=2 0 ::: 0 0 1=2 ::: 1=2 1=2 ::: 0 ::: ::: ::: ::: ::: ::: ::: ::: ::: ::: ::: 0 0 ::: 1=2 0 0 1=2 0 ::: 1=2 0 0 ::: 0 0 0 ::: 0 1=2 ::: 1=2 2 6666664 3 7777775: We check the optimality or otherwise of n0 using Equivalence Theorem 3.1....
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...The criteria are For D-optimality : Maximize Det: ½MðnÞ ¼ Det:½M11ðnÞ Det:½M22ðnÞ For A-optimality : MinimizeTr: ½MðnÞ 1 ¼ Tr:½M11ðnÞ 1 þ Tr:½M22ðnÞ 1 To get an idea of the support points of the D-optimal and A-optimal designs, we make use of the Equivalence Theorems due to Kiefer and Wolfowitz (1960) and Fedorov (1972) given below: Theorem 3.1....
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...He proved the optimality of the design with the help of the Equivalence Theorem....
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...In view of this and Theorem 3.3, we first search for the A-optimal design within the class D1 if only, say p¼ 3, and within the class D2 if p¼ q¼ 3, where designs in D1 and D2 are as follows: i. n2D1 has support points: a. (1,0,… ,0) and permutations among the first 3 components, each with mass w11; b. ð1=2, 1=2, 0, :::, 0Þ and permutations among the first 3 components each with mass w12; c. ð1=3, 1=3, 1=3, 0, :::, 0Þ with mass w13; d. ð0, :::, 0|fflffl{zfflffl} p , 1, 0, :::, 0|fflfflfflffl{zfflfflfflffl} q Þ and permutations among the last q components each with massw21; e. ð0, :::, 0|fflffl{zfflffl} p , 1=2, 1=2, 0, :::, 0|fflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflffl} q Þ and permutations among the last q components each with mass w22; where wij 0, 8i, j, 3ðw11 þ w12Þ þ w13 þ qw21ð q2 Þw22 ¼ 1; ii. n2D2 has support points a. (1,0,0,0,0,0) and permutations among the first 3 components, each with mass w11; b. ð1=2, 1=2, 0, 0, 0, 0Þ and permutations among the first 3 components each with mass w12; c. ð1=3, 1=3, 1=3, 0, 0, 0Þ and with mass w13; d. ð0, 0, 0, 1, 0, 0Þ and permutations among the last 3 components each with mass w21; e. ð0, 0, 0, 1=2, 1=2, 0Þ and permutations among the last 3 components each with mass w22; f. ð0, 0, 0, 1=3, 1=3, 1=3Þ with mass w23; where wij 0, 8i, j, 3ðw11 þ w12 þ w21 þ w22Þ þ w13 þ w23 ¼ 1: The A-optimal design n0 within D1 (or D2) is obtained by finding the masses that minimize Trace½M 1ðnÞ : Since algebraic derivations are lengthy and tedious, we numerically compute dðn0, xÞ ¼ f 0ðxÞM 2ðn0Þf ðxÞ for enumerable x 2v(with q 7 in case (i)) to observe that the conditions of Equivalence Theorem are satisfied....
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...…½MðnÞ ¼ Det:½M11ðnÞ Det:½M22ðnÞ For A-optimality : MinimizeTr: ½MðnÞ 1 ¼ Tr:½M11ðnÞ 1 þ Tr:½M22ðnÞ 1 To get an idea of the support points of the D-optimal and A-optimal designs, we make use of the Equivalence Theorems due to Kiefer and Wolfowitz (1960) and Fedorov (1972) given below: Theorem 3.1....
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"Optimum designs for parameter estim..." refers methods in this paper
...Kiefer (1961) first studied the optimality of designs in the mixture set-up, and established the D-optimality of (q, 2) simplex lattice design, which puts equal mass at the support points of the design, for estimating the parameters of the second degree model....
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