OPTIMAL DESIGNS FOR ESTIMATING PAIRS OF COEFFICIENTS IN FOURIER REGRESSION MODELS

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1 Statistica Sinica , OPTIMAL DESIGNS FOR ESTIMATING PAIRS OF COEFFICIENTS IN FOURIER REGRESSION MODELS Holger Dette 1, Viatcheslav B. Melas 2 and Petr Shpilev 2 1 Ruhr-Universität Bochu and 2 St. Petersburg State University Abstract: In the coon Fourier regression odel we investigate the optial design proble for estiating pairs of the coefficients, where the explanatory variable varies in the interval [ π, π]. L-optial designs are considered and for any iportant cases L-optial designs can be found explicitly, where the coplexity of the solution depends on the degree of the trigonoetric regression odel and the order of the ters for which the pair of the coefficients has to be estiated. Key words and phrases: Equivalence theore, Fourier regression odels, L-optial designs, paraeter subsets. 1. Introduction Fourier regression odels of the for y = β 0 + β 2j 1 sinjt + β 2j cosjt + ε, t [ π, π]. 1.1 are widely used in applications to describe periodic phenoena. Typical subject areas include engineering see e.g., McCool 1979, edicine see e.g., Kitsos, Titterington and Torsney 1988 and biology see the collection of research papers edited by Lestrel Applications of trigonoetric regression odels also appear in two diensional shape analysis see e.g., Young and Ehrlich 1977 and Currie, Ganeshananda, Noition, Garric, Shelbourne and Oragguzie 2000 aong any others. An optial design can iprove the efficiency of the statistical analysis substantially, and the proble of designing experients for Fourier regression odels has been discussed by several authors. Optial designs with respect to Kiefer s φ p -optiality criteria have been studied by Karlin and Studden 1966 and Wu 2002 aong others see also Puelshei, 1993, p.241, while Lau and Studden 1985 discuss the proble of constructing robust designs if the degree in odel 1.1 is not nown. Designs for identifying the degree have been deterined by Biederann, Dette and Hoffann 2007, Dette and Haller 1998, and Zen and Tsai More recent wor discussed the proble of constructing optial designs for the estiation of a particular coefficient

2 1588 HOLGER DETTE, VIATCHESLAV B. MELAS AND PETR SHPILEV in the Fourier regression odel 1.1 see Dette and Melas 2002, 2003, Dette, Melas and Pepelysheff 2002, and Dette, Melas and Shpilev These authors also deonstrated that unifor designs are optial for estiating a subset of the coefficients {β 2i1 1, β 2i1,..., β 2ir 1, β 2ir }, where 1 i 1 <... < i r, r {1,..., }. The ain purpose of the present paper is to obtain further insight into the construction of optial designs for estiating paraeter subsystes in the Fourier regression odel 1.1. In particular we are interested in the L-optial design proble for estiating pairs of the coefficients {β 2i1, β 2i2 } and {β 2j1 1, β 2j2 1}, where i 1, i 2 {0,..., }, j 1, j 2 {1,..., }. The precise estiation of specific pairs of coefficients is of particular interest because in any biological applications, such as two diensional shape analysis, one or two coefficients have a concrete biological interpretation see e.g., Young and Ehrlich 1977, Currie et al In Section 2, we introduce the general notation and state several preliinary results. We forulate and prove a particular version of the equivalence theore for L-optial designs, an iportant tool for the deterination of the optial designs, in Section 3. Here L-optial designs are found explicitly for several cases. Finally, several exaples are presented in Section 4 to illustrate the theoretical results. 2. L-optial Designs Consider the trigonoetric regression odel 1.1, define β = β 0, β 1,... β 2 T as the vector of unnown paraeters, and ft = f 0 t,..., f 2 t T = 1, sin t, cos t,..., sint, cost T, as the vector of regression functions. Following Kiefer 1974 we call any probability easure ξ on the design space [ π, π] with finite support an approxiate design. The support points of the design ξ give the locations where observations are taen, while the weights give the corresponding proportions of the total nuber of observations to be taen at these points. If the design ξ puts asses ξ i at the points t i i = 1,..., and n uncorrelated observations can be taen, then the quantities ξ i n are rounded to integers such that i=1 n i = n and the experienter taes n i observations at each t i i = 1,...,. In this case the covariance atrix of the least squares estiate for the paraeter β in the trigonoetric regression odel 1.1 is approxiately given by σ 2 /nm 1 ξ, where π Mξ = ftf T tdξt R π denotes the inforation atrix of the design ξ see Puelshei Note that for a syetric design ξ, after an appropriate perutation P R

