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Criterion-robust optimal designs for model discrimination and parameter estimation in Fourier regression models
Journal of Statistical Planning and Inference 124 (2004) 475 – 487
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Criterion-robust optimal designs for model discrimination and parameter estimation in Fourier regression models Mei-Mei Zen∗ , Min-Hsiao Tsai Department of Statistics, National Cheng-Kung University, Tainan 70101, Taiwan Received 30 July 2002; accepted 8 April 2003
Abstract Consider the problem of discriminating between two competitive Fourier regression on the circle [ − ; ] and estimating parameters in the models. To 5nd designs which are e6cient for both model discrimination and parameter estimation, Zen and Tsai (some criterion-robust optimal designs for the dual problem of model discrimination and parameter estimation, Indian J. Statist. 64, 322–338) proposed a multiple-objective optimality criterion for polynomial regression models. In this work, taking the same M -criterion, which puts weight (0 6 6 1) for model discrimination and 1 − for parameter estimation, and using the techniques of projection design, the corresponding M -optimal design for Fourier regression models is explicitly derived in terms of canonical moments. The behavior of the M -optimal designs is investigated under di;erent weighted selection criterion. And the extreme value of the minimum M -e6ciency of any M -optimal design is obtained at = ∗ , which results in the M∗ -optimal design to be served as a criterion-robust optimal design for the problem. c 2003 Elsevier B.V. All rights reserved. MSC: 62K05 Keywords: Canonical moments; E6ciency; Multiple-objective; M -optimal design; Projection design
1. Introduction The study of optimal designs for polynomial regression models has been extensively investigated; see Kiefer (1959, 1961, 1985), Atkinson and Donev (1992), and Pukelsheim (1993) for details. In many 5elds of sciences, such as physics, chemistry ∗
Corresponding author. Tel.: +886-6-2757575; fax: +886-6-2342469. E-mail address: [email protected] (Mei-Mei Zen).
c 2003 Elsevier B.V. All rights reserved. 0378-3758/$ - see front matter doi:10.1016/S0378-3758(03)00212-X
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and computer science, among others, the consideration of Fourier regression models is more prevalent, while the problem of optimal designs for Fourier regression received much less attention; see Mardia (1972) and Cartwright (1990). Under a Fourier regression model, if the experimenters can make certain the order of periodicity of the underlying model, then the optimal design will gain the precision of parameter estimation; see Hoel (1965), Karlin and Studden (1966), Hill (1978), Lau and Studden (1985), and Pukelsheim (1993) for discussion. Unfortunately, in many situations the experimenters have some uncertainty about the periodicity order, hence the prerequisite is model discrimination between some competitive models; see Dette and Haller (1998) for details. As mentioned in Dette and Franke (2000), usually designs are desirable to be e6cient for discriminating between several competitive models and to have good properties for the estimation of parameters in the identi5ed model. So far, most of literature focus on the study of either model discrimination or parameter estimation separately. However, the experimenters usually hope to seek some designs which could take both purposes into account. In this study, we are to seek certain nonsequential experimental designs to deal with the problem of model discrimination and parameter estimation simultaneously in Fourier regression models. The purpose of the present paper is to provide some criterion-robust designs that can be served e6ciently for the described problem. Consider the model E(Yx ) = 0 +
lc
i cos ix +
i=1
ls
i sin ix = # f (x);
(1.1)
i=1
where # =(V ; ); V =(0 ; 1 ; : : : ; lc ), =( 1 ; : : : ; ls ) are the vectors of parameters, f (x) = (c (x); s (x)), c (x) = (1; cos x; : : : ; cos lc x), s (x) = (sin x; : : : ; sin ls x) denote the vectors of regression functions, the integer max{lc ; ls } denotes the order of periodicity, and x ∈ [ − ; ]. In this study, an approximate design is a probability measure on [ − ; ]. Under (1.1), the performance of is essentially evaluated by the information matrix f M () = f (x) f (x) d(x): (1.2) [−;]
There are numerous criteria used for the construction of optimal designs; in this paper, we apply the D- and Ds -optimality criteria, s ¿ 1, to deal with the problem. As pointed out by Lau and Studden (1985), it is well known that the D-optimal designs for parameter estimation in models with lc = ls = k are also D-optimal for any lower order, e.g., lc = ls = k − 1, and they are also D2 -optimal for discriminating between order k − 1 and k. Thus we only study the problem of discriminating between Model A: lc = k − 1;
ls = k and Model B: lc = ls = k
and take regard of parameter estimation simultaneously. An alternative setting with lc = k; ls = k − 1 versus lc = ls = k is also investigated in Zen and Tsai (2002b). Note that the integer k denotes the order of periodicity.
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For polynomial situation, Pukelsheim and Rosenberger (1993) made a comparison of e6ciencies among some special experimental designs which serve all three objectives simultaneously: (1) to discriminate between Models A and B and, depending on the decision, (2) to make inferences in Model A or (3) in Model B. For multiple-objective consideration, Zen and Tsai (2002a) proposed a weighted optimality criterion and applied the maximin principle on the e6ciency to obtain a criterion-robust optimal design. Herein, we use the same criterion for Fourier regression models. As stated in Dette and Haller (1998), the determinant of (1.2) is invariant with respect to a reLection of the design at the origin; to seek optimal designs, we restrict our attention to the set of all symmetric designs on [ − ; ] hereafter. The paper will be organized as follows. In Section 2, we introduce the projection design and the M -selection criterion. Then using the theory of canonical moments, an M -optimal design can be expressed by the canonical moments of its projection. Since di;erent selection criterion results in di;erent optimal design, an appropriate selection criterion is essentially important for the problem itself. The minimum M -e6ciency of arbitrary M -optimal design and the behavior of these minimum values will be studied in Section 3. It turns out that the maximum value of the minimum M -e6ciency of any M -optimal design occurs at =∗ . Based on the maximin principle, a criterion-robust optimal design, M∗ , will be derived for any periodicity order k. Especially, the support points and weights of M∗ with the minimum M -e6ciency will be given for some small k. In Section 4, we make a comparison of e6ciencies among some special optimal designs. Section 5 gives some brief concluding remarks. Finally, the technical proof is given in the Appendix.
