Constrained optimal discrimination designs for Fourier regression models

Transcription

1 Ann Inst Stat Math (29) 6:43 57 DOI.7/s Constraine optimal iscrimination esigns for Fourier regression moels Stefanie Bieermann Holger Dette Philipp Hoffmann Receive: 26 June 26 / Revise: 9 March 27 / Publishe online: 7 July 27 The Institute of Statistical Mathematics, Tokyo 27 Abstract In this article, the problem of constructing efficient iscrimination esigns in a Fourier regression moel is consiere. We propose esigns which maximize the power of the F-test, which iscriminates between the two highest orer moels, subject to the constraints that the tests that iscriminate between lower orer moels have at least some given relative power. A complete solution is presente in terms of the canonical moments of the optimal esigns, an for the special case of equal constraints even more specific formulae are available. Keywors Constraine optimal esigns Trigonometric regression D -optimal esigns Chebyshev polynomials Canonical moments Introuction The Fourier regression or trigonometric regression moel g 2 (x) = a + g 2 (x) = a + a j sin( jx) + j= j= b j cos( jx), x [ π, π], () j= a j sin( jx) + b j cos( jx), x [ π, π], (2) j= S. Bieermann University of Southampton, School of Mathematics, Highfiel SO7 BJ, UK H. Dette (B) P. Hoffmann Ruhr-Universität Bochum, Fakultät für Mathematik, 4478 Bochum, Germany holger.ette@ruhr-uni-bochum.e; holger.ette@rub.e

2 44 S. Bieermann et al. where IN, is use to escribe perioic phenomena (see, e.g., Maria (972), Kitsos et al. (988) or the collection of research papers in biology eite by Lestrel (997)). Moreover, there are several applications of trigonometric regression moels in two-imensional shape analysis in biology. We refer to Younker an Ehrlich (977) an Currie et al. (2) for concrete examples. The value 2 in () or2 in(2) is usually enote as the egree of the Fourier regression moel. The coefficients a, a,...,a, b,...,b enote unknown parameters, which have to be estimate from the ata. The problem of esigning experiments for moels of the form () has been iscusse by several authors; see, e.g., Karlin an Stuen (966), p. 347, Feorov (972), p. 94, Hill (978), Lau an Stuen (985) for optimal esigns on the full circle, as well as Dette et al. (22a) an Dette et al. (22b) for optimal esigns on a partial circle. Most authors concentrate on the problem of etermining optimal esigns for the estimation of the full vector of unknown parameters, whereas the problem of constructing optimal esigns for moel iscrimination has been consiere by Dette an Haller (998),Dette an Melas (23) an Zen an Tsai (24). The present paper is evote to the problem of constructing optimal iscrimination esigns using constraine optimality criteria. Constraine optimal esigns have primarily been consiere by Stigler (97), Stuen (982b) an Lee (988a,b), whereas Cook an Wong (994), Dette (995) an Clye an Chaloner (996) investigate the relation between this approach an compoun optimality criteria. Although these results are interesting from a theoretical point of view constraine optimal esigns still ha to be foun numerically an explicit results coul only be inferre in rare cases. In particular, it turns out that there is a one-to-one corresponence between compoun an constraine optimal esigns, which, however, can only be exploite in rare cases to fin the constraine optimal esign from the corresponing compoun optimal esign, which is usually much simpler to calculate. Dette an Franke (2) characterize constraine optimal iscriminating esigns for polynomial regression moels utilizing the theory of canonical moments, which was introuce by Skibinsky (967) an applie by Stuen (98, 982a,b, 989) for etermining optimal esigns in polynomial regression moels. The problem of fining constraine optimal iscriminating esigns for Fourier regression moels, however, has not been consiere yet. The present paper is evote to this problem. For the construction of constraine optimal esigns we assume that the highest frequency of the moel has been fixe an etermine the esign such that the coefficient corresponing to this frequency is estimate with maximal efficiency subject to the constraints that the coefficients corresponing to the highest frequencies in the moels of lower egree can be estimate with some guarantee efficiency. The optimality criterion is carefully escribe in Sect. 2. In Sect. 3 we briefly review some facts from the theory of canonical moments (see Dette an Stuen 997), which is the basic tool for the construction of optimal iscrimination esigns. A complete characterization of the constraine optimal escriminating esigns is given in terms of their canonical moments, an in the special case of equal bouns we further specify the optimal esigns in terms of their supporting polynomials an explicit formulae for the weights. Finally, in Sect. 4 we give some concluing remarks iscussing our approach.

