comparing variances of correlated variables a test is proposed for the equality of the variances ofk 2 correlated variables pitman s test fork 2 reduces the null hypothesis to zero correlation between their sum and their difference its extension eliminating nuisance parameters by a bootstrap procedure is valid for any correlation structure between thek normally distributed variables a monte carlo study for several combinations of sample sizes and number of variables is presented comparing the level and power of the new method with previously published tests some nonnormal data are included for which the empirical level tends to be slightly higher than the nominal one the results show that our method is close in power to the asymptotic tests which are extremely sensitive to nonnormality yet it is robust and much more powerful than other robust tests

hermitian completions of band matrices and applications linear fractional descriptions are given for hermitian completions of partial banded matrices with prescribed cholesky signatures an inertia related notion the results are applied to obtain a linear fractional description for completions of partial triangular matrices with prescribed numbers of singular values 1

the comparison of sample covariance matrices using likelihood ratio tests summary the standard method for the comparison of two or more sample covariance matrices is the likelihood ratio test the purpose of the present paper is to show how this test can be made more informative by hierarchically partitioning the test statistic into three components confronted with the comparison of several sample covariance matrices most statisticians will probably consider using the likelihood ratio test seber 1984 p 449 the only reservation that needs to be made about the use of the test is its known sensitivity to nonnormality in the data layard 1974 olson 1974 manly 1986 in the present paper we assume that the data are consistent with multivariate normality the purpose of the present paper is to show that the test statistic for the standard likelihood ratio test can be partitioned into three components these reflect differences between the covariance matrices due to a matrices being proportional b changing variances and c changing correlations the components can be tested for significance

a sharper bonferroni procedure for multiple tests of significance summary a simple procedure for multiple tests of significance based on individual p values is derived this simple procedure is sharper than holm s 1979 sequentially rejective procedure both procedures contrast the ordered p values with the same set of critical values holm s procedure rejects an hypothesis only if its p value and each of the smaller p values are less than their corresponding critical values the new procedure rejects all hypotheses with smaller or equal p values to that of any one found less than its critical value

a comparative study of tests for homogeneity of variances with applications to the outer continental shelf bidding data many of the existing parametric and nonparametric tests for homogeneity of variances and some variations of these tests are examined in this paper comparisons are made under the null hypothesis for robustness and under the alternative for power monte carlo simulations of various symmetric and asymmetric distributions for various sample sizes reveal a few tests that are robust and have good power these tests are further compared using data from outer continental shelf bidding on oil and gas leases

bootstrap critical values for testing homogeneity of covariance matrices abstract bartlett s modified likelihood ratio statistic a is often suggested in multivariate analysis for testing equality of covariance matrices unfortunately the 2 approximation to the null distribution of 2 log is useful only when the data is normally distributed this article presents a pooled bootstrap procedure that replaces the 2 approximation and makes bartlett s statistic a useful tool for data analysis this procedure also applies to most quadratic form test statistics

finite dimensional wiener hopf equations and factorizations of matrices abstract let a be the algebra of all n n matrices over the real or complex numbers let a be the subalgebra of upper triangular matrices and a the subalgebra of strictly lower triangular matrices denote by p the projection of a onto a with kernel a in this paper we investigate the wiener hopf equation p ax y where y a is given and x a is a solution

a note on bootstrapping the sample median efron 1979 1982 in his treatment of the bootstrap discusses its use for estimation of the asymptotic variance of the sample median in the sampling situation of independent and identically distributed random variables with common distribution function f having a positive derivative continuous in a neighborhood of the true median mu the natural conjecture that the bootstrap variance estimator converges almost surely to the asymptotic variance is shown by an example to be false unless a tail condition is imposed on f we prove that such strong convergence does hold under the fairly nonrestrictive condition that e lbrack mid x alpha rbrack 0

estimation for a common correlation coefficient in bivariate normal distributions with missing observations summary the maximum likelihood ml estimate and the restricted or residual maximum likelihood reml estimate are considered for a common correlation coefficient among several bivariate normal distributions with different variances when some observations on either of the variables are missing the use of incomplete data in ml and reml estimation reduces mean squared errors of the correlation estimates reduction is large when the absolute value of a common correlation is large or numbers of paired observations are small an example and some simulation results are given to illustrate the characteristics of the estimates

positive definite completions of partial hermitian matrices abstract the question of which partial hermitian matrices some entries specified some free may be completed to positive definite matrices is addressed it is shown that if the diagonal entries are specified and principal minors composed of specified entries are positive then if the undirected graph of the specified entries is chordal a positive definite completion necessarily exists furthermore if this graph is not chordal then examples exist without positive definite completions in case a positive definite completion exists there is a unique matrix in the class of all positive definite completions whose determinant is maximal and this matrix is the unique one whose inverse has zeros in those positions corresponding to unspecified entries in the original partial hermitian matrix additional observations regarding positive definite completions are made