2. The M -optimal design Using the D-criterion, to compute the determinant of M f () in (1.2), a one-to-one mapping between the set of all symmetric designs on [−; ] and the set of all designs on the interval [−1; 1] is essentially necessary. Let z =cos x. For any symmetric design on [ − ; ], the projection on [ − 1; 1] is de5ned by 2(x) if 0 ¡ |x| 6 ; (2.1) (z) = (0) if x = 0: Then the determinant of M f () can be expressed in terms of the product of determinants of two information matrices for weighted polynomial regression models; see Lau and Studden (1985) and Dette and Haller (1998) for details. To discriminate between Models A and B, it is equivalent to testing the hypotheses H0 : k = 0 versus H1 : k = 0. For simplicity, let MA () and MB () denote the information matrices for Models A and B, separately. Then using the D1 -criterion, it results in the D1 -optimal design D1 which maximizes the objective function D1 ()
=
|MB ()| ; |MA ()|
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where | · | denotes the determinant. To make inferences in Model A (or B), the D-optimal design A (or B ) maximizes the objective function A ()
= |MA ()|1=2k
(or
B ()
= |MB ()|1=(2k+1) ):
(2.2)
Considering all three objectives simultaneously, mentioned in Pukelsheim and Rosenberger (1993), a multiple-objective optimal design is de5ned to maximize the following weighted function of D1 (); A () and B (): M ()
=
A () B () D1 ()
= |MA ()|(=2k)− |MB ()|[=(2k+1)]+ ;
(2.3)
where 0 6 ; ; 6 1 and ++=1. Since there is no information about which model is appropriate before model discrimination, as proposed by Zen and Tsai (2002a), it seems reasonable to put equal weights for objectives (2) and (3) (i.e. = ), then (2.3) becomes M () = [ A () B ()]
(1−)=2
[
D1 ()]
= |MA ()|(1−)=4k− |MB ()|[(1−)=2(2k+1)]+ :
(2.4)
Thus the design denoted by M , which maximizes (2.4), is called the M -optimal design. The derivation of the M -optimal design is based on canonical moments. The theory of canonical moments was introduced by Skibinsky (1967) and widely applied by Studden (1980, 1982a,b, 1989) to determine optimal designs for polynomial regression models. For more details, readers can refer to Skibinsky (1986), Lau and Studden (1985), and Dette and Studden (1997). For a given probability measure on X = [ − 1; 1], let pi denote the ith canonical moment of . Following the same arguments as in Lau and Studden (1985), () de5ned by (2.2) and (2.4) can be expressed by the canonical moments of the projection . Theorem 2.1. The canonical moments of the projection corresponding to an M -optimal design satisfy pj = 12 ; j = 1; : : : ; 2k − 1 and 1 + 1− 2 2k+1 p2k = (2.5) : 2 1 + 1− 2 2k+1 + 2k Proof. Substituting k−1 = [[(1 − )=2(2k)] − ]=[(1 − )(1=2k + 1=(2k + 1))]; k = [[(1 − )=2(2k + 1)] + ]=[(1 − )(1=2k + 1=(2k + 1))] and k = 12 in Theorem 5.1 in Lau and Studden (1985) proves the assertion. There are similar results for the canonical moments of the projection corresponding to D-optimal designs A and B with nontrivial weights k−1 = k = 12 and k = k = 12 , respectively. Note that when =1, the M1 -optimal design is the same as the D1 -optimal design, which is an optimal design for model discrimination. Moreover, taking the derivative of p2k in (2.5) with respect to , it can be shown that p2k is strictly increasing in . As mentioned in Dette and Haller (1998) [see also Skibinsky (1986)], if j denote the 5rst index with pj = 0 or 1, then the sequence of canonical moments terminates at
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this pj and the design is supposed at a 5nite number of points. Moreover, the set of probability measures on the interval [ − 1; 1] with 5rst m canonical moments equal to (p1 ; : : : ; pm ) ∈ (0; 1)m−1 × [0; 1] is a singleton if and only if pm is 0 or 1. Otherwise, there exist in5nitely many probability measures corresponding to (p1 ; : : : ; pm ). According to the terminated canonical moments, we conclude that the DA -optimal design A for parameter estimation in Model A and the D1 -optimal design D1 for model discrimination are uniquely determined, and there are in5nitely many DB -optimal designs and M -optimal designs for any 5xed 0 6 ¡ 1. Thus, for the case 0 ¡ p2k ¡ 1, we terminate the sequence of canonical moments (p1 ; : : : ; p2k ) with p2k+1 = 0 or p2k+1 = 1. This corresponds to a lower (p2k+1 = 0) or an upper (p2k+1 = 1) principal representation of (p1 ; : : : ; p2k ), and the corresponding projection has (k + 1) support points including the endpoint −1 (if p2k+1 = 0) or +1 (if p2k+1 = 1); see Skibinsky (1986). The corresponding minimal support designs are denoted by − and + , respectively. To explicitly determine (minimal support) optimal designs, it is essential to 5nd the support points and weights of the projection , which can be found in Studden (1982a). Then the inverse transformation of (2.1) gives the support points and weights for arbitrary symmetric design , including D1 ; A ; B∓ ∓ and M . 3. Criterion-robust optimal design 3.1. The e4ciencies of the M -optimal design In this section, we will consider the behavior of a design under the M -criterion, as in (2.4), for any 5xed ∈ [0; 1] 5rst. Especially, we are interested in how the M -optimal design M behaves under di;erent M -criteria. The performance of any M -optimal design M under the M -criterion is evaluated by the M -e6ciency, which is de5ned by M (M )
M -e; (M ) =
M (M )
;
; ∈ [0; 1]:
(3.1)
Direct computation gives M (M )
[ ] +((1−)=2)(1=(2k+1)) [ ] ((1−)=2)([1=(2k+1)]+1=2k) = (p2k ) (q2k ) :
Then an obvious result follows directly. Theorem 3.1. For any 6xed ∈ [0; 1] and k ∈ N, let M be any M -optimal design. Then the M -e4ciency in (3.1) can be expressed as +((1−)=2)(1=(2k+1)) (1−=2)(1=(2k+1)+1=2k) [ ] [ ] q2k p2k M -e; (M ) = ; (3.2) [] [] p2k q2k [·] where p2k is as in (2.5).
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Mei-Mei Zen, Min-Hsiao Tsai / Journal of Statistical Planning and Inference 124 (2004) 475 – 487 1.0
0.30.4 0.5 0.6
0.7
0.8
0.9
0.98
0.8
0.6
γ
1 0.4
0.2
0.98
0.0 0.0
0.2
0.9
0.8 0.4
0.7 0.6 0.5 0.4 0.3 0.6
0.8
0
1.0
γ′
Fig. 1. The contour of the M -e6ciencies for all M -optimal designs, k = 1.
3.2. Minimum M -e4ciency of the M -optimal design In order to get an insight of the behavior of the M -e6ciencies, we deal with the case of k = 1 and plot the contour of the M -e6ciencies for all M -optimal designs in Fig. 1. It is obvious that the value of M -e; (M ) is getting small as is moving apart from . Therefore, a study of the extreme value of M -e; (M ) for any 5xed is essentially important. The following lemma is necessary to the main results and the proof is given in the Appendix. Lemma 3.2. For any k ∈ N, (i) given ∈ [0; 1]; M -e; (M ), a function of , is strictly increasing on [0; ) and decreasing on ( ; 1]; (ii) given ∈ [0; 1]; M -e; (M ), a function of , is strictly increasing on [0; ) and decreasing on (; 1]. From Lemma 3.2(i), we know that for any given , the minimum value of M -e; (M ) will be attained at either = 0 or = 1. To determine which one is the exact minimum value, let h( ) =
M1 -e; (M ) M0 -e; (M )
denote the ratio of these two possible minimum e6ciencies. Substituting (3.2), it turns out 1−[1=2(2k+1)] −1=2(1=(2k+1)+1=2k) [ ] [ ] [0] q2k p2k p2k h( ) = : (3.3) [0] [0] [1] p2k q2k p2k
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481
Note that both inequalities M1 -e; (M ) ¿ M0 -e; (M ) and h( ) ¿ 1 are equivalent. The two obvious facts that [] (a) p2k in (2.5) is strictly increasing in , and 1−[1=2(2k+1)] (b) p (1 − p)−1=2(1=(2k+1)+1=2k) is strictly increasing in p
imply that h( ) is strictly increasing in . Therefore, we have the following result: Theorem 3.3. For any 6xed ∈ [0; 1] and k ∈ N, M0 -e; (M ) if ¿ ∗ ; min {M -e; (M )} = 0661 M1 -e; (M ) if ¡ ∗ ; where M0 -e; (M ) and M1 -e; (M ) are as in (3.2) and ∗ is the root of h( ) = 1 in (3.3). To 5nd a criterion-robust optimal design, the maximum value of min0661 {M -e; (M )} plays an important role. From Lemma 3.2(ii), we have the fact that, M0 -e; (M ) is strictly decreasing in and M1 -e; (M ) is strictly increasing in , which gives the following important result: Theorem 3.4. For any k ∈ N; min0661 {M -e; (M )} is increasing 6rst, then decreasing in and the maximum value of min0661 {M -e; (M )} is attained at = ∗ , i.e.