3 Discrimination esigns for Fourier regression moels 45 2 Constraine optimal esigns in Fourier regression moels We assume that we can make n 2 + inepenent observations Y,...,Y n where Y i N (g(x i ), σ 2 ), an the regression function g(x) belongs to the class of trigonometric moels {g, g,...,g 2 } where g 2l an g 2l are efine in () an (2), respectively. For k =,...,2 we efine { (, sin(x), cos(x),...,sin( jx), cos( jx)) f k (x)= T, if k = 2 j (, sin(x), cos(x),...,sin(( j )x), cos(( j )x), sin( jx)) T, if k =2 j an θ k = { (a, a, b,...,a j, b j ) T if k = 2 j (a, a, b,...,a j, b j, a j ) T if k = 2 j. Then the moels g k (x) where k =,...,2, can be written as g k (x) = f k (x) T θ k. An approximate esign is a probability measure σ with finite support on the interval [ π, π] with the interpretation that observations are taken at the support points in proportion to the corresponing masses. The information matrix in the Fourier regression moel g k (x) is given by M k (σ ) = π π f k (x) fk T (x)σ(x). (3) An optimal esign maximizes an appropriate information function of the informationmatrix(seepukelsheim 993, p. 3). There are numerous criteria which can be use for the characterization of efficient esigns. Most of these criteria focus on precise parameter estimation in a moel of given egree. In many practical situations, however, it is not known before the experiment, up to which egree a Fourier regression moel shoul be fitte. As sparse moeling is avisable we turn our attention to esigns that allow successive testing of the higher orer coefficients with high power, thus guaranteeing goo iscrimination properties of the testing proceure. Our optimality criterion for constructing iscrimination esigns is therefore base on a multiple F-test, where, starting with the given regression moel g 2 (x) in () one tests the hypotheses H (2) : b =, H (2 ) : a =, H (2 2) : b =, H (2 3) : a =,..., H () : a =, in the moels g 2, g 2,...,g, successively, an ecies for the moel g k where k is the first inex for which the hypothesis H (k ) : θ k = is rejecte. (Note that this sequence of tests can be stoppe earlier if the minimal egree of the Fourier regression moel is pre-specifie.) The statistical properties of this testing proceure are elaborately presente in Anerson (994). The quantities corresponing to the noncentrality parameter of the F-test for the hypothesis H (k) are given by δ k (σ ) = (e T k M k (σ )e k ) k =,...,2, (4)

