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Joint confidence regions in the multivariate calibration problem

JOURNAL OF STATISTICAL PLANNING AND INFERENCE 2002年第2期100卷 427-441页

作者： Mathew, T Sharma, MK Univ Maryland Dept Math & Stat Baltimore MD 21250 USA Johnson & Johnson Co LifeScan Inc Milpitas CA 95035 USA

Consider a vector valued response variable related to a vector valued explanatory variable through a normal multivariate linear model. The multivariate calibration problem deals with statistical inference on unknown values of the explanatory variable. The problem addressed is the construction of joint confidence regions for several unknown values of the explanatory variable. The problem is investigated when the variance covariance matrix is a scalar multiple of the identity matrix and also when it is a completely unknown positive definite matrix. The problem is solved in only two cases: (i) the response and explanatory variables have the same dimensions, and (ii) the explanatory variable is a scalar. In the former case, exact joint confidence regions are derived based on a natural pivot statistic. In the latter case, the joint confidence regions are only conservative. Computational aspects and the practical implementation of the confidence regions are discussed and illustrated using an example. (C) 2002 Elsevier Science B.V. All rights reserved.

关键词： conservative confidence region multivariate calibration multivariate linear model non-central chisquare distribution non-central F distribution Wishart distribution

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Statistical modeling of a former Arctic Ocean ice shelf complex using Antarctic analogies

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JOURNAL OF GEOPHYSICAL RESEARCH-EARTH SURFACE 2013年第2期118卷 1105-1117页

作者： Kirchner, N. Furrer, R. Jakobsson, M. Zwally, H. J. Robbins, J. W. Stockholm Univ Dept Phys Geog & Quaternary Geol S-10691 Stockholm Sweden Univ Zurich Inst Math CH-8001 Zurich Switzerland Stockholm Univ Dept Geol Sci S-10691 Stockholm Sweden NASA Cryospher Sci Branch Goddard Space Flight Ctr Greenbelt MD USA

Geophysical mapping and coring of the central Arctic Ocean seafloor provide evidence for repeated occurrences of ice sheet/ice shelf complexes during previous glacial periods. Several ridges and bathymetric highs shallower than present water depths of approximate to 1000m show signs of erosion from deep-drafting (armadas of) icebergs, which originated from thick outlet glaciers and ice shelves. Mapped glacigenic landforms and dates of cored sediments suggest that the largest ice shelf complex was confined to the Amerasian sector of the Arctic Ocean during Marine Isotope Stage (MIS) 6. However, the spatial extent of ice shelves can not be well reconstructed from occasional groundings on bathymetric highs. Therefore, we apply a statistical approach to provide independent support for an extensive MIS 6 ice shelf complex, which previously was inferred only from interpretation of geophysical and geological data. Specifically, we assess whether this ice shelf complex comprises a likely source of the deep-draft icebergs responsible for the mapped scour marks. The statistical modeling is based on exploiting relations between contemporary Antarctic ice shelves and their local physical environments and the assumption that Arctic Ocean MIS6 ice shelves scale similarly. Analyzing ice thickness data along the calving front of contemporary ice shelves, a peak over threshold method is applied to determine sources of deep-drafting icebergs in the Arctic Ocean MIS6 ice shelf complex. This approach is novel to modeling Arctic paleoglacial configurations. Predicted extreme calving front drafts match observed deep-draft iceberg scours if the ice shelf complex is sufficiently large.

关键词： Arctic Ocean ice shelves extreme value theory deep-draft ice berg scours multivariate linear model ICESat data in Arctic-Antarctic analogy approach

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Distinct spatial patterns of brain activity associated with memory storage and search

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NEUROIMAGE 2006年第2期33卷 794-804页

作者： Zarahn, Eric Rakitin, Brian C. Abela, Diane Flynn, Joseph Stern, Yaakov Columbia Univ Coll Phys & Surg Dept Psychiat Taub Inst Cognit Neurosci DIv New York NY 10032 USA Columbia Univ Coll Phys & Surg Dept Neurol New York NY 10032 USA Columbia Univ Coll Phys & Surg Dept Psychiat New York NY 10032 USA

The time it takes for a human participant to decide whether a given stimulus is an element of a remembered set increases approximately linearly with the number of elements in the set. Here we tested for and detected a spatial pattern of brain activity whose magnitude of expression during this memory search process correlates with set size. We then tested the idea that memory search simply involves a reactivation of neurons involved in remembering the set by statistically comparing the patterns of brain activity corresponding to memory search and set size dependent working memory maintenance. These patterns were significantly different, suggesting that memory search and working memory maintenance are mediated by distinct neural mechanisms. (c) 2006 Elsevier Inc. All rights reserved.