∗ = arg max min M -e; ( ) ; M 06 61 0661
∗
where is the root of h( ) = 1 in (3.3). To describe this main theorem graphically, Fig. 2 demonstrates the plots of min0661 {M -e; (M )} versus for k = 1 and 2; the corresponding values of ∗ are 0.52614 k=2
k =1 0.8
min {Mγ -eff (σMγ ′)
0.4
0<γ<1
min {Mγ -eff (σMγ ′)
0<γ<1
{
{
0.6
0.2
0.0
0.4 0.2 0.0
0.0
(a)
0.6
0.2
0.4
0.6 γ′
0.8
0.0
1.0
(b)
0.2
0.4
0.6 γ′
Fig. 2. The plots of min0661 {M -e; (M )} versus ; k = 1 and 2.
0.8
1.0
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Table 1 The values of ∗ and the minimum M -e6ciencies for small k k
∗
min0661 M -e; (M∗ )
min0661 M -e; (M1=2 )
min0661 M -e; (M1=3 )
1 2 3 4 5 6 7 8 9 10
0.5261 0.4752 0.4445 0.4229 0.4063 0.3931 0.3821 0.3727 0.3647 0.3575
0.7540 0.8172 0.8492 0.8697 0.8844 0.8954 0.9042 0.9114 0.9173 0.9224
0.7368 0.8050 0.8279 0.8452 0.8588 0.8699 0.8791 0.8869 0.8936 0.8994
0.6154 0.7273 0.7869 0.8247 0.8511 0.8705 0.8854 0.8972 0.9068 0.9148
and 0.47525, respectively. Note that the value of ∗ will depend on the order k and can be numerically solved only. For practical use, Table 1 gives some numerical results of ∗ and the corresponding minimum e6ciencies of M∗ ; M1=2 and M1=3 for various k, where designs M1=2 and M1=3 are shown for comparison. This is because that for small k; ∗ is near 12 , and for moderate k; ∗ is close to 13 . Note that in Table 1, ∗ is decreasing in k, and the minimum e6ciency, min0661 {M -e; (M∗ )}, is increasing in k and greater than 0.754 for all k. That means for any M -criterion, M∗ -optimal design M∗ is robust in the sense of having high minimum e6ciency. Therefore, this M∗ can be served as a criterion-robust optimal design for the problem of model discrimination and parameter estimation. The fact that ∗ is decreasing in k means that the role of model discrimination is getting less important when the order of Fourier regression model is getting large. Since di;erent selection criterion will result in di;erent optimal design, under the M∗ -selection criterion, from Theorem 2.1, we can derive the canonical moments of the projection of the criterion-robust design precisely. Although ∗ can be uniquely solved by numerical method, but there are in5nitely many M∗ -optimal designs. − + Table 2 gives the support points and weights of M and M with = ∗ ; 12 and 13 .
4. Comparison with some special designs In this section, to make a comparison of the M∗ -optimal design M∗ with some special optimal designs, we derive the M∗ -, D1 -, DA - and DB -e6ciencies 5rst. The proof of the following result is straightforward and omitted. Proposition 4.1. For any 6xed k ∈ N, the M∗ -, D1 -, DA - and DB -e4ciencies of any symmetric design whose projection design has canonical moments pj = 12 ; j 6 2k −1,
Mei-Mei Zen, Min-Hsiao Tsai / Journal of Statistical Planning and Inference 124 (2004) 475 – 487
483
Table 2 The support points and weights of M∗ ; M1=2 and M1=3 k
1
∗
− M
1 2
1 3
2
∗ 1 2
1 3
3
∗ 1 2
1 3
+ M
±3:141
±0:7167
0:2149
0:2851
±3:141
±0:7424
0:2121
0:2879
±3:141
±0:9079
0:1905
0:3095
±3:141
±1:671
±0:4106
0:1184
0:2277
0:1539
±3:141
±1:663
±0:3969
0:1189
0:2293
0:1518
±3:141
±1:726
±0:4909
0:1143
0:2179
0:1678
±3:141
±2:127
±1:138
±0:2926
0:0810
0:1605
0:1522
0:1063
±3:141
±2:121
±1:122
±0:2713
0:0814
0:1616
0:1542
0:1028
±3:141
±2:143
±1:176
±0:3368
0:0798
0:1576
0:1480
0:1146
are given as follows. M∗ -e; () =
p2k
∗ +((1−∗ )=2)(1=(2k+1))
∗]
[ p2k
D1 -e; () = p2k ; DA -e; () = (q2k )1=2k
±2:425
0
0:2851
0:4298
±2:399
0
0:2879
0:4242
±2:234
0
0:3095
0:3810
±2:731
±1:471
0
0:1539
02277
0:2368
±2:745
±1:479
0
0:1518
0:2293
0:2378
±2:651
±1:416
0
0:1678
0:2179
0:2286
±2:849
±2:004
±1:014
0
0:1063
0:1522
0:1605
0:1620
±2:870
±2:019
±1:020
0
0:1028
0:1542
0:1616
0:1628
±2:805
±1:966
±0:9983
0
0:1146
0:1480
0:1576
0:1596
q2k
∗]
[ q2k
((1−∗ )=2)(1=(2k+1)+1=2k) ;
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D1-efficiency
Mγ∗-efficiency 1.00
1.0
0.8
0.90
0.6 0.80 0.4 0.70 1
10
20
(a)
30
40
1
50
k
10
20
30
40
50
k
(b)
DA-efficiency
DB-efficiency
0.98
0.9 σMγ* σD1 σM1/2
0.8 0.94
0.7
σM1/3
0.6
σM0 σB σA
0.90 0.5 1
(c)
10
20
30
40
k
1
50
10
20
(d)
30 k
40
50
Fig. 3. The M∗ -, D1 -, DB - and DA -e6ciencies of designs M∗ , D1 ; M1=2 ; M1=3 , M0 ; B and A . Note that (1) The M∗ -, D1 -, and DB -e6ciencies of design A are all zero for k ∈ N, (2) the M∗ -, DB -, and DA -e6ciencies of design D1 are all zero for k ∈ N.
and DB -e; () = (4p2k q2k )1=(2k+1) ; where p2k is the 2kth canonical moment of the projection . Then we investigate the e6ciencies of designs D1 ; A ; B ; M0 , M1=3 ; M1=2 and M∗ , which are de5ned in Section 2. Note that M0 is an analogue of the LNauter type design which was introduced by LNauter (1974). Fig. 3 shows the plots of the e6ciencies, in Proposition 4.1, of these seven designs versus various periodicity order k. Fig. 3(a) shows that M1=2 is better than M1=3 for small k, and M1=3 is superior to M1=2 for moderate to large k; this is due to the solution of ∗ in Table 1. It is worthwhile to mention that all the M∗ -e6ciencies of D1 and A are zero for arbitrary k.
