5 Haar, R. Haar,. Antonads 994, Dogaru & Carn Kerkyacharan & Pcard 996. : Haar. Haar, y r x f rt xβ r + ε r x β r + mr k β r k ψ kx + ε r x, r,.. x [,

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1 4 Chnese Journal of Appled Probablty and Statstcs Vol.6 No. Apr. Haar,, 6,, 34 E-,,, 34 Haar.., D-, A- Q-,. :, Haar,. : O.6..,..,.. Herzberg & Traves 994, Oyet & Wens, Oyet Tan & Herzberg 6, 7. Haar Haar., L.,. Antonads & Oppenhem 995 Hardle 998. Haar 9, Haarφx Haar ψx L R,,, x < /;, x < ; φx ψx, / x < ;,,, E E S , 9.

2 5 Haar, R. Haar,. Antonads 994, Dogaru & Carn Kerkyacharan & Pcard 996. : Haar. Haar, y r x f rt xβ r + ε r x β r + mr k β r k ψ kx + ε r x, r,.. x [, ], f rt x, ψ x,, ψ mr, mr x, r,, Haar /, k/ x < k/ + / + ; ψ k x / ψ x k /, k/ + / + x < k/ + / ;,.,, m r ; k,, mr+, β rt β r, βr,, βr m r, mr, r,, β r, βr k mr+., m m. : E[ε r x], E[ε r x ε t x ] x x, E[ε r x] σ r, E[ε xε x] ρσ σ, } r, t,..3.,. D-, A- Q-... n n m + ξ n, supp{ξ n } {x,, x n }ξ n.. y r r β r + ε r, r,. r f r x,, f r x n T, y r y r x,, y r x n T, ε r ε r x,, ε r x n T, r,, r.,. Y β + ε,.

3 : Haar 53,, Y y T, y T T, β β T, β T T, ε ε T, ε T T..3 Cov [ε] Σ I n, Σ σ ρσ σ ρσ σ σ xyx y x, y x T., Gauss-Markov, β BLU β β T Σ I n T Σ I n Y, Cov [ β] T Σ I n. ξ n,, [, ], [ r mr+, mr+ mr+ r, ;,, mr+, r, r,, m m k m m +k,,, m +..ψ k x, x r, r, ;,, mr+, µ r f r x ψ k x ψ k µ r,,, m r ; k,,, / mr+, r, ;,, mr+ r. f r x f r µ r, x r, r, ;,, mr+.. r ξ n Ω r n, supp{ξ n} r {x r,, xr n r }, r, ;,, mr+. Ω n, m m k Ω n, m m +k, n m m k m m +k,,, m +.

4 54 D r mr+ r n, r H f r µ r, f r µ r,, f r µ r T, r,. mr+ D D D, H H. H, r n nr.., r D r r H, r,, D H. ψ k x, ξ n r T r H H T r H r H m r+ I mr+..3 Mξ n n Cov [ β] n T Σ I n n T HD T Σ I n D H..4 σ Σ σ, N σ σ m m +... m m,,, m +. σ D T Σ I n D T D σ D T D σ A σ A D, σ D T D σ D T D σ A T σ A.3 M ξ n n T HD T Σ I n D H n T m + H m + H T D T Σ I n D m + H m + H,.5 D T Σ I n D m + σ n m + ρσ σ n m m m + m + ρ σ N ρσ σ n T m m + ρ σ m m T. m m n

5 : Haar 55. Ξ. ξ Ξ ξ. D, A Q. : ξd Φ Dξ n M ξ n, Φ D ξ D M ξ D Mn ξ n Ξ M ξ n, ξ D D-. Mξ D Max ξ n Ξ Mξ n,.6 ξ A Φ Aξ n tr M ξ n, Φ A ξ A tr M ξ A Mn ξ n Ξ tr M ξ n,.7 ξa A-. ξq Φ Qξ n Φ Q ξq tr Dx, ξ Qdx Mn ξ n Ξ ξ Q Q-. Dx, ξ n F T xm ξ n Fx, Fx tr Dx, ξ n dx, tr Dx, ξ n dx,.8 f x f x. x,, m+, x ξ x m +.9 m + m +. D-, A- Q-. : a A m+ n, H T H m + m + m + m +, σ A σ A T σ A σ A m+ ρ σ N I m m ρ n m m T m m N m+ ρ ρ σ N n T m m N m m m+ m m k ρ m + ρ σ m + ρ σ m + m m +k, ρ.