3 OPTIMAL DESIGNS FOR ESTIMATING PAIRS OF COEFFICIENTS 1589 of the order of the regression functions, the inforation atrix 2.1 is bloc diagonal, that is Mc ξ 0 Mξ = P MξP = M s ξ with blocs given by π M c ξ = For a given atrix π cosit cosjtdξt π M s ξ = sinit sinjtdξt π L =, i,j= i, l i li T, i=0 with vectors l i R 2+1, the class Ξ L is defined as the set of all approxiate designs for which the linear cobinations of the paraeters li T β, i = 0,..., are estiable, that is l i RangeMξ; i = 0,...,. Siilarly, the sets Ξ s and Ξ c are defined as the sets of all designs for which the atrices M s ξ and M c ξ are nonsingular, respectively. Finally a design ξ is called L-optial if ξ = arg in ξ ΞL trlm + ξ, where L is a fixed and nonnegative definite atrix, and for a given atrix A the atrix A + is the pseudo-inverse see Rao The following result gives a characterization of L-optial designs that is particularly useful for designs with a singular inforation atrix. For a nonsingular inforation atrix this theore was forulated and proved in Eraov and Zhigljavsy The result is stated for a general regression odel y = β T ft + ε with regression functions and a general design space χ. Theore 2.1. Let L R denote a given and nonnegative definite atrix of the for 2.4, and assue that the set of inforation atrices is copact. 1 The design ξ is an eleent of the class Ξ L if and only if l T i M + ξmξ = l T i, i = 0,...,. 2 The design ξ is L-optial if ax t χ ϕt, ξ = trlm + ξ, 2.5 where ϕt, ξ = f T tm + ξlm + ξft. Moreover, the equality holds for any t i suppξ. ϕt i, ξ = trlm + ξ 2.6

4 1590 HOLGER DETTE, VIATCHESLAV B. MELAS AND PETR SHPILEV Proof. The first part of this theore was proved in Rao For a proof of the second part we use the following lea. Lea 2.1. If A is a nonnegative definite atrix, then A + A + 2A + A with equality if and only only if A = A + A. Proof of Lea 2.1. Let A = B T B, where B R n n. By definition of the pseudo-inverse we have A + = B + B +T, and so A + A + 2A + A = B T B + B + B +T 2B + B +T B T B = B T B + B B +T = B B +T T B B +T 0, where the second equality is obtained fro the identity B + B +T B T B = B + BB + T B = B + BB + B = B + B. Now we return to the proof of the second part of Theore 2.1. We begin by showing that a L-optial design satisfies 2.5 and 2.6. For any α 0, 1, define ξ α = 1 αξ + αξ t, where ξ t denotes the Dirac easure at the point t, and assue that the design ξ is L-optial. In this case the inequality trlm + ξ α trlm + ξ is satisfied for all α 0, 1, which iplies trlm + ξ α / α α=0+ 0. On the other hand we obtain, observing the identity the inequality LM + ξ α Mξ α α trlm + ξ α α = 0 = L M + ξ α Mξ α + LM + ξ α Mξ α α α, { = tr L M + } ξ α α=0+ α { = tr α=0+ LM + ξ α Mξ α α M + ξ α } α=0+ = trlm + ξ trlm + ξ Mξ t M + ξ = trlm + ξ ϕt, ξ 0. Therefore we have ϕt, ξ trlm + ξ for all t. Moreover, the equality ϕt, ξξdt = trlm + ξftf T tm + ξξdt = trlm + ξ is obviously fulfilled for any design ξ Ξ L. It now follows that 2.5 and 2.6 are satisfied. In general an analytical deterination of L-optial designs is very difficult. Theore 2.1 can be used to chec the optiality of a given design nuerically. We use this characterization in the following section.