关键词： fMRI memory search set size parietal prefrontal premotor working memory multivariate linear model canonical variates analysis

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Professor Heinz Neudecker and matrix differential calculus

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STATISTICAL PAPERS 2024年第4期65卷 2605-2639页

作者： Liu, Shuangzhe Trenkler, Goetz Kollo, Tonu von Rosen, Dietrich Baksalary, Oskar Maria Univ Canberra Fac Sci & Techonol Bruce Australia Dortmund Univ Technol Fac Stat Dortmund Germany Univ Tartu Inst Math & Stat Tartu Estonia Swedish Univ Agr Sci Dept Energy & Technol Uppsala Sweden Linkoping Univ Dept Math Linkoping Sweden Adam Mickiewicz Univ Fac Phys Poznan Poland

The late Professor Heinz Neudecker (1933-2017) made significant contributions to the development of matrix differential calculus and its applications to econometrics, psychometrics, statistics, and other areas. In this paper, we present an insightful overview of matrix-oriented findings and their consequential implications in statistics, drawn from a careful selection of works either authored by Professor Neudecker himself or closely aligned with his scientific pursuits. The topics covered include matrix derivatives, vectorisation operators, special matrices, matrix products, inequalities, generalised inverses, moments and asymptotics, and efficiency comparisons within the realm of multivariate linear modelling. Based on the contributions of Professor Neudecker, several results related to matrix derivatives, statistical moments and the multivariate linear model, which can literally be considered to be his top three areas of research enthusiasm, are particularly included.

关键词： Matrix differential calculus Random matrix Statistical moment Asymptotic distribution Efficiency comparison multivariate linear model

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DIAGNOSTICS FOR IDENTIFYING INFLUENTIAL CASES IN MANOVA

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COMMUNICATIONS IN STATISTICS-THEORY AND METHODS 1990年第7期19卷 2667-2681页

作者： KISH, CW CHINCHILLI, VM AH ROBINS CO INC RICHMOND VA 23261 USA

Measures of influence of multivariate cases on bilinear combinations of the estimated parameter matrix in MANOVA are extended to the generalized MANOVA (GMANOVA) model. The development is based on the confidence region resulting from the likelihood ratio criterion, and is patterned after the development of Cook's univariate DImeasure. Influence measures corresponding to the other 3 common multivariate test criteria are presented. The measures are shown to be invariant with respect to the conditioning matrices associated with the tests of goodness-of-fit (GOF) and general linear hypotheses. The measures provide a method of assessment of whether the outcome of the GOF test results from the general functional form or the impact of some highly influential cases. The methodology is illustrated in 2 numerical examples.

关键词： influence measures multivariate linear model Cook's DI confidence regions Wilk's likelihood ratio Hotelling-Lawley trace Pillai's Trace Roy's largest root goodness-of-fit-test invariance

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Rank estimation of accelerated lifetime models with dependent censoring

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JOURNAL OF THE AMERICAN STATISTICAL ASSOCIATION 2006年第475期101卷 1085-1093页

作者： Peng, Limin Fine, Jason P. Emory Univ Rollins Sch Publ Hlth Dept Biostat Atlanta GA 30322 USA Univ Wisconsin Dept Stat Madison WI 53792 USA Univ Wisconsin Dept Biostat & Med Informat Madison WI 53792 USA

Under independent censoring, estimation of the covariate effects in the accelerated lifetime model may be based on censored data rank tests. Similar rank methodology has been developed with bivariate accelerated lifetime models for dependent censoring but uses artificial censoring, which may lead to substantial information loss. We present a new artificial censoring technique using pairwise ranking and establish the asymptotic properties of a pairwise rank estimator. Simulations show that the pairwise approach achieves large reductions in artificial censoring and large efficiency gains over the existing rank estimator. The simulations evidence moderate efficiency gains under independent censoring over a rank estimator that is semiparametric efficient under independent censoring. An AIDS data analysis illustrates the practical utility of the inferential procedures.

关键词： artificial censoring multivariate linear model pairwise comparison semicompeting risks data semiparametric efficiency U-statistic

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Alternative measures of the explanatory power of general multivariate regression models

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JOURNAL OF MATHEMATICAL SOCIOLOGY 2004年第2期28卷 125-146页

作者： Spiess, M Tutz, G DIW Berlin GSOEP D-14195 Berlin Germany Univ Munich Dept Stat Munich Germany

In this paper R-2-type measures of the explanatory power of multivariate linear and categorical probit models proposed in the literature are reviewed and their deficiencies discussed. It is argued that a measure of the explanatory power should take into account the components which are explicitly modelled when a regression model is estimated while it should be indifferent to components not explicitly modelled. Based on this view three different measures for multivariate probit models are proposed. Results of a simulation study are presented, designed to compare two measures in various situations, to evaluate the BCa bootstrap technique for testing the hypothesis that the corresponding measure is zero, and to calculate approximate confidence intervals. The BC, bootstrap technique turned out to work quite well for a wide range of situations, but may lead to misleading results if the true values of the corresponding measure are close to zero.