6 56, Mξ n n T HD T Σ I n D H n DT Σ I n D H T H m + + m + σ A σ A σ A T n σ A σ A H T H m n + + m + m + m + m + m + m + σ ρ σ n m + m + n m + n, mr+.. n n r n m m k m m +k n, r, ;,, mr+, n r n r n r, r, Mξ mr+ n, Φ D ξ n.., Ω n, ξ n, Fx Fx x.,,, m +, Ω n, x D. b, x, x. D,.9 tr M ξ n tr n T HD T Σ I n D H tr{nd T Σ I n D H T H } { m+ n m + tr σ + m+ m + tr nσ m + m + n σ n + ρ σ m + m + n ρ σ N + n m + n m + + ρ σ m + + ρ σ } m m T m m n ρ σ m m m + k m m +k + ρ σ n m m.., n r n r n r, r, Φ mr+ A ξ n. A,.9A. c x,,, m+ x,, [/ m m ] +,., f r x r H, x, H f x e,

7 : Haar 57 H f x e. e,. x, xŷ e Dx, ξ n F T xm T ξ n Fx n [D T Σ I n D] e. e T e,,, m+ ; [/ m m ] + x, { m + tr Dx, ξ n n tr e T σ e + tr e T m + σ n + ρ σ n tr Dx, ξ n dx m+ m+ m+ n ρ σ N + ρ σ tr Dx, ξ n dx σ n + ρ σ n σ n + ρ σ n σ n + ρ σ m + m + σ n + ρ σ m + m + + ρ σ } m m T m m e n.. + ρ σ dx + ρ σ m m n n m + + ρ σ m + + ρ σ m + m + m +,.3. Φ Q ξ n Φ A ξ n, Q- A-..9 Q-.. Haar.,,.9D-, A- Q-.,.,. [] Antonads, A. and Oppenhem, G., Wavelets and statstcs, Papers from the Conference Held n Vllard de Lans, Noverber 6-8, Sprnger, Berln, New York, 995.

8 58 [] Antonads, A., Gregore, G. and McKeague, I.W., Wavelet method for curve estmaton, J. Am. Stat. Assoc., 89994, [3] Dogaru, T. and Carn, L., Applcaton of Haar-wavelet-based multresoluton tme-doman schemes to electromagnetc scatterng problems, IEEE Trans. Antennas Propag., 5, [4] Haar, A., Zur theore der orthogonalen Funktonensysteme, Math. Ann., 699, [5] Hardle, W., Kerkyacharan, G., Pcard, D. and Tsybaov, A., Wavelets, Approxmaton and Statstcal Applcatons, Sprnger, Berln, New York, 998. [6] Herzberg, A.M. and Traves, W.N., An optmal expermental desgn for the Haar regresson model, Can. J. Stat., 994, [7] Kerkyacharan, G. and Pcard, D., Estmatng nonquadratc functonals of a densty usng Haar wavelets, Ann. Statst., 4996, [8] Oyet, A.J., Mnmax A- and D-optmal nteger-valued wavelet desgns for estmaton, Can. J. Stat., 3, [9] Oyet, A.J. and Wens, D.P., Robust desgns for wavelet approxmatons of regresson models, J. Nonparametrc Stat.,, [] Tan, Y.G. and Herzberg, A.M., Estmaton and optmal desgns for lnear Haar wavelet models, Metrka, 657, [] Tan, Y.G. and Herzberg, A.M., A-mnmax and D-mnmax robust optmal desgns for approxmately lnear Haar wavelet models, Computatonal Statstcs & Data Analyss, 56, Optmal Desgns for Dual Response Lnear Haar-Wavelet Regresson Model Lu n College of Scence, Donghua Unversty, Shangha, 6 Yue Rongxan College of Mathematcs, Shangha Normal Unversty, Shangha, 34 Scentfc Computng Key Laboratory of Shangha Unverstes and E-Insttute of Shangha Unverstes, Shangha, 34 Ths paper dscusses optmal desgn problems for the lnear regresson model wth two responses, n whch each response s expressed by a lnear combnaton of Haar-wavelets. It s shown that the obtaned desgn s D-, A- and Q-optmal smultaneously. Moreover, ths desgn s ndependent of the covarance matrx of the responses. Keywords: Dual response, Haar-wavelet, optmal desgns. AMS Subect Classfcaton: 6K5.

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