5 OPTIMAL DESIGNS FOR ESTIMATING PAIRS OF COEFFICIENTS Analytical Solutions of the L-optial Design Proble In the present section we develop explicit solutions of the L-optial design proble in the Fourier regression odel 1.1 for several paraeter subsystes. We begin with the proble of constructing optial designs for estiating two coefficients corresponding to the sinus- or cosinus functions. More precisely, define e j R 2+1 as the jth unit vector and consider the atrices L 2 /2 1, 4 /2 1 = e 2 /2 1 e T 2 /2 1 + e 4 /2 1e T 4 /2 1 L 2 /2, 4 /2 = e 2 /2 e T 2 /2 + e 4 /2 e T 4 /2. We call an L-optial design with the atrix L 2 /2 1, 4 /2 1 or L 2 /2, 4 /2 an L-optial design for estiating the pair of coefficients β 2 2 1, β 4 /2 1 or β 2 /2, β 4 /2, respectively. Theore 3.1. Consider the trigonoetric regression odel 1.1 with > 3. 1 The design ξ2 /2 1, 4 /2 1 = tn t n 1... t 1 t 1... t n n 2n... 2n 2n n with n = 2 /2, t i = 2 i/2 π/n + 1 i 1 x, x = 2 arctan 4 5/n is L- optial for estiating the pair of coefficients β 2 /2 1, β 4 /2 1. Moreover, trlm + ξ 2 /2 1, 4 /2 1 = 5/2 + 3/2. 2 For any α [0, ω n ] the design ξ2 /2, π 4 /2 = tn 1... t 1 0 t 1... t n 1 π ω n α ω n 1... ω 1 ω 0 ω 1... ω n 1 α with n = 2 /2, t i = i 1π/n, i = 2,..., n, ω 0 = 5ω 1, ω 1 = 5 1/4n, ω i = ω i 2, i = 2,..., n, is L-optial for estiating the pair of coefficients β 2 /2, β 4 /2. Moreover, trlm + ξ 2 /2, 4 /2 = trlm + ξ 0,2 /2 = 5/2 + 3/2. 3 The L-optial design ξ0, 2 /2 for estiating the pair of coefficients β 0, β 2 /2 coincides with the design ξ2 /2,4 /2 defined in part 2. Proof. We only prove the first part of the theore for the case where the degree of the Fourier regression odel is even. All other stateents of the theore are treated siilarly. In this case /2 = /2 and the design in part 1 of Theore 3.1 can be rewritten as ξ 1, 2 1 = t t 1... t 1 t 1... t , 2,

6 1592 HOLGER DETTE, VIATCHESLAV B. MELAS AND PETR SHPILEV with t i = i/2 2π/ + 1 i 1 x, x = 2/ arctan 4 5. The proof is based on the characterization of L-optial designs given in Theore 2.1. Recall the definition of the atrix M s = M s ξ R at 2.3, and let s[i,j] = s[i,j] ξ denote the eleent of the atrix M s in the position i, j. We consider the syste of equations s[j, /2] = 0, j = 1,...,, j 2, 3.1 s[j, ] = 0, j = 1,..., 1, 3.2 s[/2, /2] = sin 2 2 x, s[,] = sin 2 x. 3.3 We show below that the design ξ 1, 2 1 satisfies Under this assuption we obtain for the function ϕt, ξ in Theore 2.1 the representation ϕt, ξ = f T tm + ξ LM + ξ ft = sin2 /2t sin 4 /2x + sin2 t sin 4 x. Now a straightforward calculation shows trlm + ξ = 5/2 + 3/2, and it is easy to prove that 2.5 is satisfied calculating the solution of the equation[ ϕt, ξ / t] = 0. Moreover, a siilar calculation shows that trlm + ξ = ϕt i, ξ = [sin 2 /2x] + [1/sin 2 x] = 5/2 + 3/2, and it reains to show For a proof, we note that s[j, /2] = 2 sinjt sin 2 t ω = 1 sinjt sin 2 t =1 and that sin/2t = 1 1/2 sin/2x, which yields s[j, /2] = [sin/2x/] =1 1 1/2 sinjt. Next we prove that =1 1 1/2 sinjt = 0 if j /2. For this purpose we use the definition of t and obtain 1 1/2 sinjt =1 =1 2πj 2πj = sinjx + sin cosjx cos sinjx 2πj 2πj + 1 sin cosjx + cos sinjx + 2πj + 1 /2 1 sin 2 1 cosjx + cos + 1 j+1+/2 1 sinjx [ /2 1 = sinjx 2 1 cos 2πj ] + 1j+/2 1. =0 2πj 2 1 sinjx

