Alessandra Giovagnoli

G-majorization with applications to matrix orderings

Abstract A vector x is said to G -majorize a vector y if y lies in the convex hull of the orbit of x under a group G . The present paper contains a straightforward account with two important statements equivalent to G -majorization. In certain cases, for example when G is a finite reflection group, one equivalent condition reduces to a finite set of linear inequalities representing a cone ordering in the fundamental region of the group. The other condition is that every convex G -invariant function of y is less than the same function of x . Upper and lower weak majorizations ( GW -majorizations) are introduced by combining G with a second ordering compatible with it. The results are applied where possible to matrix orderings where A ≺ G B is G -majorization when G is a subgroup of the orthogonal group O n acting by congruence, i.e. g ( B )= QBQ T with Q the matrix representation for an element of O n . When G is the symmetric group of permutation matrices this defines a new ordering and generalizes a proposition of Kiefer in the design of experiments.

Properties of frequency distributions induced by general ‘up-and-down’ methods for estimating quantiles

Abstract In this paper a general definition of an up-and-down algorithm for estimating given quantiles of a binary response curve is suggested. Conditions are given for the distribution to which the design converges to be unimodal, for its mode to be ‘next’ to the quantile specified and for the distribution to be more concentrated around the mode. Results of Derman (1957) and of Durham and Flournoy (1994) are extended.

On the Large Sample Optimality of Sequential Designs for Comparing Two or More Treatments

Abstract The present paper deals with optimal (in Kiefers sense) response-adaptive designs for parametric inference on v ≥ 2 treatments. Sometimes (e.g., for nonlinear models) a sequential estimation procedure combined with an adaptive experiment suggests itself as the “natural” best design. One of the questions is whether, since we proceed sequentially, we should infer conditionally on the design. Another question is whether such an adaptive design is really optimal for the chosen type of inference. The main purpose of this paper is to give proofs of the asymptotic optimality for inferring both conditionally and unconditionally of a large class of such designs, incorporating response-adaptive randomization as well. The asymptotic optimality of the Maximum Likelihood design, namely that based on the step-by-step updating of the parameter estimates by maximum likelihood, is proved for responses belonging to the exponential family. Under this procedure the MLEs retain the strong consistency and asymptotical normality properties. Furthermore, such properties still hold approximately for suitable inverse sampling stopping rules.

Optimum Continuous Block Designs

The incidence matrix of a block design is replaced by a normalized version, N, in which the entries are non-negative numbers whose sum is unity. The so-called C-matrix, the information matrix for estimation of treatm ent contrasts, is similarly replaced by the normalized analogue C(N). We study the set of ordered eigenvalues of all C(N) and give a complete specification for three treatments (rows). For any number of treatm ents we characterize the eigenvalues of an im portant subclass of designs for which the non-zero entries in any given block are equal. It is suggested that the natural ordering between designs is upper weak majorization of the eigenvalues. Using this we show how to improve a given N-matrix and this leads to several optimality statements.

Robust design via simulation experiments: a modified dual response surface approach

The existing procedures for robust design, devised for physical experiments, may be too limiting when the system can be simulated by a computer model. In this paper we introduce a modification of the dual response surface modelling, which incorporates the option of stochastically simulating some of the noise factors when their probabilistic behaviour is known. Our method generalizes both the crossed and the combined array approaches and finds a natural application to integrated parameter and tolerance design. The method appears suitable for designing complex measurement systems and in this paper is applied to the design of a high-precision optical profilometer. Copyright

Experimental designs for mean and variance estimation in variance components models

For a mixed linear model E(Y) = Xa, V(Y) = v, with V a known function of an unknown vector parameter 8, under normality assumptions, maximum likelihood can be employed to estimate a and B simultaneously by iterative procedures. If the design is factorial, an important practical problem is the choice of the number of observations for each level of the random factors (including error). In this paper we modify logdet of the exptected Fisher information matrix, introducing weights to allow for different emphasis on estimation of mean and variance, and choose this as our optimality criterion. We show that for the case of only one random factor plus error, balanced designs are optimal, and we find the optimal number of replications. We also consider the possibility that the costs of replications and of different levels of the random effect may differ, and find the expression of the optimal replication number, keeping the total cost of the experiment fixed. Not suprisingly, this expression depends on the variance ratio, which is unknown, thus a pseudo-Bayesian approach is required.

GROUP INVARIANT ORDERINGS AND EXPERIMENTAL DESIGNS

Abstract Recent work by Giovagnoli and Wynn and by Eaton develops the theory of G-majorization with application to matrix orderings. Using this theory much of the work begun by Kiefer on ‘universally’ optimal designs of experiments can be better understood. The technique is to combine a group ordering (G-majorization) with another invariant ordering, such as the Loewner ordering, to define upper weak G-majorization on the information matrices of the experiments. Using an idea from previous work of Giovagnoli and Wynn combined with work by Pukelsheim and Styan on the matrix concavity of information matrices a general theory of weak G-majorization for linear models is developed which includes orderings for subsets of estimable functions.

Optimal Sampling Design with Random Size Clusters for a Mixed Model with Measurement Errors

Our investigation concerns sampling in epidemiological studies, in the presence of both strata and clusters; the problem is to choose the number of clusters to sample in each stratum given that the size of the clusters in general is a random variable. The issue of unplanned randomness in the design seems to have been scarcely addressed in the survey sampling literature. We were motivated by a sample survey — carried out in 1990–1995 by the Italian National Institute of Nutrition (INN-CA) — on the food habits of the Italian population, divided into four geographical areas: the household came in both as random factor which influenced the individual response and — due to the varying number of its members — as a random component of the design which affected the sample size. In this paper we assume various mixed models under different hypothesis on measurement errors (typically correlated) in the response and for each of them find the optimal designs under several optimality criteria, namely the determinant, the trace, the maximum eigenvalue of the unconditional Fisher information of the fixed effect parameters. In all the models we deal with in the present paper, the optimal design depends on just one unknown parameter τ, a given function of the variance components and correlation coefficients. The dependence of the design on τ is investigated through some simulations. The solutions given for the special cases motivated by the INN-CA study should be applicable to a wider variety of situations.

Cyclic majorization and smoothing operators

Abstract Recent work by M. L. Eaton and the present authors reviews and generalizes work on majorization and group majorization. The standard material on majorization was extended from the symmetric group to more general groups in the important paper of Eaton and Perlman (1977). The present paper studies one special nonreflection group, namely the cyclic group on n elements. We say that a vector x cyclically majorizes a vector y , written y C x , if it lies in the convex hull of all vectors which can be obtained from x by cyclic permutation. The class of order-preserving functions is studied, and the theory gives an ordering on the smoothness of periodic functions with possible application to time series analysis and also an ordering of smoothing operators.

A group majorization ordeting for correlation matrices

Abstract We compare multivariate distributions from the point of view of the “strength” of linear relationships among the random variables. To this purpose we define a group majorization ordering for correlation matrices based on the permutation group and the sign-change group. This partial ordering has many intuitively appealing properties. We show some of its implications in terms of multiple, partial, and canonical correlation coefficients, as well as for sample correlation.