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485
Fig. 3(b) shows that all the D1 -e6ciencies of M∗ , M1=2 ; M1=3 and M0 are increasing in k. Moreover, for arbitrary k the D1 -e6ciencies of B and A are always 12 and 0, respectively. Fig. 3(c) shows that the DB -e6ciencies of M∗ ; M1=2 and M1=3 are decreasing 5rst, then increasing in k. Overall, the DB -e6ciencies are higher for arbitrary k. Note that the DB -e6ciency of M0 is increasing in k, and it is the second highest for most of k; the DB -e6ciencies of D1 and A are all zero for arbitrary k. Fig. 3(d) shows that the DA -e6ciencies are increasing, except D1 and A whose are 0 and 1, respectively. The DA -e6ciencies of M∗ , M1=2 and M1=3 are somewhat conservative for small k. 5. Concluding remarks Using a multiple-objective selection criterion, the M -criterion, for the problem of model discrimination and parameter estimation in the Fourier regression models, we obtain the maximum value of the minimum M -e6ciency of any M -optimal designs at = ∗ . The corresponding M∗ -optimal design can be served as a criterion-robust optimal design. And its minimum M -e6ciency is given for small order of periodicity k. In particular, this criterion-robust optimal design could be used practically with any M -e6ciency greater than 0.754. Note that in polynomial regression situation, for any degree k, any M -e6ciency of the criterion-robust optimal design is greater than 0.952, shown in Zen and Tsai (2002a). Also for small k; ∗ in polynomial regression is near 0.35, but ∗ in Fourier regression is near 0.5. The comparison of ∗ in both polynomial and Fourier models shows that one should put a little bit more weight on model discrimination in Fourier than in polynomial regression models. Acknowledgements The authors would like to thank the referees for their helpful comments; this research was partially supported by NSC91-2118-M006-001 from the National Science Council, ROC. Appendix Proof of Lemma 3.2. Let 1 ; 2 ; be any number in [0; 1], from (3.2), we have M1 -e; (M ) M2 -e; (M ) (1 −2 ) [ ] (1−1=2(2k+1)) [2 ] 1=2(1=(2k+1)+1=2k) q p 2k 2k = × M1 -e; (M2 ); [2 ] [ ] p2k q2k (A.1)
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[·] where p2k is as in (2.5). Note that the e6ciency of any design is between 0 and 1.
(i) For any 5xed , we prove the results in two steps:
[2 ] [ ] [·] is strictly increasing in , we have p2k ¿ p2k . (a) Choose 1 ; 2 ∈ [0; ). Since p2k [ ] [2 ] [2 ] [ ] This implies that both ratios p2k =p2k and q2k =q2k are greater than 1, which results in that M1 -e; (M ) ¡ M2 -e; (M ), if 1 ¡ 2 . That is, for any given , M -e; (M ) is strictly increasing in ∈ [0; ).
[ ] [2 ] ¡ p2k , which implies that (b) Choose 1 ; 2 ∈ ( ; 1]. In this case, we have p2k [ ] [2 ] [2 ] [ ] p2k =p2k and q2k =q2k are less than 1. Therefore, we have that M1 -e; (M ) ¡ M2 -e; (M ), if 1 ¿ 2 . That is, for any given , M -e; (M ) is strictly decreasing in ∈ ( ; 1].
(ii) For any 5xed , multiply both sides of (A.1) by M2 -e; (M )=M1 -e; (M2 ), and replace by 1 , 1 by , we thus obtain M -e; (M1 ) M -e; (M2 ) (−2 ) [1 ] (1−1=2(2k+1)) [2 ] 1=2(1=(2k+1)+1=2k) q p 2k 2k × M2 -e; (M1 ): = [2 ] [1 ] p2k q2k Similar argument as in (i) proves the assertion. References Atkinson, A.C., Donev, A.N., 1992. Optimum Experimental Designs. Oxford University Press, Oxford. Cartwright, M., 1990. Fourier Methods for Mathematicians, Scientists and Engineers. Ellis Horwood, New York. Dette, H., Franke, T., 2000. Constrained D- and D1 -optimal designs for polynomial regression. Ann. Statist. 28, 1702–1727. Dette, H., Haller, G., 1998. Optimal designs for the identi5cation of the order of a Fourier regression. Ann. Statist. 26, 1496–1521. Dette, H., Studden, W.J., 1997. The Theory of Canonical Moments with Applications in Statistics, Probability, and Analysis. Wiley, New York. Hill, P.D.H., 1978. A note on the equivalence of D-optimal design measures for three rival linear models. Biometrika 65, 666–667. Hoel, P.G., 1965. Minimax designs in two-dimensional regression. Ann. Math. Statist. 36, 1097–1106. Karlin, S., Studden, W.J., 1966. Tchebyche; Systems: With Applications in Analysis and Statistics. Interscience, New York. Kiefer, J., 1959. Optimum experimental designs (with discussion). J. Roy. Statist. Soc. Ser. B 21, 272–319. Kiefer, J., 1961. Optimum designs in regression problems, II. Ann. Math. Statist. 32, 298–325. Kiefer, J., 1985. Jack Carl Kiefer Collected Papers III. Design of Experiments. Springer, New York. Lau, T.S., Studden, W.J., 1985. Optimal designs for trigonometric and polynomial regression using canonical moments. Ann. Statist. 13, 383–394. LNauter, E., 1974. Experimental design in a class of models. Math. Oper. Statist. 5, 379–398. Mardia, K., 1972. The Statistics of Directional Data. Academic Press, New York. Pukelsheim, F., 1993. Optimal Design of Experiments. Wiley, New York.
