Friedrich Pukelsheim

Optimal design of experiments

Experimental Designs in Linear Models Optimal Designs for Scalar Parameter Systems Information Matrices Loewner Optimality Real Optimality Criteria Matrix Means The General Equivalence Theorem Optimal Moment Matrices and Optimal Designs D-, A-, E-, T-Optimality Admissibility of Moment and Information Matrices Bayes Designs and Discrimination Designs Efficient Designs for Finite Sample Sizes Invariant Design Problems Kiefer Optimality Rotatability and Response Surface Designs Comments and References Biographies Bibliography Index.

The Three Sigma Rule

Abstract For random variables with a unimodal Legesgue density, the 3[sgrave] rule is proved by elementary calculus. It emerges as a special case of the Vysochanskiĭ-Petunin inequality, which in turn is based on the Gauss inequality.

On linear regression designs which maximize information

Abstract Necessary and sufficient conditions are established when a continuous design contains maximal information for a prescribed s-dimensional parameter in a classical linear model. The development is based on a thorough study of a particular dual problem and its interplay with the optimal design problem, extending partial results and earlier approaches based on differential calculus, game theory, and other programming methods. The results apply in particular to a class of information functionals which covers c-, D-, A-, L-optimality, they include a complete account of the non-differentiable criterion of E-optimality, and provide a constructive treatment of those situations in which the information matrix is singular. Corollaries pertain to the case of s out of k parameters, simultaneous optimality with respect to several criteria, multiplicity of optimal designs, bounds on their weights, and optimality which is induced by admissibility.

The distance between two random vectors with given dispersion matrices

For two p-dimensional random vectors X and Y with dispersion matrices Σ11 and Σ22, respectively, we determine that covariance matrix Ψ0 of X and Y that minimizes the L2-distance between X and Y. There is a dual to this problem that is of interest in another context.

Seat biases of apportionment methods for proportional representation

Abstract In proportional representation systems, an important issue is whether a given apportionment method favors larger parties at the expense of smaller parties. For an arbitrary number of parties, ordered from largest to smallest by their vote counts, we calculate (apparently for the first time) the expected differences between the seat allocation and the ideal share of seats, separately for each party, as a function of district magnitude, with a particular emphasis on three traditional apportionment methods. These are (i) the quota method with residual fit by greatest remainders, associated with the names of Hamilton and Hare, (ii) the divisor method with standard rounding (Webster, Sainte-Lague), and (iii) the divisor method with rounding down (Jefferson, Hondt). For the first two methods the seat bias of each party turns out to be practically zero, whence on average no party is advantaged or disadvantaged. On the contrary, the third method exhibits noticeable seat biases in favor of larger parties. The theoretical findings are confirmed via empirical data from the German State of Bavaria, the Swiss Canton Solothurn, and the US House of Representatives.

Some properties of matrix partial orderings

Abstract The matrix partial orderings considered are: (1) the star ordering and (2) the minus ordering or rank subtractivity, both in the set of m × n complex matrices, and (3) the Lowner ordering, in the set of m × m matrices. The problems discussed are: (1) inheriting certain properties under a given ordering, (2) preserving an ordering under some matrix multiplications, (3) relationships between an ordering among direct (or Kronecker) and Hadamard products and the corresponding orderings between the factors involved, (4) orderings between generalized inverses of a given matrix, and (5) preserving or reversing a given ordering under generalized inversions. Several generalizations of results known in the literature and a number of new results are derived.

Experimental Designs for Model Discrimination

Abstract We present designs that perform well for several objectives simultaneously. Three different approaches are discussed: to augment a given design in an optimal way, to evaluate a mixture of the various criteria, and to optimize one objective subject to achieving a prescribed efficiency level for the others. Our sample designs are for the situation of discriminating between a second- and third-degree polynomial fit, under the D-criterion and geometric mixtures of D-criteria.

Estimating variance components in linear models

Estimation of variance components in linear model theory is presented as an application of estimation of the mean by introducing a dispersion-mean correspondence. Without any further computations, this yields most general representations of minimum variance-minimum bias-invariant quadratic estimates, estimates from MINQUE theory, and Ridge-type estimates of the variance components.

Another look at rotatability

Rotatability is one of many desirable characteristics of a response-surface design. Recent work (Draper and Guttman 1988; Khuri 1988) has, for the first time, provided ways to measure “how rotatable” a design may be when it is not perfectly rotatable. This had previously been assessed by the viewing of tediously obtained contour diagrams. This article provides a criterion that is easy to compute and is invariant under design rotation. It also easily extends to higher degree models.

On the History of the Kronecker Product

History reveals that what is today called the Kronecker product should be called the Zehfuss product.