7 OPTIMAL DESIGNS FOR ESTIMATING PAIRS OF COEFFICIENTS 1593 Now, applying standard trigonoetric forulae, it is easy to calculate the su in the last expression if j /2, /2 1 2 = =0 1 cos 1 cosjπ/ 2πj jπ cos + cos 3jπ cos 3jπ + cos 3jπ 1jπ /2 1 cos + cos = j+/2 1. 5jπ It follows that =1 1 1/2 sinjt = sinjx /2 1+j + 1 j+/2 1 = 0, which shows that 3.1 is satisfied. The equality s[j,] = 0 j follows by siilar arguents, that is s[j,] = sinjx 2 /2 1 =0 cos2πj/ + 1 j 1 = 0. Finally we obtain 3.3 by a direct calculation. This copletes the proof of Theore 3.1. Rear 3.1. Note that Theore 3.1 holds for trigonoetric regression odels of degree > 3. If = 2, the L-optial design ξ1,3 has equal asses at the points π+x, x, x, π x, where x = arctan 4 5 and trlms 1 ξ = 3+ 5/2. For any α [0, 5 5/8], the L-optial design ξ0,2 has asses [5 5/8] α, 5 1/8, 5 5/8, 5 1/8, and α at the points π, π/2, 0, π/2, and π, respectively, with trlmc 1 ξ = 3 + 5/2. If = 3, the L-optial design ξ1,3 has equal asses at the points π + x, x, x, π x, where x = arctan 4 5 and trlm s + ξ = 3 + 5/2, while the L-optial design ξ0,2 has asses 1/2 2z α, z, z, 1/2 2z, z,, z, and α at the points π, π + x, x, 0, x, π x, and π, respectively, where z , x , and trlmc 1 ξ The optiality of the designs can be easily checed with Theore 2.1. Rear 3.2. Note that, by Theore 3.1, the su of variances of the estiates for the corresponding coefficients in a Fourier regression odel of degree > 3 is 5/2 + 3/2 for the L-optial design, while the D-optial design yields 4 for this su. Thus the reduction in the su of variances obtained by the L-optial design is approxiately 35%. There are two other cases where L-optial designs for the trigonoetric regression odel 1.1 can be constructed explicitly, see the following two theores. For the sae of brevity only a proof of Theore 3.3 is given here.

8 1594 HOLGER DETTE, VIATCHESLAV B. MELAS AND PETR SHPILEV Table 1. The solutions of x and z of the the syste 3.4. The L-optial design for estiating the specific pair is specified in the first part of Theore 3.2. = 3 {2 1, 4 1} {2 1, 6 1} {4 1, 6 1} x arctan π/10 z 1/ /4 Table 2. The solutions of x and z of the the syste 3.5. The L-optial design for estiating the specific pair is specified in the first part of Theore 3.2. = 3 {0, 2} {0, 4} {0, 6} {2, 4} {2, 6} {4, 6} x π/2 π/ z /4 1/ Theore 3.2. Consider the trigonoetric regression odel 1.1 with = 3. The design ξ t t = 1 t 1 t 1 t ω ω 1 ω 1 ω 1 ω with t 1 = π/2 x/, t 2 = π/2, t 3 = π/2 + x/, t i = t i 3 + π/, i = 4, 5...,, ω 1 = z/, ω 2 = 1 4z/2, ω 3 = z/, ω i = ω i 3, i = 4, 5...,, is L-optial for estiating one of the pairs of the coefficients {β 2 1, β 4 1 }, {β 2 1, β 6 1 }, {β 4 1, β 6 1 }, where only the values x and z depend on the particular pair under consideration, and satisfy trlms 1 ξ = 0 x trlms 1 ξ = z Siilarly, for any α [0, ω ] the design ξ π t 1 t = 1 0 t 1 t 1 π ω α ω 1 ω 1 ω 0 ω 1 ω 1 α with t 0 = 0, t 1 = x/, t 2 = π x/, t i = t i 3 + π/, i = 3, 4..., 1, ω 0 = 1 4z/2, ω 1 = z/, ω 2 = z/, ω i = ω i 3, i = 3, 4...,, is L-optial for estiating one of the pairs {β 0, β 2 }, {β 0, β 4 }, {β 0, β 6 }, {β 2, β 4 }, {β 2, β 6 }, {β 4, β 6 }, where only the values x and z depend on the particular pair under consideration and satisfy trlmc 1 ξ = 0 x trlmc 1 ξ = z Soe nuerical values for the paraeters x and z in Theore 3.2 are presented in Tables 1 and 2.

