Asymptotically minimax estimation of order-constrained parameters and eigenfunctions of the laplacian on the ball
Annales de l'I.H.P. Probabilités et statistiques, Tome 38 (2002) no. 2, pp. 193-206.
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     author = {Kor\'anyi, Adam and MacGibbon, K. Brenda},
     title = {Asymptotically minimax estimation of order-constrained parameters and eigenfunctions of the laplacian on the ball},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     pages = {193--206},
     publisher = {Elsevier},
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     number = {2},
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     zbl = {1010.62009},
     language = {en},
     url = {http://www.numdam.org/item/AIHPB_2002__38_2_193_0/}
}
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Korányi, Adam; MacGibbon, K. Brenda. Asymptotically minimax estimation of order-constrained parameters and eigenfunctions of the laplacian on the ball. Annales de l'I.H.P. Probabilités et statistiques, Tome 38 (2002) no. 2, pp. 193-206. http://www.numdam.org/item/AIHPB_2002__38_2_193_0/

[1] P. Berard, Spectres et groupes cristallographiques I: Domaines euclidiens, Inventiones Mathematicae 58 (1980) 179-199. | MR | Zbl

[2] P. Berard, G. Besson, Spectres et groupes cristallographiques II: Domaines spheriques, Annales de l'Institut Fourier 30 (1980) 237-248. | Numdam | MR | Zbl

[3] J.O. Berger, Statistical Decision Theory and Bayesian Analysis, Springer-Verlag, New York, 1985. | MR | Zbl

[4] P.E. Berkin, B.Ya. Levit, Second-order asymptotically minimax estimates for the mean of a normal population, Problemy Peredachi Informatsii 16 (3) (1980) 60-79, Translation: Problems of Information Transmission (1981) 212-229. | Zbl

[5] P.J. Bickel, Minimax estimation of the mean of a normal distribution when the parameter space is restricted, Ann. Statist. 9 (1981) 1301-1309. | MR | Zbl

[6] N. Bourbaki, Éléments de mathématique, Fascicule XXXIV, Groupes et algèbres de Lie, Chapitres IV, V et VI, Hermann, Paris, 1969.

[7] L.D. Brown, Statistical Decision Theory, Mimeographed Notes, Cornell University, Ithaca, NY, 1979.

[8] L.D. Brown, L. Gajek, Information inequalities for the Bayes risk, Ann. Statist. 18 (1990) 1578-1594. | MR | Zbl

[9] G. Casella, W. Strawderman, Estimating a bounded normal mean, Ann. Statist. 9 (1981) 868-876. | MR | Zbl

[10] D.L. Donoho, R.C. Liu, B. Macgibbon, Minimax risk over hyperrectangles and implications, Ann. Statist. 18 (1990) 1416-1437. | MR | Zbl

[11] A. Erdélyi, The Bateman Manuscript Project, Higher Transcendental Functions (3 volumes), McGraw-Hill, New York, 1953-1955, (Erdélyi A. (Ed.)).

[12] T.S. Ferguson, Mathematical Statistics, A Decision Theoretic Approach, Academic, New York, 1967. | MR | Zbl

[13] G.B. Folland, Introduction to Partial Differential Equations, Princeton University Press, 1995. | MR | Zbl

[14] D. Gilbarg, N.S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin, 1983. | MR | Zbl

[15] G. Harder, A Gauss-Bonnet formula for discrete arithmetically defined groups, Ann. Sci. Ec. Norm. Super. 4 (1971) 409-445. | Numdam | Zbl

[16] I.M. Johnstone, K.B. Macgibbon, Minimax estimation of a constrained Poisson vector, Ann. Statist. 20 (1992) 807-831. | MR | Zbl

[17] I.M. Johnstone, K.B. Macgibbon, Asymptotically minimax estimation of a constrained Poisson vector via polydisc transforms, Annales de l'Institut Henri Poincaré, Probabilité et Statistiques 29 (1993) 289-319. | Numdam | MR | Zbl

[18] P.J. Kempthorne, Numerical specification of discrete least favorable prior distributions, SIAM J. Sci. Statist. Comput. 8 (1987) 171-184. | MR | Zbl

[19] B.Ya. Levit, On asymptotic minimax estimates of the second order, Theor. Probab. Appl. 25 (1980) 552-568, (Translation). | Zbl

[20] B.Ya. Levit, Minimax estimation and positive solutions of elliptic equations, Theor. Probab. Appl. 27 (1982) 563-586, (Translation). | MR | Zbl

[21] B.Ya. Levit, Second order asymptotic optimality and positive solutions of Schrödinger's equation, Theor. Probab. Appl. 30 (1985) 333-363, (Translation). | MR | Zbl

[22] B.Ya. Levit, Evaluation of the minimax risk, in: Proc. Fourth International Vilnius Conf. Prob. Theory Math. Stat., Vilnius, Vol. 4, 1985, pp. 181-183.

[23] B.Ya. Levit, On second order admissibility in simultaneous estimation, in: Prohorov Yu.A., Sazonov V.V. (Eds.), Proc. 1st. World Congress Bernoulli Soc. Tashkent, USSR, VNU Sc. Press, Ultrecht, 1986. | MR | Zbl

[24] P.G. Macdonald, The volume of a compact Lie Group, Inventiones Mathematicae 56 (1980) 93-95. | MR | Zbl

[25] T. Robertson, F.T. Wright, R.L. Dykstra, Order Restricted Statistical Inference, Wiley, New York, 1988. | MR | Zbl

[26] D.A. Schoenfeld, Confidence bounds for normal means under order restrictions, with applications to dose-response curves, toxicology experiments, and low-dose extrapolation, J. Amer. Statist. Assoc. 81 (1986) 186-195. | Zbl

[27] H. Urakawa, Bounded domains which are isospectral but not congruent, Ann. Sci. Ec. Norm. Super. IV (15) (1982) 441-456. | Numdam | MR | Zbl

[28] H. Urakawa, Reflection groups and the eigenvalue problems of vibrating membranes with mixed boundary conditions, Tohoku Math. J. II (36) (1984) 175-183. | MR | Zbl

[29] A. Wald, Statistical Decision Functions, Wiley, New York, 1950. | MR | Zbl

[30] G. Watson, A Treatise on the Theory of Bessel Functions, Cambridge University Press, Cambridge, 1962. | JFM | MR

[31] E. Zinzius, Minimaxschatzer für den Mittelwert υ einer normalverteilten Zufallgröße mit bekannter Varianz bei vorgegebener oberer und unterer Schranke für υ, Math. Operationsforsch. Statist. Ser. Statist. 12 (1981) 551-557. | Zbl