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Pukelsheim, F., Rosenberger, J.L., 1993. Experimental designs for model discrimination. J. Amer. Statist. Assoc. 88, 642–649. Skibinsky, M., 1967. The range of the (n + 1)-th moment for distributions on [0; 1]. J. Appl. Probab. 4, 543–552. Skibinsky, M., 1986. Principal representations and canonical moment sequences for distribution on an interval. J. Math. Anal. Appl. 120, 95–120. Studden, W.J., 1980. Ds -optimal designs for polynomial regression using continued fractions. Ann. Statist. 8, 1132–1141. Studden, W.J., 1982a. Some robust-type D-optimal designs in polynomial regression. J. Amer. Statist. Assoc. 77, 916–921. Studden, W.J., 1982b. Optimal designs for weighted polynomial regression using canonical moments. In: Gupta, S.S., Berger, J.O. (Eds.), Statistical Decision Theory and Related Topics III, Vol. 2, Academic Press, New York, pp. 335 –350. Studden, W.J., 1989. Note on some $p -optimal designs for polynomial regression. Ann. Statist. 17, 618–623. Zen, M.M., Tsai, M.H., 2002a. Some criterion-robust optimal designs for the dual problem of model discrimination and parameter estimation. Indian J. Statist. 64, 322–338. Zen, M.M., Tsai, M.H., 2002b. Criterion-robust optimal designs for model discrimination and parameter estimation in Fourier regression models. Technical Report No. 44, Department of Statistics, National Cheng-Kung University.
www.elsevier.com/locate/jspi
Criterion-robust optimal designs for model discrimination and parameter estimation in Fourier regression models Mei-Mei Zen∗ , Min-Hsiao Tsai Department of Statistics, National Cheng-Kung University, Tainan 70101, Taiwan Received 30 July 2002; accepted 8 April 2003
Abstract Consider the problem of discriminating between two competitive Fourier regression on the circle [ − ; ] and estimating parameters in the models. To 5nd designs which are e6cient for both model discrimination and parameter estimation, Zen and Tsai (some criterion-robust optimal designs for the dual problem of model discrimination and parameter estimation, Indian J. Statist. 64, 322–338) proposed a multiple-objective optimality criterion for polynomial regression models. In this work, taking the same M -criterion, which puts weight (0 6 6 1) for model discrimination and 1 − for parameter estimation, and using the techniques of projection design, the corresponding M -optimal design for Fourier regression models is explicitly derived in terms of canonical moments. The behavior of the M -optimal designs is investigated under di;erent weighted selection criterion. And the extreme value of the minimum M -e6ciency of any M -optimal design is obtained at = ∗ , which results in the M∗ -optimal design to be served as a criterion-robust optimal design for the problem. c 2003 Elsevier B.V. All rights reserved. MSC: 62K05 Keywords: Canonical moments; E6ciency; Multiple-objective; M -optimal design; Projection design
1. Introduction The study of optimal designs for polynomial regression models has been extensively investigated; see Kiefer (1959, 1961, 1985), Atkinson and Donev (1992), and Pukelsheim (1993) for details. In many 5elds of sciences, such as physics, chemistry ∗
Corresponding author. Tel.: +886-6-2757575; fax: +886-6-2342469. E-mail address: [email protected] (Mei-Mei Zen).
c 2003 Elsevier B.V. All rights reserved. 0378-3758/$ - see front matter doi:10.1016/S0378-3758(03)00212-X
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and computer science, among others, the consideration of Fourier regression models is more prevalent, while the problem of optimal designs for Fourier regression received much less attention; see Mardia (1972) and Cartwright (1990). Under a Fourier regression model, if the experimenters can make certain the order of periodicity of the underlying model, then the optimal design will gain the precision of parameter estimation; see Hoel (1965), Karlin and Studden (1966), Hill (1978), Lau and Studden (1985), and Pukelsheim (1993) for discussion. Unfortunately, in many situations the experimenters have some uncertainty about the periodicity order, hence the prerequisite is model discrimination between some competitive models; see Dette and Haller (1998) for details. As mentioned in Dette and Franke (2000), usually designs are desirable to be e6cient for discriminating between several competitive models and to have good properties for the estimation of parameters in the identi5ed model. So far, most of literature focus on the study of either model discrimination or parameter estimation separately. However, the experimenters usually hope to seek some designs which could take both purposes into account. In this study, we are to seek certain nonsequential experimental designs to deal with the problem of model discrimination and parameter estimation simultaneously in Fourier regression models. The purpose of the present paper is to provide some criterion-robust designs that can be served e6ciently for the described problem. Consider the model E(Yx ) = 0 +
lc
i cos ix +
i=1
ls
i sin ix = # f (x);
(1.1)
i=1
where # =(V ; ); V =(0 ; 1 ; : : : ; lc ), =( 1 ; : : : ; ls ) are the vectors of parameters, f (x) = (c (x); s (x)), c (x) = (1; cos x; : : : ; cos lc x), s (x) = (sin x; : : : ; sin ls x) denote the vectors of regression functions, the integer max{lc ; ls } denotes the order of periodicity, and x ∈ [ − ; ]. In this study, an approximate design is a probability measure on [ − ; ]. Under (1.1), the performance of is essentially evaluated by the information matrix f M () = f (x) f (x) d(x): (1.2) [−;]
There are numerous criteria used for the construction of optimal designs; in this paper, we apply the D- and Ds -optimality criteria, s ¿ 1, to deal with the problem. As pointed out by Lau and Studden (1985), it is well known that the D-optimal designs for parameter estimation in models with lc = ls = k are also D-optimal for any lower order, e.g., lc = ls = k − 1, and they are also D2 -optimal for discriminating between order k − 1 and k. Thus we only study the problem of discriminating between Model A: lc = k − 1;
ls = k and Model B: lc = ls = k
and take regard of parameter estimation simultaneously. An alternative setting with lc = k; ls = k − 1 versus lc = ls = k is also investigated in Zen and Tsai (2002b). Note that the integer k denotes the order of periodicity.
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For polynomial situation, Pukelsheim and Rosenberger (1993) made a comparison of e6ciencies among some special experimental designs which serve all three objectives simultaneously: (1) to discriminate between Models A and B and, depending on the decision, (2) to make inferences in Model A or (3) in Model B. For multiple-objective consideration, Zen and Tsai (2002a) proposed a weighted optimality criterion and applied the maximin principle on the e6ciency to obtain a criterion-robust optimal design. Herein, we use the same criterion for Fourier regression models. As stated in Dette and Haller (1998), the determinant of (1.2) is invariant with respect to a reLection of the design at the origin; to seek optimal designs, we restrict our attention to the set of all symmetric designs on [ − ; ] hereafter. The paper will be organized as follows. In Section 2, we introduce the projection design and the M -selection criterion. Then using the theory of canonical moments, an M -optimal design can be expressed by the canonical moments of its projection. Since di;erent selection criterion results in di;erent optimal design, an appropriate selection criterion is essentially important for the problem itself. The minimum M -e6ciency of arbitrary M -optimal design and the behavior of these minimum values will be studied in Section 3. It turns out that the maximum value of the minimum M -e6ciency of any M -optimal design occurs at =∗ . Based on the maximin principle, a criterion-robust optimal design, M∗ , will be derived for any periodicity order k. Especially, the support points and weights of M∗ with the minimum M -e6ciency will be given for some small k. In Section 4, we make a comparison of e6ciencies among some special optimal designs. Section 5 gives some brief concluding remarks. Finally, the technical proof is given in the Appendix.
