Classical multivariate analysis rests on observations ( ) Y n k × having n k > mutually independent rows, with dispersion matrix as a direct product ( ) n V Y I = ⊗ Σ , supported in turn by a rich literature.That independence may fail is modeled here on taking the rows of Y to be exchangeably dependent such that ( ) V Υ = Ω ⊗ Σ where exchangeability rests on the choice for ( ) n n Ω × .Three choices are considered; each interjects additional parameters into the model; and it remains to ask which, if any, of findings widely known under independence, might apply also under exchangeable dependence.Conventional inferences for the location and scale parameters ( ) , µ Σ are reconsidered.Excluding µ these are found to carry over in large part to include the exchangeable errors of this study.

Linear inference is the foundation stone for much of theoretical and applied statistics. In practice errors often have excessive tails and are lacking the moments required in conventional usage. For random vector responses such errors often are modeled via spherical alpha-stable distributions with stability index alpha is an element of (0, 2], arising in turn through central limit theory but converging to non-Gaussian limits. Earlier work [Jensen, D.R. (2018). Biom. Biostat. Int. J. 7: 205-210] reexamined conventional linear models under n-dimensional alpha-stable responses, to the effect that Ordinary Least Square (0 LS) solutions and residual vectors under alpha-stable errors also have alpha-stable distributions, whereas F ratios remain exact in level and power as for Gaussian errors. The present study generalizes those findings to include multivariate linear models having matrix responses of order (n x k). Topics in inference focus on both location and scale matrices, the latter in connection with analogs of simple, multiple, and canonical correlations without benefit of second moments, seen nonetheless to gauge degrees of association under alpha-stable symmetry.

Linear inference remains pivotal in statistical practice, despite errors often having excessive tails and thus deficient of moments required in conventional usage.Such errors are modeled here via spherical α -stable measures on n  with stability index (0,2], α∈ arising in turn through multivariate central limit theory devoid of the second moments required for Gaussian limits.This study revisits linear inference under α -stable errors, focusing on aspects to be salvaged from the classical theory even without moments.Critical entities include Ordinary Least Squares ( ) OLS solutions, residuals, and conventional F ratios in inference.Closure properties are seen in that OLS solutions and residual vectors under α -stable errors also have α -stable distributions, whereas F ratios remain exact in level and power as for Gaussian errors.Although correlations are undefined for want of second moments, corresponding scale parameters are seen to gauge degrees of association under α -stable symmetry.

In linear models having near collinear columns of X, ridge and surrogate estimators often are used to mitigate collinearity. A new class of estimators is based on mixtures, either of X and a design minimal in an ordered class or of the Fisher information and a scalar matrix. Comparisons are drawn among choices for the mixing parameter, and the estimators are found to be admissible relative to ordinary least squares. Case studies demonstrate that selected mixture designs are perturbed from the original design to a lesser extent than are those of the surrogate method, while retaining reasonable efficiency characteristics.

Given observations { Y i ; 1 ≤ i ≤ n { with dispersion matrix Σ , a pervading issue is whether shifts have occurred in designated subsets of the observations. Early work on single shifts used order statistics or the R-Student t i statistics as diagnostics, initially derived under i.i.d. Gaussian assumptions. These diagnostics recently have been shown to remain exact in level and power under equicorrelation and more general dispersion structures, and under star-contoured mixtures supplanting Gaussian errors, with an accounting for irregularities engendered by shifts at other than the designated cases. Extensions here pertain to outlying subsets using the R -Fisher diagnostics F I , showing invariance of its distribution and of related diagnostics under more general dispersion structures and mixtures over these. Shifts occurring at cases other than those designated induce doubly noncentral F distributions. These elicit profound disturbances in operating characteristics of the diagnostics, serving in turn to explain masking and swamping, and the discovery of hidden “regression effects” among outliers. Evidence for anomalies arising from denominator noncentralities rests on two-sided rejection rules to be given. Numerical studies serve to illuminate the essence of the findings in practice.

Given a model {Y = X beta + epsilon} with Fisher information matrix Xi = X'X, a principal objective is to find information enhancing transformations T for which T Xi T' >=(L) Xi under the positive definite ordering, so as to improve essentials in linear inference. This is achieved through properties of congruences together with basic orderings of linear spaces. These foundations in turn support a new class of geometric mixture models on "mixing" the original design with another to assume the role of "target," to the following effects. Ridge, surrogate, and other solutions are often used to mitigate the effects of ill-conditioned models. Instead, in this study an ill-conditioned design matrix X is mixed with a well-conditioned design as target, leveraging the former toward the latter as the mixing parameter evolves, thus offering an alternative approach to ill-conditioning. The methodology is demonstrated with case studies from the literature, where the geometric mixtures are compared with the ridge, surrogate, and recently found arithmetic mixture models.