9 OPTIMAL DESIGNS FOR ESTIMATING PAIRS OF COEFFICIENTS 1595 Theore 3.3. Consider the trigonoetric regression odel 1.1 with = 4. The design ξ t t 1 t 1 t 1 t = ω ω 1 ω 1 ω 1 ω with t 1 = x 1 /, t 2 = x 2 /, t 3 = π x 2 /, t 4 = π x 1 /, t i = t i 4 + π/, i = 5, 6...,, ω 1 = z 1 /, ω 2 = 1 4z 1 /, ω 3 = 1 4z 1 /, ω 4 = z 1 /, ω i = ω i 4, i = 5, 6...,, is L-optial for estiating one of the pairs of coefficients {β 2 1, β 4 1 }, {β 2 1, β 6 1 }, {β 2 1, β 8 1 }, {β 4 1, β 6 1 }, {β 4 1, β 8 1 }, {β 6 1, β 8 1 }, where only the values x 1, x 2 and z 1 depend on the particular pair under consideration and satisfy trlm 1 s ξ x 1 = 0, trlm 1 s ξ x 2 = 0, trlm 1 s ξ z 1 = Siilarly, if n = 5/4, then for any α [0, ω n ] the design ξ π t n 1 t 1 0 t 1 t n 1 π = ω n α ω n 1 ω 1 ω 0 ω 1 ω n 1 α with t 0 = 0, t 1 = x 1 /, t 2 = x 2 /, t 3 = π x 2 /, t 4 = π x 1 /, t i = t i 5 + π/, i = 5, 6..., n 1, ω 0 = 1 4z 1 4z 2 /2, ω 1 = ω 4 = z 1 /, ω 2 = ω 3 = z 2 /, ω i = ω i 5, i = 5, 6,..., n, is optial for estiating any of the pairs of the coefficients {β 0, β 4 }, {β 0, β 6 }, {β 0, β 8 }, {β 2, β 4 }, {β 2, β 6 }, {β 2, β 8 }, {β 4, β 6 }, {β 4, β 8 }, {β 6, β 8 }, where only the values x 1, x 2, z 1, and z 2 depend on the particular pair under consideration and satisfy trlmc 1 ξ = 0, x 1 trlmc 1 ξ = 0, z 1 trlmc 1 ξ = 0 x 2 trlmc 1 ξ = 0. z The nuerical values of the quantities x 1, x 2, z 1, and z 2 are listed in the Tables 3 and 4. Proof of Theore 3.3. We only prove the first part of Theore 3.3, the second part is treated siilarly. We begin with the case = 1, i.e., = 4, for which it is easy to chec by direct calculations that the design ξ defined in the first part of Theore 3.3 is L-optial. The corresponding nuerical values of the quantities x 1, x 2, and z 1 can be found as the solution of 3.6. Now let 2 and = 4.

10 1596 HOLGER DETTE, VIATCHESLAV B. MELAS AND PETR SHPILEV Table 3. The solutions x 1, x 2 and z 1 of the the syste 3.6. The L-optial design for estiating the specific pair of coefficients in the Fourier regression odel 1.1 is specified in the first part of Theore 3.3. = 4 {2 1, 4 1} {2 1, 6 1} {2 1, 8 1} x 1 π/ x 2 π/2 π/ z 1 6 6/ = 4 {4 1, 6 1} {4 1, 8 1} {6 1, 8 1} x arctan 4 5/ x π arctan 4 5/ z / Table 4. The solutions of x 1, x 2, z 1 and z 2 of the the syste 3.7. The L-optial design for estiating the specific pair of coefficients in the Fourier regression odel 1.1 is specified in the second part of Theore 3.3. = 4 {0,4} {0,6} {0,8} {2,4} {2,6} x 1 π/4 π/3 π/4 π/ x 2 π/2 π/3 π/2 π/ z /6 1/ / z /6 1/8 4 2/ = 4 {2,8} {4,6} {4,8} {6,8} x π/ x 2 π/2 π/2 π/2 π/2 z We consider the syste of equations s[,j] = 0, j = 1,..., 4, j, j 3, s[4,j] = 0, j = 1,..., 4 1, j 2, 1 s[,] = 4 z 1 sin 2 x z 1 sin 2 x 2, 1 s[,3] = 4 z 1 sinx 1 sin3x z 1 sinx 2 sin3x 2, 1 s[4,2] = 4 z 1 sin2x 1 sin4x z 1 sin2x 2 sin4x 2, 1 s[4,4] = 4 z 1 sin 2 4x z 1 sin 2 4x 2, 3.8 where s[i,j] = s[i,j] ξ is the eleent of the atrix M s ξ R in the ith row and jth colun. We prove below that these equalities are satisfied for the