2. The M -optimal design Using the D-criterion, to compute the determinant of M f () in (1.2), a one-to-one mapping between the set of all symmetric designs on [−; ] and the set of all designs on the interval [−1; 1] is essentially necessary. Let z =cos x. For any symmetric design on [ − ; ], the projection on [ − 1; 1] is de5ned by 2(x) if 0 ¡ |x| 6 ; (2.1) (z) = (0) if x = 0: Then the determinant of M f () can be expressed in terms of the product of determinants of two information matrices for weighted polynomial regression models; see Lau and Studden (1985) and Dette and Haller (1998) for details. To discriminate between Models A and B, it is equivalent to testing the hypotheses H0 : k = 0 versus H1 : k = 0. For simplicity, let MA () and MB () denote the information matrices for Models A and B, separately. Then using the D1 -criterion, it results in the D1 -optimal design D1 which maximizes the objective function D1 ()
=
|MB ()| ; |MA ()|
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where | · | denotes the determinant. To make inferences in Model A (or B), the D-optimal design A (or B ) maximizes the objective function A ()
= |MA ()|1=2k
(or
B ()
= |MB ()|1=(2k+1) ):
(2.2)
Considering all three objectives simultaneously, mentioned in Pukelsheim and Rosenberger (1993), a multiple-objective optimal design is de5ned to maximize the following weighted function of D1 (); A () and B (): M ()
=
A () B () D1 ()
= |MA ()|(=2k)− |MB ()|[=(2k+1)]+ ;
(2.3)
where 0 6 ; ; 6 1 and ++=1. Since there is no information about which model is appropriate before model discrimination, as proposed by Zen and Tsai (2002a), it seems reasonable to put equal weights for objectives (2) and (3) (i.e. = ), then (2.3) becomes M () = [ A () B ()]
(1−)=2
[
D1 ()]
= |MA ()|(1−)=4k− |MB ()|[(1−)=2(2k+1)]+ :
(2.4)
Thus the design denoted by M , which maximizes (2.4), is called the M -optimal design. The derivation of the M -optimal design is based on canonical moments. The theory of canonical moments was introduced by Skibinsky (1967) and widely applied by Studden (1980, 1982a,b, 1989) to determine optimal designs for polynomial regression models. For more details, readers can refer to Skibinsky (1986), Lau and Studden (1985), and Dette and Studden (1997). For a given probability measure on X = [ − 1; 1], let pi denote the ith canonical moment of . Following the same arguments as in Lau and Studden (1985), () de5ned by (2.2) and (2.4) can be expressed by the canonical moments of the projection . Theorem 2.1. The canonical moments of the projection corresponding to an M -optimal design satisfy pj = 12 ; j = 1; : : : ; 2k − 1 and 1 + 1− 2 2k+1 p2k = (2.5) : 2 1 + 1− 2 2k+1 + 2k Proof. Substituting k−1 = [[(1 − )=2(2k)] − ]=[(1 − )(1=2k + 1=(2k + 1))]; k = [[(1 − )=2(2k + 1)] + ]=[(1 − )(1=2k + 1=(2k + 1))] and k = 12 in Theorem 5.1 in Lau and Studden (1985) proves the assertion. There are similar results for the canonical moments of the projection corresponding to D-optimal designs A and B with nontrivial weights k−1 = k = 12 and k = k = 12 , respectively. Note that when =1, the M1 -optimal design is the same as the D1 -optimal design, which is an optimal design for model discrimination. Moreover, taking the derivative of p2k in (2.5) with respect to , it can be shown that p2k is strictly increasing in . As mentioned in Dette and Haller (1998) [see also Skibinsky (1986)], if j denote the 5rst index with pj = 0 or 1, then the sequence of canonical moments terminates at
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this pj and the design is supposed at a 5nite number of points. Moreover, the set of probability measures on the interval [ − 1; 1] with 5rst m canonical moments equal to (p1 ; : : : ; pm ) ∈ (0; 1)m−1 × [0; 1] is a singleton if and only if pm is 0 or 1. Otherwise, there exist in5nitely many probability measures corresponding to (p1 ; : : : ; pm ). According to the terminated canonical moments, we conclude that the DA -optimal design A for parameter estimation in Model A and the D1 -optimal design D1 for model discrimination are uniquely determined, and there are in5nitely many DB -optimal designs and M -optimal designs for any 5xed 0 6 ¡ 1. Thus, for the case 0 ¡ p2k ¡ 1, we terminate the sequence of canonical moments (p1 ; : : : ; p2k ) with p2k+1 = 0 or p2k+1 = 1. This corresponds to a lower (p2k+1 = 0) or an upper (p2k+1 = 1) principal representation of (p1 ; : : : ; p2k ), and the corresponding projection has (k + 1) support points including the endpoint −1 (if p2k+1 = 0) or +1 (if p2k+1 = 1); see Skibinsky (1986). The corresponding minimal support designs are denoted by − and + , respectively. To explicitly determine (minimal support) optimal designs, it is essential to 5nd the support points and weights of the projection , which can be found in Studden (1982a). Then the inverse transformation of (2.1) gives the support points and weights for arbitrary symmetric design , including D1 ; A ; B∓ ∓ and M . 3. Criterion-robust optimal design 3.1. The e4ciencies of the M -optimal design In this section, we will consider the behavior of a design under the M -criterion, as in (2.4), for any 5xed ∈ [0; 1] 5rst. Especially, we are interested in how the M -optimal design M behaves under di;erent M -criteria. The performance of any M -optimal design M under the M -criterion is evaluated by the M -e6ciency, which is de5ned by M (M )
M -e; (M ) =
M (M )
;
; ∈ [0; 1]:
(3.1)
Direct computation gives M (M )
[ ] +((1−)=2)(1=(2k+1)) [ ] ((1−)=2)([1=(2k+1)]+1=2k) = (p2k ) (q2k ) :
Then an obvious result follows directly. Theorem 3.1. For any 6xed ∈ [0; 1] and k ∈ N, let M be any M -optimal design. Then the M -e4ciency in (3.1) can be expressed as +((1−)=2)(1=(2k+1)) (1−=2)(1=(2k+1)+1=2k) [ ] [ ] q2k p2k M -e; (M ) = ; (3.2) [] [] p2k q2k [·] where p2k is as in (2.5).
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Mei-Mei Zen, Min-Hsiao Tsai / Journal of Statistical Planning and Inference 124 (2004) 475 – 487 1.0
0.30.4 0.5 0.6
0.7
0.8
0.9
0.98
0.8
0.6
γ
1 0.4
0.2
0.98
0.0 0.0
0.2
0.9
0.8 0.4
0.7 0.6 0.5 0.4 0.3 0.6
0.8
0
1.0
γ′
Fig. 1. The contour of the M -e6ciencies for all M -optimal designs, k = 1.
