Comparison Of Linear Experiments Research Articles

Quotient of information matrices in comparison of linear experiments for quadratic estimation

AbstractThe ordering of normal linear experiments with respect to quadratic estimation, introduced by Stępniak in [Ann. Inst. Statist. Math. A 49 (1997), 569-584], is extended here to the experiments involving the nuisance parameters. Typical experiments of this kind are induced by allocations of treatments in the blocks. Our main tool, called quotient of information matrices, may be interesting itself. It is known that any orthogonal allocation of treatments in blocks is optimal with respect to linear estimation of all treatment contrasts. We show that such allocation is, however, not optimal for quadratic estimation.

On a problem with singularity in comparison of linear experiments

For consistency, the parameter space in the Gauss–Markov model with singular covariance matrix is usually restricted by observation vector. This restriction arises some difficulties in comparison of linear experiments. To avoid it we reduce the problem of comparison from singular to nonsingular case.

Order relations for linear models: A survey on recent developments

In this survey the most applicable order relations between linear experiments are studied. For linear normal experiments the cases of known and unknown variances require sophisticated arguments from linear algebra and some tools from convexity theory. The comparison of linear experiments also casts some new light on the popular statistical notions of sufficiency and deficiency.

On a matrix result in comparison of linear experiments

A useful statistical relation between normal linear experiments N( Aβ,σ V) and N( Bβ,σ W) is reduced in [Stȩpniak, Statistics 30 (1998) 279–289] to the matrix conditions A ′( V+ AA ′) − A− B ′( W+ BB ′) − B⩾0 and r( V+ AA ′)− r( A)⩾ r( W+ BB ′)− r( B). The aim of this note is to present a characterization of these conditions in terms of linear transformations. This characterization may be useful for matrix theory and practice.

Matrix loss in comparison of linear experiments

The usual ordering of linear experiments was induced by estimation of some scalar parameters under quadratic risk. New orderings, based on estimation of vector parameters under quadratic and matrix risk, respectively, are defined and considered here.

Stochastic ordering and schur-convex functions in comparison of linear experiments

The usual ordering of linear experiments is defined by quadratic risk of attainable linear estimators. It is shown that under normality assumption this ordering can be introduced in a risk-free way by stochastic ordering of the estimators. Moreover an application of Schur-convex functions to design of experiments is presented.

Comparison of Linear Experiments with Known Covariances

For two linear experiments $d_1 = L(X_1\beta, V_1)$ and $d_2 = L(X_2\beta, V_2)$ where the covariances $V_1$ and $V_2$ are known and can be singular or nonsingular, we characterize the following relations: $d_1$ at least as good as $d_2, d_1$ better than $d_2$, and $d_1$ equivalent to $d_2$. Sometimes only a subset of parameters is of interest to the experimenter. We extend the above relations between $d_1$ and $d_2$ to estimation of a common subset of parameters and give analogous characterizations. Three examples are given.