关键词： Pseudo-R-2 multivariate linear model multivariate probit model panel model simulation study bootstrap confidence intervals

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A unified matrix model including both CCA and F matrices in multivariate analysis: The largest eigenvalue and its applications

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BERNOULLI 2018年第4B期24卷 3447-3468页

作者： Han, Xiao Pan, Guangming Yang, Qing Nanyang Technol Univ Sch Phys & Math Sci Singapore 637371 Singapore

Let Z(M1xN) = T-1/2 X where (T-1/2)(2) = T is a positive definite matrix and X consists of independent random variables with mean zero and variance one. This paper proposes a unified matrix model Omega = (ZU(2)U(2)(T)Z(T))(-1)ZU(1)U(1)(T)Z(T), where U-1 and U-2 are isometric with dimensions N x N-1 and N x (N - N-2) respectively such that (U1U1)-U-T = IN1, (U2U2)-U-T = I-N - N-2 and (U1U2)-U-T = 0. Moreover, U-1 and U-2 (random or non-random) are independent of Z(M1xN) and with probability tending to one, rank(U-1) = N1 and rank(U-2) = N - N-2. We establish the asymptotic Tracy-Widom distribution for its largest eigenvalue under moment assumptions on X when N-1, N-2 and M-1 are comparable. The asymptotic distributions of the maximum eigenvalues of the matrices used in Canonical Correlation Analysis (CCA) and of F matrices (including centered and non-centered versions) can be both obtained from that of Omega by selecting appropriate matrices U-1 and U-2. Moreover, via appropriate matrices U-1 and U-2, this matrix Omega can be applied to some multivariate testing problems that cannot be done by both types of matrices. To see this, we explore two more applications. One is in the MANOVA approach for testing the equivalence of several high-dimensional mean vectors, where U-1 and U-2 are chosen to be two nonrandom matrices. The other one is in the multivariate linear model for testing the unknown parameter matrix, where U-1 and U-2 are random. For each application, theoretical results are developed and various numerical studies are conducted to investigate the empirical performance.

关键词： canonical correlation analysis F matrix largest eigenvalue MANOVA multivariate linear model random matrix theory Tracy-Widom distribution

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Approximating covariance matrices estimated in multivariate models by estimated auto- and cross-covariances

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JOURNAL OF GEODESY 2010年第6期84卷 383-397页

作者： Koch, K. R. Kuhlmann, H. Schuh, W. -D. Univ Bonn Inst Geodesy & Geoinformat D-53115 Bonn Germany

Quantities like tropospheric zenith delays or station coordinates are repeatedly measured at permanent VLBI or GPS stations so that time series for the quantities at each station are obtained. The covariances of these quantities can be estimated in a multivariate linear model. The covariances are needed for computing uncertainties of results derived from these quantities. The covariance matrix for many permanent stations becomes large, the need for simplifying it may therefore arise under the condition that the uncertainties of derived results still agree. This is accomplished by assuming that the different time series of a quantity like the station height for each permanent station can be combined to obtain one time series. The covariance matrix then follows from the estimates of the auto- and cross-covariance functions of the combined time series. A further approximation is found, if compactly supported covariance functions are fitted to an estimated autocovariance function in order to obtain a covariance matrix which is representative of different kinds of measurements. The simplification of a covariance matrix estimated in a multivariate model is investigated here for the coordinates of points of a grid measured repeatedly by a laserscanner. The approximations are checked by determining the uncertainty of the sum of distances to the points of the grid. To obtain a realistic value for this uncertainty, the covariances of the measured coordinates have to be considered. Three different setups of measurements are analyzed and a covariance matrix is found which is representative for all three setups. Covariance matrices for the measurements of laserscanners can therefore be determined in advance without estimating them for each application.

关键词： multivariate linear model Positive semidefinite function Compactly supported covariance function Auto- and cross-covariance function Monte Carlo simulation Laserscanner

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APPROXIMATE NULL DISTRIBUTION OF THE LARGEST ROOT IN multivariate ANALYSIS

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ANNALS OF APPLIED STATISTICS 2009年第4期3卷 1616-1633页

作者： Johnstone, Iain M. Stanford Univ Dept Stat Stanford CA 94305 USA

The greatest root distribution occurs everywhere in classical multivariate analysis, but even under the null hypothesis the exact distribution has required extensive tables or special purpose software. We describe a simple approximation, based on the Tracy-Widom distribution, that in many cases can be used instead of tables or software, at least for initial screening. The quality of approximation is studied, and its use illustrated in a variety of setttings.

关键词： Canonical correlation characteristic root equality of covariance matrices greatest root statistic largest eigenvalue MANOVA multivariate linear model Tracy-Widom distribution

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