Abstract
Within the framework of classical linear regression model optimal design criteria of stochastic nature are considered. The particular attention is paid to the shape criterion. Also its limit behaviour is established which generalizes that of the distance stochastic optimality criterion. Examples of the limit maximin criterion are considered and optimal designs for the line fit model are found.
References
Eaton ML, Perlman MD (1991) Concentration inequalities for multivariate distributions: I. Multivariate normal distributions. Statistics and Probability Letters 12:487–504
Giovagnoli A, Wynn HP (1995) Multivariate dispersion orderings. Statistics and Probability Letters 22:325–332
Hall RL, Kanter M, Perlman MD (1980) Inequalities for the probability content of a rotated square and related convolutions. Annals of Probability 8(4):802–813
Hwang JT (1985) Universal domination and stochastic domination: estimation simultaneously under a broad class of loss functions. Annals of Statistics 13:295–314
Liski E, Luoma A, Mandal NK, Sinha BK (1998) Pitman nearness, distance criterion and optimal regression designs. Calcutta Statistical Association Bulletin 48(191–192):179–194
Liski E, Luoma A, Zaigraev A (1999) Distance optimality design criterion in linear models. Metrika 49:193–211
Liski E, Zaigraev A (2001) A stochastic characterization of Loewner optimality design criterion in linear models. Metrika 53:207–222
Mandal NK, Shah KR, Sinha BK (2000) Comparison of test vs. control treatments using distance optimality criterion. Metrika 52:147–162
Marshall AW, Olkin I (1979) Inequalities: Theory of majorization and its applications. Academic Press, New York
Mathew T, Nordström K (1997) Inequalities for the probability content of a rotated ellipse and related stochastic domination results. Annals of Applied Probability 7(4):1106–1117
Pukelsheim F (1993) Optimal design of experiments. Wiley, New York
Rao CR (1973) Linear statistical inference and its applications, 2nd edition. Wiley, New York
Shah KR, Sinha BK (1989) Theory of optimal designs. Springer-Verlag Lecture Notes in Statistics Series, No. 54
Sherman S (1955) A theorem on convex sets with applications. Annals of Mathematical Statistics 26:763–766
Sinha BK (1970) On the optimality of some design. Calcutta Statistical Association Bulletin 20:1–20
Steȩpniak C (1989) Stochastic ordering and Schur-convex functions in comparison of linear experiments. Metrika 36:291–298
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Zaigraev, A. Shape optimal design criterion in linear models. Metrika 56, 259–273 (2002). https://doi.org/10.1007/s001840100179
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DOI: https://doi.org/10.1007/s001840100179