4 46 S. Bieermann et al. where e k enotes the (k + )th unit vector in R k+ an the esign σ is assume to have at least (2 + ) support points (see Pukelsheim 993, p. 7). A esign σk is calle D -optimal for the moel g k or D k-optimal if it maximizes δ k.thisisin fact equivalent to maximizing the power of the corresponing test. Collecting ata following a D k -optimal sampling scheme therefore gives the best results for iscriminating between moels g k an g k. To iscriminate between more than two ifferent moels, one has to construct an optimality criterion base on several functions δ k (σ ), k = 2, 2,...,. As these quantities are of ifferent scalings we stanarize them by using the corresponing efficiencies to make them comparable in size. The expression eff k (σ ) := δ k(σ ) δ k (σk k =,...,2 (5) ), is calle the D k -efficiency of the esign σ in the Fourier regression moel g k(x). Dette an Haller (998) propose to maximize a weighte p-mean of the efficiencies eff,...,eff 2 for the construction of an optimal esign for iscriminating between the moels {g,...g 2 }. In the present paper, we consier an alternative optimality criterion to obtain efficient iscriminating esigns. This approach is attractive if the main interest of the experimenter is in iscriminating between the two moels of highest egree, while at the same time the optimal esign shoul allow for an efficient iscrimination between the moels of lower egrees. We consier two criteria for etermining a constraine optimal iscriminating esign for the Fourier regression moel. The first approach consiers the highest cosine frequency as most important an a constraine optimal iscriminating esign σ maximizes eff 2 (σ ) subject to eff k (σ ) c k, k = 2, 2 2,...,2 2 j (6) for some j {,..., }. The secon criterion, however, etermines the esign which maximizes eff 2 (σ ) subject to the constraints eff k (σ ) c k, k =2, 2 2,...,2 2 j, (7) an j {,..., }. For both criteria, the quantities c 2 2 j,...,c 2 (, ) are given by the experimenter reflecting the esire minimal relative power of testing H (k), k = 2 2 j,...,2. A necessary conition for the existence of optimal esigns is c 2 2 j + c 2 2 j. Unlike criterion (6), criterion (7) correspons to the situation where the highest sine frequency is regare as most important, i.e. the power of testing H (2 ) is maximize whereas the preceing test of H (2) (an all the other hypotheses H (k) ) have some prespecifie minimal relative power. The situation where the experimenter prefers to start with moel (2) for some practical reason, an therefore start the testing proceure with H (2 ), can be incorporate in criterion (7) by putting c 2 =. For the solution of the constraine optimization problems (6) an (7) we nee several tools, which will be explaine in what follows.

5 Discrimination esigns for Fourier regression moels 47 It follows by stanar arguments (see Pukelsheim 993, Chap. 4, 5) that eff k is a concave function on the set of esigns on the interval [ π, π] an invariant with respect to a reflection of the esign σ at the origin. Consequently, if there exists a constraine optimal iscriminating esign, then there also exists an optimal esign in the set of all symmetric esigns on the interval [ π, π]. We note that these symmetric esigns inuce esigns ξ σ on the interval [, ] by the projection ξ σ (cos x) = { 2σ(x) = 2σ( x) if < x π σ() if x = (8) for any symmetric esign σ. The corresponing set of the measures ξ σ on [, ] will be enote by [,].ItwasshowninDette an Haller (998) that for any σ δ k (σ ) = where B (ξ σ ) = A (ξ σ ) =, an 2 2( j ) A j (ξ σ ) A j (ξ σ ) if k = 2 j 2 2( j ) B j (ξ σ ) B j (ξ σ ) if k = 2 j (9) ( A k (ξ σ ) = ) k z i+ j ξ σ (z) i, j= () an ( ) k B k (ξ σ ) = ( z 2 )z i+ j ξ σ (z) i, j= () enote the information matrices of the esign ξ σ on the interval [, ] for a homosceastic an a heterosceastic polynomial regression moel with efficiency function λ(z) = ( z 2 ) (see Karlin an Stuen 966), respectively. Consequently, the problem of etermining constraine optimal iscriminating esigns for the Fourier regression moel can be solve by maximizing a certain function over the set of probability measures on the interval [, ] an transforming the maximizing measure back via (8). The problem of maximizing the right han sie of (9) over the set [,] if k = 2 j is in fact the D -optimal esign for the orinary polynomial regression moel, while for o values of k the right han sie of (9) correspons to the weighte polynomial regression with efficiency function σ 2 (x) = σ 2 /( x 2 ), x (, ). The solutions of these problems are well known (see Stuen 968, 982b an yiel δ k (σk ) = max δ k (σ ) = (k =,...,2), an therefore the efficiency of a symmetric esign σ efine in (5) can be rewritten σ as eff k (σ ) = 2 2( j ) A j (ξ σ ) A j (ξ σ ) if k = 2 j 2 2( j ) B j (ξ σ ) B j (ξ σ ) if k = 2 j. (2)