Lower and upper spectral bounds are known for matrices X X(k x k) under Loewner [Uber monotone matrixfunktionen. Math Z. 1934; 38: 177-216] order, as are corresponding bounds for the factor X(n x k) under an induced order. Least upper bounds for the latter give designs with dominating Fisher Information, with consequent gains in linear inference; see Jensen DR, Ramirez DE [Enhanced design efficiency through least upper bounds. J Stat Comput Simul. 2016; 86: 1798-1817]. The present study examines properties on ordering the singular values of a design matrix using majorization as in Marshall and Olkin [Inequalities: theory of majorization and its applications. New York: Academic Press; 1979]. The principal focus includes conditioning through condition numbers, variance inflation factors, and lengths and efficiencies of OLS solutions. Functions monotone under the induced order are identified; equivalence classes of designs are displayed preserving a dispersion matrix or its eigenvalues; a minimal element Xm(n x k) is characterized; as are equivalence classes of (A, D, E)optimal designs showing the latter not to be unique. Algorithms to achieve enhanced designs are given on modifying a single design, or on amalgamating two designs, with essential consequences in linear inference. A collateral procedure, based on mixtures of Fisher information matrices, serves effectively to ameliorate the ill effects of near collinearity. Case studies illustrate gains to be made in practice, to include a substantial improvement in the analysis of classically ill-conditioned data from the literature.

Given Gaussian observation vectors $[\seqcl{\BY}{n}]$ having a common mean and dispersion matrix, a pervading issue is to identify shifted observations of type $\{\BYi\!\to\!\BYi\!+\!\bdeli\}.$ Conventional usage enjoins Hotelling's $\Tisq$ diagnostics, derived and applied under the mutual independence of $[\seqcl{\BY}{n}]$. Independence often fails, yet the need to identify outliers nonetheless persists. Accordingly, the present study reexamines $\Tisq$ under dependencies to include equicorrelations and more general matrices. Such dependencies are found in the analysis of calibrated vector measurements and elsewhere. In addition, mixtures of these distributions having star--shaped contours arise on occasion in practice. Nonetheless, the $\Tisq$ diagnostics are shown to remain exact in level and power for all such mixtures. Moreover, further matrix distributions, not necessarily having finite moments, are seen to generalize $n$--dimensional spherical symmetry to include non--Gaussian matrices of order $(n\!\times\!k)$ supporting $\Tisq.$ For these the use of $\Tisq$ remains exact in level. These findings serve to expand considerably the range of applicability of $\Tisq$ in practice, to include matrix Cauchy and other heavy tailed distributions intrinsic to econometric and other studies. Case studies serve to illuminate the methodology.

Tests for vector hypothesesdiffering only in their first two coordinates.These are called directed alternatives.The spectral decomposition of supports the identification of one-dimensional alternatives least likely and most likely to be discerned, to complement conventional data analysis.Applications are drawn in the use of Hotelling's 2 Τ and of F -tests in linear inference.Moreover, it is seen that a given design may be recast so as to reverse the least likely and most likely alternatives.Numerical examples serve to illustrate the findings.

Let Y0=[Y1, . . . ,Yn] be k–dimensional Gaussian vectors having the common mean μ and dispersion matrix Σ. To identify outlying shifts of type {Yi→Yi+δ}, Hotelling’s Ti diagnostics properly account for dependencies among the k elements of Yi. This study reconsiders the use of Ti under failure of the classical venue that [Y1, . . . ,Yn] themselves are mutually independent. Diagnostics for the case k=1 include those of Dixon (1950), Grubbs (1950) and Ferguson (1961) based on order statistics, and the R– Student deletion diagnostics ti , all predicated on independent observations. To relax the latter, Jensen and Ramirez (2015) recently showed these procedures to remain exact in level and power under specified dependencies among [Y1, . . . , Yn], and for scale mixtures of these dependent distributions. These findings are in keeping with correlated data, as found in the analysis of calibrated measurements, for example. Specifically, Hotelling’s Ti diagnostics are shown here to remain exact in level and power for dispersion mixtures of matrix Gaussian errors having star–shaped contours, to considerably enhance their range of applicability. AMS Subject Classification: 62E15, 62H15, 62J20

Lower and upper spectral bounds are known for positive-definite (k x k) matrices in (S-k(+), >=(L)) under Loewner (Uber monotone Matrixfunktionen. Math Z. 1934; 38: 177-216) ordering. Lower and upper singular bounds for matrices of order (n x k) in (F-nxk, >=) derive under an induced ordering. These orderings are combined here to the following effects. Given two first-order experimental designs (X, Z) in (F-nxk, >=), their upper singular bound XM enhances both X and Z in that its Fisher Information matrix dominates those for both X and Z, thus ordering essentials in Gauss-Markov estimation. Moreover, if Sigma, Omega, and Xi are dispersion matrices for linear estimators under X, Z, and X-M, respectively, then Xi is the spectral lower bound for (Sigma, Omega) in (S-k(+), >=(L)). In essence this algorithm identifies elements in Z complementary to those of X, and combines these into X-M. Case studies illustrate gains to be made thereby in first and second-order designs. Specifically, two examples demonstrate that designs optimal under separate criteria may be combined into a single design dominating both. In addition, selected examples demonstrate that classical second-order designs may be improved inter se.

