Summary
A linear regression setup is considered, where the parameters of interest are only part of or a linear function of the full set of unknown parameters. Under the standard statistical assumptions the problem of finding an optimal approximate design leads to a convex minimization problem of the following type:
Minimize Φ(M) = φ(J(M)) over the set of all information matrices M of those approximate designs which allow a linear unbiased estimator for a given ℝS — valued linear function of the unknown parameters.
The optimality criterion φ is a convex and decreasing real-valued function on the set of all positive definite (s × s)-matrices, and J(M) is the reduced information matrix of M. The possibility of an optimal design with a singular M has caused a great many difficulties when trying to obtain characterizations for optimality via directional derivatives as in the “regular case”. Recently the author has given a description of directional derivatives by means of shorted matrices and minimum seminorm generalized inverses. The present paper gives some further results. The set of all subgradients of the objective function Φ is found. As a result, a general equivalence theorem for optimality can be derived directly using a major theorem of convex analysis, as pointed out by Pukelsheim and Titterington (1983). A saddle point, characterization for optimal designs is proved which has a nice interpretation in terms of a least favourable transformation of the linear regression setup.
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References
Anderson, W.N. (1971). Shorted operators. SIAM J. Appl. Math. 20, 520–525.
Gaffke, N. (1981). Some classes of optimality criteria and optimal designs for complete two-way layouts. Ann. Statist. 9, 893–898.
Gaffke, N. (1904). Directional derivatives of optimality criteria at singular matrices in convex design theory. Math. Operationsforsch. Statist., Ser Statistics. (To appear).
Gaffke, N., Krafft, O. (1977). Optimum properties of latin square designs and a matrix inequality. Math. Operationsforsch. Statist., Ser. Statistics 8, 345–350.
Gaffke, N., Krafft, O. (1982). Matrix inequalities in the Löwner ordering. In: Körte, B. (ed.). Modern Applied Mathematics, Optimization and Operations Research. North-Holland, Amsterdam, 595–622.
Kiefer, J. (1961). Optimum designs in regression problems, II. Ann. Math. Statist. 32, 298–325.
Kiefer, J. (1974). General equivalence theory for optimum designs (approximate theory). Ann. Statist. 2, 849–879.
Kiefer, J. (1975). Construction and optimality of generalized Youden designs. In: Srivastava, J.N. (ed.). A survey of statistical design and linear models. North-Holland, Amsterdam, 333–353.
Krein, M. (1947). The theory of self-adjoint extensions of semi-bounded Hermitian transformations and its applications I. Mat. Sbornik N.S. 20 (62), 431–495, and
Krein, M. (1947). The theory of self-adjoint extensions of semi-bounded Hermitian transformations and its applications I. Mat. Sbornik N.S. 21 (63), 365–404. (Russian, with English summary).
Mitra, S. K., Puri, M.L. (1979). Shorted operators and generalized inverses of matrices. Lin. Algebra Appl. 25, 45–56.
Näther, W., Reinsch, V. (1981). D-optimality and Whittle’s equivalence theorem. Math. Operationsforsch. Statist., Ser. Statistics 12, 307–316.
Pukelsheim, F. (1980). On linear regression designs which maximize information. J. Statist. Plann. Infer. 4, 339–364.
Pukelsheim, F., Styan, G.P.H. (1983). Convexity and monotonicity properties of dispersion matrices of estimators in linear models. Scand. J. Statist. 10, 145–149.
Pukelsheim, F., Titterington, D.M. (1983). General differential and Lagrangian theory for optimal experimental design. Ann. Statist. 11, 1060–1068.
Rao, C.R., Mitra, S.K. (1971). Generalized Inverse of Matrices and its Applications, Wiley. New York.
Rockafellar, R.T. (1972). Convex Analysis. Second printing, Princeton.
Silvey, S.D. (1978). Optimal design measures with singular information matrices. Biometrika 65, 553–559.
Silvey, S.D. (1980). Optimal Design. Chapman-Hall, London.
Whittle, P. (1973). Some general points in the theory of optimal experimental design. J. Roy. Statist. Soc. B 35, 123–130.
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Gaffke, N. (1985). Singular Information Matrices, Directional Derivatives, and Subgradients in Optimal Design Theory. In: Caliński, T., Klonecki, W. (eds) Linear Statistical Inference. Lecture Notes in Statistics, vol 35. Springer, New York, NY. https://doi.org/10.1007/978-1-4615-7353-1_6
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DOI: https://doi.org/10.1007/978-1-4615-7353-1_6
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