3.2. Minimum M -e4ciency of the M -optimal design In order to get an insight of the behavior of the M -e6ciencies, we deal with the case of k = 1 and plot the contour of the M -e6ciencies for all M -optimal designs in Fig. 1. It is obvious that the value of M -e; (M ) is getting small as is moving apart from . Therefore, a study of the extreme value of M -e; (M ) for any 5xed is essentially important. The following lemma is necessary to the main results and the proof is given in the Appendix. Lemma 3.2. For any k ∈ N, (i) given ∈ [0; 1]; M -e; (M ), a function of , is strictly increasing on [0; ) and decreasing on ( ; 1]; (ii) given ∈ [0; 1]; M -e; (M ), a function of , is strictly increasing on [0; ) and decreasing on (; 1]. From Lemma 3.2(i), we know that for any given , the minimum value of M -e; (M ) will be attained at either = 0 or = 1. To determine which one is the exact minimum value, let h( ) =
M1 -e; (M ) M0 -e; (M )
denote the ratio of these two possible minimum e6ciencies. Substituting (3.2), it turns out 1−[1=2(2k+1)] −1=2(1=(2k+1)+1=2k) [ ] [ ] [0] q2k p2k p2k h( ) = : (3.3) [0] [0] [1] p2k q2k p2k
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481
Note that both inequalities M1 -e; (M ) ¿ M0 -e; (M ) and h( ) ¿ 1 are equivalent. The two obvious facts that [] (a) p2k in (2.5) is strictly increasing in , and 1−[1=2(2k+1)] (b) p (1 − p)−1=2(1=(2k+1)+1=2k) is strictly increasing in p
imply that h( ) is strictly increasing in . Therefore, we have the following result: Theorem 3.3. For any 6xed ∈ [0; 1] and k ∈ N, M0 -e; (M ) if ¿ ∗ ; min {M -e; (M )} = 0661 M1 -e; (M ) if ¡ ∗ ; where M0 -e; (M ) and M1 -e; (M ) are as in (3.2) and ∗ is the root of h( ) = 1 in (3.3). To 5nd a criterion-robust optimal design, the maximum value of min0661 {M -e; (M )} plays an important role. From Lemma 3.2(ii), we have the fact that, M0 -e; (M ) is strictly decreasing in and M1 -e; (M ) is strictly increasing in , which gives the following important result: Theorem 3.4. For any k ∈ N; min0661 {M -e; (M )} is increasing 6rst, then decreasing in and the maximum value of min0661 {M -e; (M )} is attained at = ∗ , i.e.
∗ = arg max min M -e; ( ) ; M 06 61 0661
∗
where is the root of h( ) = 1 in (3.3). To describe this main theorem graphically, Fig. 2 demonstrates the plots of min0661 {M -e; (M )} versus for k = 1 and 2; the corresponding values of ∗ are 0.52614 k=2
k =1 0.8
min {Mγ -eff (σMγ ′)
0.4
0<γ<1
min {Mγ -eff (σMγ ′)
0<γ<1
{
{
0.6
0.2
0.0
0.4 0.2 0.0
0.0
(a)
0.6
0.2
0.4
0.6 γ′
0.8
0.0
1.0
(b)
0.2
0.4
0.6 γ′
Fig. 2. The plots of min0661 {M -e; (M )} versus ; k = 1 and 2.
0.8
1.0
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Mei-Mei Zen, Min-Hsiao Tsai / Journal of Statistical Planning and Inference 124 (2004) 475 – 487
Table 1 The values of ∗ and the minimum M -e6ciencies for small k k
∗
min0661 M -e; (M∗ )
min0661 M -e; (M1=2 )
min0661 M -e; (M1=3 )
1 2 3 4 5 6 7 8 9 10
0.5261 0.4752 0.4445 0.4229 0.4063 0.3931 0.3821 0.3727 0.3647 0.3575
0.7540 0.8172 0.8492 0.8697 0.8844 0.8954 0.9042 0.9114 0.9173 0.9224
0.7368 0.8050 0.8279 0.8452 0.8588 0.8699 0.8791 0.8869 0.8936 0.8994
0.6154 0.7273 0.7869 0.8247 0.8511 0.8705 0.8854 0.8972 0.9068 0.9148
and 0.47525, respectively. Note that the value of ∗ will depend on the order k and can be numerically solved only. For practical use, Table 1 gives some numerical results of ∗ and the corresponding minimum e6ciencies of M∗ ; M1=2 and M1=3 for various k, where designs M1=2 and M1=3 are shown for comparison. This is because that for small k; ∗ is near 12 , and for moderate k; ∗ is close to 13 . Note that in Table 1, ∗ is decreasing in k, and the minimum e6ciency, min0661 {M -e; (M∗ )}, is increasing in k and greater than 0.754 for all k. That means for any M -criterion, M∗ -optimal design M∗ is robust in the sense of having high minimum e6ciency. Therefore, this M∗ can be served as a criterion-robust optimal design for the problem of model discrimination and parameter estimation. The fact that ∗ is decreasing in k means that the role of model discrimination is getting less important when the order of Fourier regression model is getting large. Since di;erent selection criterion will result in di;erent optimal design, under the M∗ -selection criterion, from Theorem 2.1, we can derive the canonical moments of the projection of the criterion-robust design precisely. Although ∗ can be uniquely solved by numerical method, but there are in5nitely many M∗ -optimal designs. − + Table 2 gives the support points and weights of M and M with = ∗ ; 12 and 13 .
4. Comparison with some special designs In this section, to make a comparison of the M∗ -optimal design M∗ with some special optimal designs, we derive the M∗ -, D1 -, DA - and DB -e6ciencies 5rst. The proof of the following result is straightforward and omitted. Proposition 4.1. For any 6xed k ∈ N, the M∗ -, D1 -, DA - and DB -e4ciencies of any symmetric design whose projection design has canonical moments pj = 12 ; j 6 2k −1,
Mei-Mei Zen, Min-Hsiao Tsai / Journal of Statistical Planning and Inference 124 (2004) 475 – 487
483
Table 2 The support points and weights of M∗ ; M1=2 and M1=3 k
1
∗
− M
1 2
1 3
2
∗ 1 2
1 3
3
∗ 1 2
1 3
+ M
±3:141
±0:7167
0:2149
0:2851
±3:141
±0:7424
0:2121
0:2879
±3:141
±0:9079
0:1905
0:3095
±3:141
±1:671
±0:4106
0:1184
0:2277
0:1539
±3:141
±1:663
±0:3969
0:1189
0:2293
0:1518
±3:141
±1:726
±0:4909
0:1143
0:2179
0:1678
±3:141
±2:127
±1:138
±0:2926
0:0810
0:1605
0:1522
0:1063
±3:141
±2:121
±1:122
±0:2713
0:0814
0:1616
0:1542
0:1028
±3:141
±2:143
±1:176
±0:3368
0:0798
0:1576
0:1480
0:1146
are given as follows. M∗ -e; () =
p2k
∗ +((1−∗ )=2)(1=(2k+1))
∗]
[ p2k
D1 -e; () = p2k ; DA -e; () = (q2k )1=2k
±2:425
0
0:2851
0:4298
±2:399
0
0:2879
0:4242
±2:234
0
0:3095
0:3810
±2:731
±1:471
0
0:1539
02277
0:2368
±2:745
±1:479
0
0:1518
0:2293
0:2378
±2:651
±1:416
0
0:1678
0:2179
0:2286
±2:849
±2:004
±1:014
0
0:1063
0:1522
0:1605
0:1620
±2:870
±2:019
±1:020
0
0:1028
0:1542
0:1616
0:1628
±2:805
±1:966
±0:9983
0
0:1146
0:1480
0:1576
0:1596
q2k
∗]
[ q2k
((1−∗ )=2)(1=(2k+1)+1=2k) ;
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Mei-Mei Zen, Min-Hsiao Tsai / Journal of Statistical Planning and Inference 124 (2004) 475 – 487
D1-efficiency
Mγ∗-efficiency 1.00
1.0
0.8
0.90
0.6 0.80 0.4 0.70 1
10
20
(a)
30
40
1
50
k
10
20
30
40
50
k
(b)
DA-efficiency
DB-efficiency
0.98
0.9 σMγ* σD1 σM1/2
0.8 0.94
0.7
σM1/3
0.6
σM0 σB σA
0.90 0.5 1
(c)
10
20
30
40
k
1
50
10
20
(d)
30 k
40
50
Fig. 3. The M∗ -, D1 -, DB - and DA -e6ciencies of designs M∗ , D1 ; M1=2 ; M1=3 , M0 ; B and A . Note that (1) The M∗ -, D1 -, and DB -e6ciencies of design A are all zero for k ∈ N, (2) the M∗ -, DB -, and DA -e6ciencies of design D1 are all zero for k ∈ N.