6 48 S. Bieermann et al. 3 The solution of the constraine optimal esign problem For the characterization of the measure ξ σ [,] corresponing to the constraine optimal iscriminating esign σ by the relation (8) we require some basic facts about the theory of canonical moments which has been introuce by Stuen (98, 982a,b) in the context of optimal esign. We will only give a very brief heuristical introuction an refer to the monograph of Dette an Stuen (997)formore etails. It is well known that a probability measure on the interval [, ], say ξ, is etermine by its sequence of moments (m, m 2,...).Skibinsky (967) efine a one to one mapping from the sequences of orinary moments onto sequences (p, p 2,...) whose elements vary inepenently in the interval [, ]. For a given probability measure on the interval [, ] the element p j of the corresponing sequence is calle the jth canonical moment of ξ. If j is the first inex for which p j {, } then the sequence of canonical moments terminates at p j, an the measure is supporte at a finite number of points. The support points an corresponing masses can be foun explicitly by evaluating certain orthogonal polynomials (see Dette an Stuen 997, Chap. 3). The set of probability measures on the interval [, ] with first k canonical moments equal to (p,...,p k ) (, ) k [, ] is a singleton if an only if p k {, }. Otherwise there exists an uncountable number of probability measures corresponing to (p,...,p k ) (see Skibinsky 986). It turns out that the eterminants in (2) can be escribe in terms of the canonical moments p, p 2,...of the measure ξ σ (see Stuen 982b), that is k A k (ξ σ ) =2 k(k+) (q 2l 2 p 2l q 2l p 2l ) k l+ (3) l= k B k (ξ σ ) =2 k(k+) (p 2l 2 q 2l p 2l q 2l ) k l+ (4) l= where p =, q = an q j = p j for j. Observing (2), (3) an (4), we fin that the efficiencies are increasing functions of the terms p 2l q 2l, an consequently the o canonical moments of the optimal projection esign ξ σ satisfy p 2l = 2 l =,...,. (5) Therefore we can restrict ourselves to esigns with this property, an (2) reuces to eff k (σ ) = 2 2 j 2 p 2 j j l= q 2l p 2l if k = 2 j 2 2 j 2 q 2 j j l= q 2l p 2l if k = 2 j where p 2, p 4,... enote the canonical moments of even orer of the esign ξ σ [,] satisfying (5) an corresponing to the measure σ via (8). Our main result gives a characterization of the canonical moments of ξ σ. (6)

7 Discrimination esigns for Fourier regression moels 49 Theorem (a) If there exists a constraine optimal iscriminating esign for the vector (c 2 2 j,...,c 2 ) in (6), then there also exists a symmetric optimal iscriminating esign σ. The canonical moments up to the orer 2 of the corresponing projection ξ σ are etermine by the system of equations p 2n = 2 n =,..., p 2n = 2 n =,..., j max 2, c 2 2 j+2n 2 2n j+n l= j p 2l q 2l, p 2 2 j+2n = max 2, c 2 2 j+2n 2 2n j+n p 2l q 2l, p 2 = c j l= j p 2lq 2l. l= j if c 2 2 j+2n >c 2 2 j+2n if c 2 2 j+2n c 2 2 j+2n n =,..., j (b) If there exists a constraine optimal iscriminating esign for the vector (c 2 2 j,..., c 2 2, c 2 ) in (7), then there also exists a symmetric constraine optimal iscriminating esign σ. The canonical moments up to the orer 2 ofthe corresponing projection ξ σ are etermine by the system of equations p 2n = 2 n =,..., p 2n = 2 n =,..., j max 2, c 2 2 j+2n 2 2n j+n l= j p 2l q 2l, p 2 2 j+2n = max 2, c 2 2 j+2n 2 2n j+n p 2l q 2l, p 2 = c j l= j p 2lq 2l. l= j if c 2 2 j+2n >c 2 2 j+2n if c 2 2 j+2n c 2 2 j+2n n =,..., j A necessary conition for the existence of optimal esigns with respect to either criterion ((6) or (7)) is given by c 2 2 j + c 2 2 j.