11 OPTIMAL DESIGNS FOR ESTIMATING PAIRS OF COEFFICIENTS 1597 design ξ. In this case it follows that the quantities trlm 1 s ξ and ϕt, ξ = f T tm + ξ LM + ξ ft in Theore 2.1 are independent of the value note that the atrix L is a given diagonal atrix where the non-vanishing entries depend on the particular pair of paraeters under consideration. Consequently it is sufficient to prove Theore 3.3 in the case = 1, which has been done in the previous paragraph. In order to prove that the equalities 3.8 are satisfied we note that for i = 1,..., 4 1, i, 2, 3, we have s[,i] = 2 4 = 2 sinx 1 z 1 sint j sinit j ω j 1 j 1 sinit 4j 3 + sinit 4j sinx 2 4 z 1 1 j 1 sinit 4j 2 + sinit 4j 1. For the first su on the right side we obtain 1j 1 sinit 4j 3 +sinit 4j = 0. Siilarly, it follows substituting x 1 for x 2 that the second su also vanishes, which iplies s[,i] = 0 for i = 1,..., 4 1, i, 2, 3. For the eleent s[4,i] we find for i = 1,..., 4 1, i, 2, 3, s[4,i] = 2 sinx 1 z sinx 2 4 z 1 A straightforward calculation now yields = sinit 4j 3 sinit 4j cos ix1 iπ sinit 4j 3 sinit 4j sinit 4j 2 sinit 4j 1. iπ cos ix 1 iπ + iπ + cos ix 1 iπ cos ix 1 + iπ 2 sin = 0, iπ and the sae arguents show that the second su also vanishes, which iplies s[4,i] = 0. To conclude the proof it reains to calculate s[i,j], i, j = 1,..., 4,

12 1598 HOLGER DETTE, VIATCHESLAV B. MELAS AND PETR SHPILEV for which we obtain s[i,j] = 2z 1 r=1 sinix 1 + r 1π sinjx 1 + r 1π + sini x 1 + rπ sinj x 1 + rπ z sinix2 sinjx i+jr i+jr r=1 2 sinix1 sinjx 1 z = + 4 z sinix2 sinjx i+jr i+jr. r=1 Fro this representation it is obvious that s[i,j], i, j = 1,..., 4, have the values specified by the syste of equations, and the theore has been proved. Rear 3.3. Note that for any with /2 <, and for any β [0, 1/2], the design ξ0,2 with equal asses at the points π, π + π/,..., pi + 2 1/π, π is L-optial for estiating the pair of coefficients {β 0, β 2 }. Moreover, in this case it follows that trlmc 1 ξ0,2 = 2 see Dette and Melas 2003, Lea Exaples In this section we present several exaples that illustrate the theoretical results obtained in Section 3. Exaple 4.1. We present the L-optial design for estiating the coefficients β 3, β 7 i.e., the coefficients of sin2t and sin4t in the trigonoetric regression odel of degree 4. The L-optial design ξ3,7 is directly obtained fro Theore 3.1 and has equal asses at the points π + x, π/2 x, π/2 + x, x, x, π/2 x, π/2 + x, π x, where x = 1/2 arctan The corresponding support points of the design ξ3,7 are depicted in the left part of Figure 1. A straightforward calculation shows that the function ϕt, ξ3,7 is given explicitly by ϕt, ξ 3,7 = f T tm + ξ 3,7 LM + ξ 3,7 ft = sin 2 2t sin 2 4t.