Consider [Y1,…, Yn] as Gaussian observations with common mean μ and dispersion matrix Σ. Approaches for detecting outlying observations include the R-Student statistics in regression diagnostics, as well as tests due to Grubbs, Dixon, and Ferguson using order statistics. All are known to be valid under Σ = σ2In; Grubbs’s test also holds under an equicorrelated matrix Σ(ρ) and the more general structure \(\sum \,(\xi ) = {\sigma ^2}[{I_n} + {1_n}\xi ' + \xi {1'_n} - \bar \xi {1_n}{1'_n}]\). Dispersion mixtures of Gaussian errors having Σ(ρ) and Σ(ξ) are studied in detail; their densities have star-shaped contours as encountered on occasion in practice. Under these mixtures, the aforementioned diagnostics all are shown to be exact in significance level and in power as for the case where Σ = σ2In. This expands considerably their range of applicability in practice. Case studies serve to illustrate essentials of the findings.

Corrections are given to common misconceptions regarding the use of centered and uncentered regressors, and of meanings to be ascribed to A -, D -and E -criteria in evaluating two such designs.

Shifts in responses typically are obscured from users, so that regression proceeds as if unshifted. At issue is the infusion of such shifts into classical analysis. On projecting outliers into the “Regressor” and “Error” spaces of a model, findings here are that shifts in responses may account for shifts in the OLS solutions, or for inflated residuals, or both. These in turn impact estimation, prediction, and hypothesis tests, all of vital interest to users, and all considered here. Tools for identifying shifts are given. Case studies illustrate effects of shifts on regression, to include a reexamination of studies from the literature.

Anomalies persist in the use of deletion diagnostics in regression. Tests for outliers under subset deletions utilize the R-Fisher FI statistics, each having a noncentral F-distribution with noncentrality parameter λ as a function of shifts only at deleted rows in the index set I. Numerous studies examine empirical outcomes of these diagnostics in random experiments. In contrast, studies here are probabilistic, examining distributions behind those empirical outcomes and tracking the effects of shifts at nondeleted rows. By allowing shifts at nondeleted rows in a set J, in addition to traditional shifts at deleted rows in I, FI is shown to have a doubly noncentral F-distribution. By removing the unnecessary restriction that shifts occur only at deleted rows, these findings support constructs akin to power curves in tracking probabilities of masking or swamping as shifts evolve. In addition, “regression effects” among outliers may have unforeseen consequences. A dichotomy of shifts is discovered as projections into the “regressor” and “error” spaces of a model. Hidden shifts at nondeleted rows can obfuscate not only meanings ascribed to traditional outlier diagnostics, but also to subset influence diagnostics corresponding one-to-one with FI. In short, despite wide usage abetted by software support, deletion diagnostics in current vogue no longer can be recommended to achieve objectives traditionally cited. Case studies illustrate the debilitating effects of these anomalies in practice, together with conclusions misleading to prospective users.

Single-case deletion diagnostics rest on a single outlier at a point in the regressor space. Residuals are basic. Our objectives are to show (i) that shifts serve to inflate or deflate stochastically the ordinary squared residuals, even for non-outlying data; and (ii) that outliers generate anomalies as doubly noncentral distributions for both outlier and influence diagnostics, these specific to a given design. Outliers account for masking and swamping, with probabilities as evaluated here. Despite wide acceptance, software support, and routine useage, these anomalies despoil meanings historically ascribed to deletion diagnostics, thus abrogating objectives traditionally cited for their use. Case studies document some misdirected and unintended consequences of their continued use in practice.

AbstractMultivariate distributions arise throughout statistics and applied probability and they are defined on finite‐dimensional spaces. They serve as probabilistic models for dependent outcomes of random experiments. Biometric data typically comprises observations on multiple characteristics for each experimental subject, and joint distributions are central to the modeling and analyses of such data. From multivariate distributions, the distributions of various sample statistics of note in statistical inference can be derived. Multivariate distributions also characterize the behavior of stochastic processes through properties of their finite‐dimensional projections.

Multivariate versions of the Student's t distribution derive through Studentization from multivariate normal models. Multivariate t distributions are useful to model errors of a random experiment offering greater flexibility and heavier tails than multivariate normal models. There are two types of multivariate t distributions. Type I distributions are derived by scaling each component of a normal vector by a single random scalar. Type II distributions involve separate scalings by elements of a further random vector, itself having a joint multivariate distribution.