8 5 S. Bieermann et al. Proof Because both parts are proven similarly, we restrict ourselves to a proof of part (a). By the previous iscussion the canonical moments of o orer, 3,...,2 must be /2. Note that j+n eff 2 2 j+2n (σ ) = p 2 2 j+2n 2 2( j+n ) l= p 2l q 2l, n =,..., j. In orer to maximize these efficiencies we have to choose the canonical moments such that the proucts p 2l q 2l are as large as possible. This can be accomplishe by choosing p 2l as close as possible to the value /2 such that the constraints in (6) are satisfie. Since there are no restrictions on the efficiencies eff (σ ),..., eff 2 2 j 2 (σ ) we obtain p 2 = = p 2 2 j 2 = 2. Substituting this choice into the formulae for the higher orer efficiencies, (6) reuces to j+n eff 2 2 j+2n (σ ) = q 2 2 j+2n 2 2n l= j j+n eff 2 2 j+2n (σ ) = p 2 2 j+2n 2 2n l= j p 2l q 2l p 2l q 2l. We start with the case n =, for which the representations eff 2 2 j (σ ) = q 2 2 j, eff 2 2 j (σ ) = p 2 2 j yiel the constraints p 2 2 j c 2 2 j, q 2 2 j c 2 2 j. Consequently any esign ξ σ for which p 2 2 j [c 2 2 j, c 2 2 j ] satisfies the constraints of orer 2 2 j an 2 2 j. We therefore assume that c 2 2 j + c 2 2 j in what follows to ensure the existence of such a esign. If 2 [c 2 2 j, c 2 2 j ] one can choose p 2 2 j = 2 to maximize p 2 2 jq 2 2 j.else we have either c 2 2 j 2 or c 2 2 j 2, an we choose p 2 2 j = c 2 2 j or p 2 2 j = c 2 2 j, respectively. If n > we note that the constraints eff 2 2 j+2n (σ ) c 2 2 j+2n an eff 2 2 j+2n (σ ) c 2 2 j+2n reuce to p 2 2 j+2n q 2 2 j+2n c 2 2 j+2n 2 2n =: c j+n 2 2 j+2n p 2l q 2l l= j c 2 2 j+2n 2 2n j+n l= j p 2l q 2l =: c 2 2 j+2n Therefore the same arguments as presente for the case n = yiel the corresponing result for p 2 2 j+2n. Finally we consier the case n = j, where there is only one constraint eff 2 (σ ) c 2, which can be rewritten as p 2 c l= p 2lq 2l.

9 Discrimination esigns for Fourier regression moels 5 In orer to maximize this expression one has to choose p 2 such that there is equality. This proves the final assertion of part (a) in Theorem. Remark Note that Theorem characterizes the canonical moments up to the orer 2 of the projection ξ σ of the (symmetric) constraine optimal iscriminating esign σ. In general p 2 {, } an in these cases there exists an infinite number of probability measures on the interval [, ] with the canonical moments p,...p 2 (see Skibinsky 986). Each of these measures correspons to a constraine optimal iscriminating esign by the projection (8). One possible choice among these esigns with a reasonable small support will be illustrate later in the proof of Theorem 2, Eq. (24). In what follows, we present two further results, where a solution of the constraine optimal esign problem can be foun explicitly. For this purpose let T j (x) an U j (x) enote the jth Chebyshev polynomial of the first an secon kin, respectively (see Rivlin 974). Theorem 2 Consier the constraine optimal esign problem in (6) where c 2 2 j = =c 2 2 = c (, ),c l < c(l= 2 2 j,...,2 ). If there exists a constraine optimal iscriminating esign, then there also exists a symmetric constraine optimal iscriminating esign σ. (a) If c > /2, efine κ = κ(c 2, c) = 2 c 2 2c + 4c 2((2c ) j 2c), P+ (x) = (xu j(x) 2κU j (x))t j (x) 2c(xU j (x) 2κU j 2 (x))t j (x), (7) (x) = (xu j (x) 2κU j (x))u j (x) 2c(xU j (x) 2κU j 2 (x))u j 2 (x) (8) The polynomial P + (x) has + istinct roots x,...x in the interval (, ), an the esign ξ σ with masses λ k = (x k ) x P + (x), k =..., (9) x=x k at x,...,x correspons to a constraine optimal iscriminating esign by the projection (8) for the optimization problem (6). (b) If c < /2, efine P+ (x) = T +(x) ( 2c)T (x), (x) = U (x) ( 2c)U 2 (x). The polynomial P + (x) has + istinct roots x,...,x in the interval (, ), an the esign ξ σ with masses (9) at x,...,x correspons to a constraine optimal iscrimination esign by the projection (8) for the optimization problem (6)

