D.R. Jensen

Vector efficiency in multiparameter estimation

Abstract Relative efficiencies are studied through invariance in multiparameter estimation. Scalar efficiency indices, together with lower and upper bounds, are generated via invariant monotone functions, and are augmented to include tripartite efficiency numbers. Bounds on directed Fisher efficiencies emerge through generalized Rayleigh quotients. Applications are noted in improving estimators through conditioning, in comparing regular estimators with efficient estimators achieving their minimal dispersion bounds, and in comparing two second-order experimental designs.

Minimal Properties of Moore-Penrose Inverses

Abstract Let Ax = y be consistent; let x0 = Gy be any minimum-norm solution satisfying (AG)′ = AG; and let A+ be the Moore-Penrose inverse of A. It is shown that φ(G) ⩾ φ(A+) for any φ in a class Φ containing the unitarily invariant matrix norms. The conditioning of the system Ax = y is studied via condition numbers Cφ(A, G). It is shown that Cφ(A, G) ⩾ Cφ(A, A+) for every φ∈ Φ. Moreover, bounds on Cφ(A, G) are given in terms of singular values. Parallel results are found when A and G are symmetric, with applications to linear models of less than full rank.

Properties of selected subset diagnostics in regression

Anomalies are cited in the use of subset regression diagnostics. Two leverage diagnostics are unified through canonical leverages. Other diagnostics relate one-to-one with the R-Fisher statistic, supporting exact but equivalent tests at level [alpha]. Three distance diagnostics exhibit ad hoc scalings; their distributions are characterized and ordering properties noted; one is studied further through simulation; and a properly scaled version is given. Case studies reexamine the use of selected subset diagnostics in practice.

The use of studentized diagnostics in regression

Abstract. Recent work by LaMotte (1999) uncovered redundancies and inconsistencies in the current practice of selected deletion diagnostics in regression. The present study extends earlier work to include further diagnostics on using different methods. Benchmarks adjusted to the scale of each diagnostic are given to assure consistency across diagnostics. Case studies illustrate anomalies in the use of these diagnostics as currently practiced. Alternative diagnostics are given to gauge effects of single-case deletions on variances and biases in prediction and estimation.

Structured Dispersion and Validity in Linear Inference

Abstract Structured matrices ∑(γ) = [I n + eγ′ + γe′ − ggee′] arise in nonstandard linear models, where e ′ = [1, …, 1], γ ′ = [ γ 1 , …, γ n ], and gg = (γ 1 + … + γ n ) n . Their properties are studied, including expressions for eigenvalues, conditions for positive definiteness, and conditioning of bE(γ) as γ varies. It is shown that if γ majorizes γ 0 , then the condition numbers are ordered as c φ (∑( γ )) ⩾ c φ (∑( γ 0 )) for every condition number { c φ (·); φ ∈ ϱ } generated by the unitarily invariant matrix norms. Applications are noted in linear inference and in outlier detection.

RECOVERING INFORMATION FROM INCREMENTAL RESPONSES

Summary Some yields analysed and reported in the literature have been adjusted by subtracting a control. It is found that full information can be recovered for estimable parameters and the error variance using these incremental responses, in comparison with unadjusted data. These findings are of practical importance, and they supplement materials usually found in a graduate course in linear inference. The issues are illustrated using a case study from the literature.

Bounds on Mahalanobis norms and their applications

Abstract Local and global bounds for ratios of norms, and minimal and maximal norms, are constructed for pairs and ensembles of quadratic norms of R k , with corresponding results for Mahalanobis distance functions. These support envelopes for distributions of certain quadratic forms in Gaussian variates. Applications are noted in the use of quadratic classification rules and in assessing hit probabilities in ballistic systems.