and DB -e; () = (4p2k q2k )1=(2k+1) ; where p2k is the 2kth canonical moment of the projection . Then we investigate the e6ciencies of designs D1 ; A ; B ; M0 , M1=3 ; M1=2 and M∗ , which are de5ned in Section 2. Note that M0 is an analogue of the LNauter type design which was introduced by LNauter (1974). Fig. 3 shows the plots of the e6ciencies, in Proposition 4.1, of these seven designs versus various periodicity order k. Fig. 3(a) shows that M1=2 is better than M1=3 for small k, and M1=3 is superior to M1=2 for moderate to large k; this is due to the solution of ∗ in Table 1. It is worthwhile to mention that all the M∗ -e6ciencies of D1 and A are zero for arbitrary k.
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Fig. 3(b) shows that all the D1 -e6ciencies of M∗ , M1=2 ; M1=3 and M0 are increasing in k. Moreover, for arbitrary k the D1 -e6ciencies of B and A are always 12 and 0, respectively. Fig. 3(c) shows that the DB -e6ciencies of M∗ ; M1=2 and M1=3 are decreasing 5rst, then increasing in k. Overall, the DB -e6ciencies are higher for arbitrary k. Note that the DB -e6ciency of M0 is increasing in k, and it is the second highest for most of k; the DB -e6ciencies of D1 and A are all zero for arbitrary k. Fig. 3(d) shows that the DA -e6ciencies are increasing, except D1 and A whose are 0 and 1, respectively. The DA -e6ciencies of M∗ , M1=2 and M1=3 are somewhat conservative for small k. 5. Concluding remarks Using a multiple-objective selection criterion, the M -criterion, for the problem of model discrimination and parameter estimation in the Fourier regression models, we obtain the maximum value of the minimum M -e6ciency of any M -optimal designs at = ∗ . The corresponding M∗ -optimal design can be served as a criterion-robust optimal design. And its minimum M -e6ciency is given for small order of periodicity k. In particular, this criterion-robust optimal design could be used practically with any M -e6ciency greater than 0.754. Note that in polynomial regression situation, for any degree k, any M -e6ciency of the criterion-robust optimal design is greater than 0.952, shown in Zen and Tsai (2002a). Also for small k; ∗ in polynomial regression is near 0.35, but ∗ in Fourier regression is near 0.5. The comparison of ∗ in both polynomial and Fourier models shows that one should put a little bit more weight on model discrimination in Fourier than in polynomial regression models. Acknowledgements The authors would like to thank the referees for their helpful comments; this research was partially supported by NSC91-2118-M006-001 from the National Science Council, ROC. Appendix Proof of Lemma 3.2. Let 1 ; 2 ; be any number in [0; 1], from (3.2), we have M1 -e; (M ) M2 -e; (M ) (1 −2 ) [ ] (1−1=2(2k+1)) [2 ] 1=2(1=(2k+1)+1=2k) q p 2k 2k = × M1 -e; (M2 ); [2 ] [ ] p2k q2k (A.1)
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[·] where p2k is as in (2.5). Note that the e6ciency of any design is between 0 and 1.
(i) For any 5xed , we prove the results in two steps:
[2 ] [ ] [·] is strictly increasing in , we have p2k ¿ p2k . (a) Choose 1 ; 2 ∈ [0; ). Since p2k [ ] [2 ] [2 ] [ ] This implies that both ratios p2k =p2k and q2k =q2k are greater than 1, which results in that M1 -e; (M ) ¡ M2 -e; (M ), if 1 ¡ 2 . That is, for any given , M -e; (M ) is strictly increasing in ∈ [0; ).
[ ] [2 ] ¡ p2k , which implies that (b) Choose 1 ; 2 ∈ ( ; 1]. In this case, we have p2k [ ] [2 ] [2 ] [ ] p2k =p2k and q2k =q2k are less than 1. Therefore, we have that M1 -e; (M ) ¡ M2 -e; (M ), if 1 ¿ 2 . That is, for any given , M -e; (M ) is strictly decreasing in ∈ ( ; 1].
(ii) For any 5xed , multiply both sides of (A.1) by M2 -e; (M )=M1 -e; (M2 ), and replace by 1 , 1 by , we thus obtain M -e; (M1 ) M -e; (M2 ) (−2 ) [1 ] (1−1=2(2k+1)) [2 ] 1=2(1=(2k+1)+1=2k) q p 2k 2k × M2 -e; (M1 ): = [2 ] [1 ] p2k q2k Similar argument as in (i) proves the assertion. References Atkinson, A.C., Donev, A.N., 1992. Optimum Experimental Designs. Oxford University Press, Oxford. Cartwright, M., 1990. Fourier Methods for Mathematicians, Scientists and Engineers. Ellis Horwood, New York. Dette, H., Franke, T., 2000. Constrained D- and D1 -optimal designs for polynomial regression. Ann. Statist. 28, 1702–1727. Dette, H., Haller, G., 1998. Optimal designs for the identi5cation of the order of a Fourier regression. Ann. Statist. 26, 1496–1521. Dette, H., Studden, W.J., 1997. The Theory of Canonical Moments with Applications in Statistics, Probability, and Analysis. Wiley, New York. Hill, P.D.H., 1978. A note on the equivalence of D-optimal design measures for three rival linear models. Biometrika 65, 666–667. Hoel, P.G., 1965. Minimax designs in two-dimensional regression. Ann. Math. Statist. 36, 1097–1106. Karlin, S., Studden, W.J., 1966. Tchebyche; Systems: With Applications in Analysis and Statistics. Interscience, New York. Kiefer, J., 1959. Optimum experimental designs (with discussion). J. Roy. Statist. Soc. Ser. B 21, 272–319. Kiefer, J., 1961. Optimum designs in regression problems, II. Ann. Math. Statist. 32, 298–325. Kiefer, J., 1985. Jack Carl Kiefer Collected Papers III. Design of Experiments. Springer, New York. Lau, T.S., Studden, W.J., 1985. Optimal designs for trigonometric and polynomial regression using canonical moments. Ann. Statist. 13, 383–394. LNauter, E., 1974. Experimental design in a class of models. Math. Oper. Statist. 5, 379–398. Mardia, K., 1972. The Statistics of Directional Data. Academic Press, New York. Pukelsheim, F., 1993. Optimal Design of Experiments. Wiley, New York.
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