10 52 S. Bieermann et al. Remark 2 A necessary conition for the existence of a symmetric constraine optimal iscrimination esign for the esign problem (6) is κ>, which ensures that the value of the canonical moment p 2 will be within the interval (, ).Ifκ it is therefore recommene to moify the choices of c an c 2 accoringly so that κ attains a positive value, before starting to calculate the optimal esign. Proof We only prove part (a) of the Theorem. Part (b) follows by similar (an even simpler) arguments. Note that the canonical moments of the constraine optimal iscriminating esign can be obtaine by Theorem. The canonical moments of o orer satisfy p 2n =, n =,...,, (2) 2 while the canonical moments of even orer less or equal than 2 2 j 2aregiven by p 2n =, n =,..., j. (2) 2 For the next canonical moment of even orer we have from Theorem { } p 2 2 j = max 2, c = c, an it can be shown by a straightforwar inuction that p 2 2 j+2t = (2c )t 2c, t =,..., j. (22) 2 (2c )(t + ) 2c Note that this representation implies 2 < c < j +, 2 j because the canonical moments vary in the interval (, ). The remaining canonical moment of orer 2 is obtaine by a irect calculation, that is p 2 = c 2 (2c ) j 2c 2 c (2c )( j + ) 2c. (23) It follows from a straightforwar but teious calculation that p 2 (, ) κ>, which proves Remark 2. Note that (2) (23) o not etermine a esign on the interval [, ] (except in the case c 2 =, which is exclue). In orer to obtain a esign with finite support we exten this sequence by p 2+ = 2, p 2+2 =. (24) The esign ξσ on the interval [, ] with canonical moments (2) (24) is uniquely etermine an has + support points not incluing or (seeskibinsky 986),

11 Discrimination esigns for Fourier regression moels 53 entailing that the esign σ will be supporte on points. For the calculation of the support points an corresponing weights we apply Theorem 3.6. in Dette an Stuen (997). By this result the esign ξ σ has weights λ k = (x k ) x P + (x) (25) x=x k at the roots x,...,x of the polynomial P+ (x), where P + (x) an P() (x) are obtaine from the recursion W k+ (x) = xw k (x) q 2k 2 p 2k W k (x) (26) (note that p 2 j = 2 for j =,..., + ) with ifferent initial conitions, that is P+ (x) = W +(x) for W (x), W (x) (27) (x) = W + (x) for W (x), W (x). (28) We now calculate these polynomials using (2) (24) an begin with P+ (x). From the initial conition in (27) an (2) we obtain by a straightforwar calculation W j (x) = Observing (22) an 2 j T j(x), W j (x) = 2 j 2 T j (x) (29) ( q 2l 2 p 2l = ) (2c )(l + j) 2c (2c )(l + j) 2c 2 (2c )(l + j) 2c 2 (2c )(l + j + ) 2c = (2c )(l + j + ) 2c 4 (2c )(l + j + ) 2c = 4 ( j < l ), we obtain the recursion W j+ = xw j (x) 2 cw j (x) W l+ (x) = xw l (x) 4 W l (x) if j < l. Now a straightforwar inuction yiels W j+l (x) = 2 l+ j (U l(x)t j (x) 2cU l (x)t j (x)), l =,..., j.

