Olaf Krafft

D-optimal designs for a multivariate regression model

Considered is a linear regression model with a one-dimensional control variable and an m-dimensional response variable y. The components of y may be correlated with known covariance matrix. Let B be the covariance matrix of the Gauss-Markoff estimator for the unknown parameter vector of the model. Under rather mild assumptions on the set of regression functions a factorization lemma for det B is proved which implies that D-optimal designs do not depend on the covariance matrix of y. This allows the use of recent results of Dette to determine approximate D-optimal designs for polynomial regression. A partial result for exact D-optimal designs is given too.

MEAN PASSAGE TIMES FOR TRIDIAGONAL TRANSITION MATRICES AND A TWO-PARAMETER EHRENFEST URN MODEL

A two-parameter Ehrenfest urn model is derived according to the approach taken by Karlin and McGregor [7] where Krawtchouk polynomials are used. Furthermore, formulas for the mean passage times of finite homogeneous Markov chains with general tridiagonal transition matrices are given. In the special case of the Ehrenfest model they have quite a different structure as compared with those of Blom [2] or Kemperman [9].

Optimum properties of latin square designs and a matrix inequality

The paper deals with uniform and D-optimality of designs in the two-way elimination of heterogeneities. It is shown that designs which are optimum for the hypothesis that all treatment effects are equal are optimum for some other hypotheses, too. The Proof is based on a new matrix- and determinantal inequality.

A-optimal connected block designs with nearly minimal number of observations

Abstract A-optimal block designs are determined in the class of connected block designs with constant block size under the restriction that the number of observations is v + b where v is the number of treatments and b is the number of blocks. Graph-theoretic methods are employed to derive the results. In particular, the Moore-Penrose inverse of the C -matrix is obtained in terms of a distance matrix of a graph associated with the design.

Optimum designs in complete two-way layouts

Abstract In the usual two-way layout of ANOVA (interactions are admitted) let nij ⩾ 1 be the number of observations for the factor-level combination( i , j ). For testing the hypothesis that all main effects of the first factor vanish numbers n ∗ ij are given such that the power function of the F -test is uniformly maximized (U-optimality), if one considers only designs ( nij ) for which the row-sums ni are prescribed. Furthermore, in the (larger) set of all designs for which the total number of observations is given, all D-optimum designs are constructed.

An arithmetic—Harmonic-mean inequality for nonnegative definite matrices

Abstract A generalization of the arithmetic—harmonic-mean inequality is presented, making use of the concept of the parallel sum of matrices.

Convergence of the powers of a circulant stochastic matrix

Abstract Let C = circ( c 1 ,…, c s ) be a circulant stochastic matrix. The convergence of C n as n tends to infinity is characterized in terms of the set U = { j : c j 0}. The results are applied to Markov chains having C as transition matrix.

On universally optimal designs for regression setups

Abstract The set of optimality criteria to which Kiefers maximum trace principle is applicable is discussed. Furthermore, for a linear regression model y ( x ) = a′f(x) let ξ ∗ be an (approximate) D-optimal design for estimating a by its BLUE. Then gx ∗ is universally optimal for estimating b = K a by its BLUE, where K is a regular matrix determined by ξ ∗ .

A maximin linear estimator for linear parameters under restrictions in form of inequalities

For the linear model y =Xβ+e with β restricted to β′Hβ≦1 a maximin-linear estimator for a parameter γ = Cβ is derived. The optimaiity criterion is the mean square error.

A Refined Geometric-Arithmetic Means Inequality for Integers

A series of problems in integer programming seems to be solvable by known — but for special sets of integers — improved inequalities. Considered are t-norms Mt(x,a) of x = (x.j,...,xn), where the xi are taken from the first N integers and not all xi equal. Multiplying Mt(x,a) by a suitable weight function ga(t) one gets inequalities which are better than in case x ∈ ℝ + n Especially a refined harmonic-geometric-arithmetic means inequality is obtained.