13 OPTIMAL DESIGNS FOR ESTIMATING PAIRS OF COEFFICIENTS π π π π Figure 1. Left part: Support points of the L-optial design for estiating the coefficients β 3, β 7 in the Fourier regression odel of degree 4. Right part: support points of the L-optial design for estiating β 4, β 6. Figure 2. Left part: the function ϕt, ξ3,7 defined in Exaple 4.1. Right part: The function ϕt, ξ4,6 defined in Exaple 4.2. Then 2.6 is checed by a direct calculation, and the function ϕt, ξ 3,7 is depicted in the right part of Figure 2. Exaple 4.2. Consider the trigonoetric regression odel of degree = 4, and use Theore 3.3 to deterine the L-optial design for estiating the pair of coefficients β 4 and β 6 that correspond to the ters cos2t and cos3t. The L-optial design ξ4,6 for estiating these coefficients has asses α, 0.09, 0.145, 0.09, 0.175, 0.09, 0.145, 0.09, α at the points π, 2.13, π/2, 1.02, 0, 1.02, π/2, 2.13, π where α [0, 0.175]. The support points of the optial design are depicted in the right part of Figure 1. We finally note that a straightforward calculation yields, for the function ϕt, ξ4,6 in Theore 2.1, ϕt, ξ4,6 = f T tm + ξ4,6 LM + ξ4,6 ft = cos2t cos4t cos6t cos8t.

14 1600 HOLGER DETTE, VIATCHESLAV B. MELAS AND PETR SHPILEV Acnowledgeents The support of the Deutsche Forschungsgeeinschaft SFB 475, Koplexitätsredution in ultivariaten Datenstruturen is gratefully acnowledged. The wor of the authors was also supported in part by an NIH grant award IR01GM072876:01A1 and by the BMBF project SKAVOE. References Biederann, S., Dette, H. and Hoffann, P Constrained optial discriinating designs for Fourier regression odels. Ann. Statist. Math., in press. Currie, A. J., Ganeshananda, S., Noition, D. A., Garric, D., Shelbourne, C. J. A. and Oragguzie, N Quantiative evalution of apple Malus x doestica Borh. fruit shape by principle coponent analysis of Fourier descriptors. Euphytica 111, Dette, H. and Haller, G Optial designs for the identification of the order of a Fourier regression. Ann. Statist. 26, Dette, H. and Melas, V. B E-optial designs for Fourier regression odels. Math. Methods Statist. 11, Dette, H. and Melas, V. B Optial designs for estiating individual coefficients in Fourier regression odels. Ann. Statist. 31, Dette, H., Melas, V. B. and Pepelysheff, A D-Optial designs for trigonoetric regression odels on a partial circle. Ann. Inst. Statist. Math. 54, Dette, H., Melas, V. B. and Shpilev, P. V Optial designs for estiating the coefficients of the lower frequencies in trigonoetric regression odels. Ann. Inst. Statist. Math. 59, Eraov, M. and Zhigljavsy, A Matheatical Theory of Optial Experient. Naua, Moscow. in Russian. Karlin, S. and Studden, W. J Tchebycheff Systes: With Applications in Analysis and Statistics. Interscience, New Yor. Kiefer, J. C General equivalence theory for optiu designs approxiate theory. Ann. Statist. 2, Kitsos, C. P., Titterington, D. M. and Torsney, B An optial design proble in rhythoetry. Bioetrics 44, Lestrel, P. E Fourier Descriptors and Their Applications in Biology. Cabridge University Press, Cabridge. Lau, T. S. and Studden, W. J Optial designs for trigonoetric and polynoial regression. Ann. Statist. 13, McCool, J. I Systeatic and rando errors in least squares estiation for circular contours. Precision Engineering 1, Puelshei, F Optial Design of Experients. Wiley, New Yor. Rao, C. R Linear Statistical Inference and Its Applications. Wiley, New Yor. Wu, H Optial designs for first-order trigonoetric regression on a partial circle. Statist. Sinica 12, Young, J. C. and Ehrlich, R Fourier bioetrics: haronic aplitudes as ultivariate shape descriptors. Systeatic Zoology 26,

15 OPTIMAL DESIGNS FOR ESTIMATING PAIRS OF COEFFICIENTS 1601 Zen, M.-M. and Tsai, M.-H Criterion-robust optial designs for odel discriination and paraeter estiation in Fourier regression odels. J. Statist. Plann. Inference 124, Ruhr-Universität Bochu, Faultät für Matheati, Bochu, Gerany. E-ail: St. Petersburg State University, Departent of Matheatics, St. Petersburg, Russia. E-ail: St. Petersburg State University, Departent of Matheatics, St. Petersburg, Russia. E-ail: Received January 2008; accepted May 2008