12 54 S. Bieermann et al. We finally note that by (22) an (23) wehaveq 2 2 p 2 = κ, from which it follows that P+ (x) = [ 2 (xu j (x) 2κU j (x))t j (x) 2c(xU j (x) ] 2κU j 2 (x))t j (x) using (26) an (27). Observing the initial conitions in (28) it follows that the polynomial (x) can be calculate analogously, where (29) is replace by W j (x) = 2 j U j (x), W j (x) = 2 j 2 U j 2(x). Consequently, by a straightforwar inuction we obtain (x) = [ 2 (xu j (x) 2κU j (x))u j (x) ] 2c(xU j (x) 2κU j 2 (x))u j 2 (x), an the assertion (a) of Theorem follows from Theorem 3.6. in Dette an Stuen (997). We conclue this section with an analogue for the optimization problem (7). The proof is similar an omitte for brevity. Theorem 3 Consier the constraine optimal esign problem in (7) where c 2 2 j = = c 2 2 = c 2 = c (, ), c l < c (l = 2 2 j,...,2 3). If there exists a constraine optimal iscriminating esign then there also exists a symmetric constraine optimal iscriminating esign σ. (a) If c > /2, efine κ = κ(c 2, c) = c 2 4c, an consier for this κ the polynomials P+ (x) an P() (x) efine by (7) an (8), respectively. The polynomial P+ (x) has + istinct roots x,...,x in the interval (, ), an the esign ξ σ which has masses (9) at the points x,...,x correspons to a constraine optimal iscriminating esign for the optimization problem (7) by the projection (8). (b) If c < /2, efine P+ (x) = T +(x) + ( 2c)T (x), (x) = U (x) + ( 2c)U 2 (x).

13 Discrimination esigns for Fourier regression moels 55 The polynomial P + (x) has + istinct roots x,...,x in the interval (, ), an the esign ξ σ with masses (9) correspons to a constraine optimal iscrimination esign for the optimization problem (7) by the projection (8). 4 Concluing remarks an iscussion We have applie the theory of canonical moments to erive constraine optimal esigns for iscriminating between Fourier moels of ifferent egree. For general constraints, we have foun explicit recursive relations (see Theorem ) for the first 2 canonical moments of the optimal esigns, from which optimal esigns can be compute applying Theorem 3.6. in Dette an Stuen (997). For some special cases (with respect to the constraints) we present explicit formulae for the supporting polynomials an the weights, which allow the irect computation of one of the optimal esigns (see Theorems 2 an 3). The choice of the lower bouns c l, l = 2 2 j,...,2, for the efficiencies is up to the experimenter accoring to his interest in a specific testing problem. There is, however, no guarantee that there exists an optimal esign satisfying these constraints. Apart from the necessary conitions that κ > (see Remark 2) an c 2 2 j+2n +c 2 2 j+2n, where n =,..., j an c k are efine in the proof of Theorem there seems to be no simple check if a particular choice of values c l will be amissible. Naturally, the experimenter woul like to have the bouns as large as possible, which might, however, contraict the existence of an optimal esign. Once the bouns have been chosen, we recommen to either compute the optimal canonical moments by the recursive relations given in Theorem or the optimal supporting polynomials P+ (x) via Theorems 2 or 3 (if appropriate). If any of the canonical moments is outsie the open interval (, ) or the roots of the polynomial P+ (x) are either outsie (, ) or not istinct then there exists no optimal esign with respect to these constraints. In this situation, the choice of the constraints has to be moifie by lowering the values of the less important bouns. If the experimenter oes not favour any testing problem over another, a natural approach to choose the bouns is c l = /2, l {2 2j,...,2} for some j. Applying Theorem yiels that in this situation all canonical moments up to the orer 2 equal /2. A somewhat relate approach woul be to maximize the minimal efficiency eff l (σ ) for any l within some subset L of {, 2,...,2}. This criterion has alreay been consiere in Dette an Haller (998), Sect. 4, Theorem 4.3. From part (b) of this theorem, it follows for example that for the choice L ={2 2j,...,2}, the optimal esigns also have canonical moments p k = /2, k =,...,2. These esigns therefore coincie with the constraine optimal esigns where all values c l, l L, are equal to /2. This correspons to intuition since neither approach gives preference to any testing problem within L over another. For ifferent choices of L the reaer is referre to Sect. 4 in Dette an Haller (998). In practice, however, there will be higher interest in testing the higher orer coefficients in most situations. A challenging problem for future consierations will be the generalization of our results an techniques to moels in more than one imension. In this situation, one approach may be to fin optimal esigns within the class of prouct esigns [see, e.g.,

14 56 S. Bieermann et al. Dette an Stuen (997), Sect. 5.8, for some results on multivariate polynomials]. Another metho worth trying will be a lattice esign approach as escribe in Riccomagno et al. (997), i.e. embe the higher imensional problem into a oneimensional structure an exploit results in one imension, incluing those in the present paper. Acknowlegments This work of the authors was supporte by the SFB 475 (Komplexitätsreuktion in multivariaten Datenstrukturen). The work of H. Dette was supporte in part by a NIH grant awar IRGM72876:A. The authors woul also like to thank I. Gottschlich, who type this paper with consierable technical expertise. We are also grateful to two referees for their constructive comments, which le to a substantial improvement of an earlier version of this paper. References Anerson, T. W. (994). The statistical analysis of time series. Classics Library, New York: Wiley. Currie, A. J., Ganeshananam, S., Noiton, D. A., Garrick, D., Shelbourne, C. J. A., Oraguzie, N. (2). Quantitative evaluation of apple (Malus omestica Borkh.) fruit shape by principle component analysis of Fourier escriptors. Euphytica,, 29E27. Clye, M., Chaloner, K. (996). The equivalence of constraine an weighte esigns in multiple objective esign problems. Journal of the American Statistical Association, 9, Cook, R. D., Wong, W. K. (994). On the equivalence of constraine an compoun optimal esigns. Journal of the American Statistical Association, 89, Dette, H. (995). Discussion of the paper Constraine optimization of experimental esign by V. Feorov an D. Cook. Statistics, 26, Dette, H., Franke, T. (2). Constraine D-anD -optimal esigns for polynomial regression. Annals of Statistics, 28, Dette, H., Haller, G. (998). Optimal esigns for the ientification of the orer of a Fourier regression. Annals of Statistics, 26, Dette, H., Melas, V. B. (23). Optimal esigns for estimating iniviual coefficients in Fourier regression moels. Annals of Statistics, 3, Dette, H., Melas, V. B., Bieermann, S. (22a). A functional-algebraic etermination of D-optimal esigns for trigonometric regression moels on a partial circle. Statistics & Probability Letters, 58(4), Dette, H., Melas, V. B., Pepelyshev, A. (22b). D-optimal esigns for trigonometric regression moels on a partial circle. Annals of the Institute of Statistical Mathematics, 54(4), Dette, H., Stuen, W. J. (997). The theory of canonical moments with applications in statistics, probability an analysis. New York: Wiley. Feorov, V. V. (972). Theory of optimal experiments. New York: Acaemic Press. Hill, P. D. H. (978). A note on the equivalence of D-optimal esign measures for three rival linear moels. Biometrika, 65, Karlin, S., Stuen, W. J. (966). Tchebycheff systems: with applications in analysis an statistics. New York: Interscience. Kitsos, C. P., Titterington, D. M., Torsney, B. (988). An optimal esign problem in rhythmometry. Biometrics, 44, Lau, T. S., Stuen, W. J. (985). Optimal esigns for trigonometric an polynomial regression. Annals of Statistics, 3, Lee, C. M. S. (988a). Constraine optimal esigns. Journal of Statistical Planninng an Inference, 8, Lee, C. M. S. (988b). D-optimal esigns for polynomial regression, when lower egree parameters are more important. Utilitas Mathematica, 34, Lestrel, P. E. (997). Fourier escriptors an their applications in biology. Cambrige: Cambrige University Press. Maria, K. (972). The statistics of irectional ata. New York: Acaemic Press. Pukelsheim, F. (993). Optimal esign of experiments. New York: Wiley. Riccomagno, E., Schwabe, R., Wynn, H.P. (997). Lattice-base D-optimum esigns for Fourier regression moels. Annals of Statistics, 25(6), Rivlin, T. J. (974). Chebyshev polynomials. New York: Wiley.

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