shift invariant subspaces factorization and interpolation for matrices i the canonical case abstract this article begins a program of classifying subspaces of block matrices that are invariant under multiplication on the right by the algebra of block upper or lower triangular matrices this gives a unifying point of view to several general kinds of factorization problems including lu qr and cholesky factorization a second application is to some matrix completion problems such as that of filling in the upper triangular entries of a partially defined matrix to make the result of norm less than or equal to 1 included is a linear fractional parametrization of the set of all solutions of this problem

comparing treatment variances in repeated measures bioavailability trials in repeated measures bioavailability bioequivalence studies involving two formulations the underlying measurements such as the area under the plasma concentration time curve auc and the maximum plasma concentration cmax typically have a block compound symmetric covariance matrix to test the equality of the variances in such a covariance matrix we derive in this paper an exact test and four asymptotic tests we compare these tests using simulation to determine those that have good performance in terms of controlling the type i error rate as well as providing good power we provide an example to demonstrate the proposed method copyright 1999 john wiley sons ltd

an improved bonferroni procedure for multiple tests of significance summary a modification of the bonferroni procedure for testing multiple hypotheses is presented the method based on the ordered p values of the individual tests is less conservative than the classical bonferroni procedure but is still simple to apply a simulation study shows that the probability of a type i error of the procedure does not exceed the nominal significance level a for a variety of multivariate normal and multivariate gamma test statistics for independent tests the procedure has type i error probability equal to a the method appears particularly advantageous over the classical bonferroni procedure when several highly correlated test statistics are involved

spectral models for covariance matrices summary a new model for the simultaneous eigenstructure of multiple covariance matrices is proposed the model is much more flexible than existing models and subsumes most of them as special cases a fisher scoring algorithm for computing maximum likelihood estimates of the parameters under normality is given asymptotic distributions of the estimators are derived under normality as well as under arbitrary distributions having finite fourth order cumulants special attention is given to elliptically contoured distributions likelihood ratio tests are described and sufficient conditions are given under which the test statistics are asymptotically distributed as chi squared random variables procedures are derived for evaluating bartlett corrections under normality some conjectures made by flury 1988 are verified others are refuted a small simulation study of the adequacy of the bartlett correction is described and the new procedures are illustrated on two datasets

a general theory for jackknife variance estimation the delete 1 jackknife is known to give inconsistent variance estimators for nonsmooth estimators such as the sample quantiles this well known deficiency can be rectified by using a more general jackknife with d the number of observations deleted depending on a smoothness measure of the point estimator our general theory explains why jackknife works or fails it also shows that i for sufficiently smooth estimators the jackknife variance estimators with bounded d are consistent and asymptotically unbiased and ii for nonsmooth estimators d has to go to infinity at a rate explicitly determined by a smoothness measure to ensure consistency and asymptotic unbiasedness improved results are obtained for several classes of estimators in particular for the sample p quantiles the jackknife variance estimators with d satisfying n 1 2 d rightarrow 0 and n d rightarrow infty are consistent and asymptotically unbiased

a stagewise rejective multiple test procedure based on a modified bonferroni test summary simes 1986 has proposed a modified bonferroni procedure for the test of an overall hypothesis which is the combination of n individual hypotheses in contrast to the classical bonferroni procedure it is not obvious how statements about individual hypotheses are to be made for this procedure in the present paper a multiple test procedure allowing statements on individual hypotheses is proposed it is based on the principle of closed test procedures marcus peritz gabriel 1976 and controls the multiple level a

testing for variance homogeneity of correlated variables summary four large sample methods are proposed for testing the equality of the variances of a p variate normal population two of the methods are asymptotically distribution robust against departures from the multinormality assumption of the parent population in applying the tests no restrictive assumptions are made about the off diagonal structure of the covariance matrics and no random splitting of the sample into subsamples is required an example of the use of the tests is provided

on a theorem of castelnuovo and the equations defining space curves introduction 491 w notat ion and conventions 493 w a regularity theorem 494 w a rationality theorem 498 w the existence of secant lines 501 w open problems 505 references 506

a bootstrap comparison of genetic covariance matrices summary an important problem in evolutionary biology is how population bottlenecks affect additive genetic variance va when only additive effects are present a decrease in va is predicted whereas with many forms of nonadditive interactions an increase is predicted we present an analysis of the changes in variance that occur as a result of a bottleneck in a population of the red flour beetle tribolium castaneum in this experiment the additive genetic covariance matrix for four traits in two sexes was measured in the ancestral population the population then went through a bottleneck and the genetic covariance matrix was again measured these data were analyzed by first examining the univariate estimates for va and second examining the covariance matrices for the univariate measures the statistic we used was the change in va for the multivariate comparisons we used three measures the difference in rank the difference in the number of positive eigenvalues to test changes in the number of heritable traits a multivariate equivalent of bartlett s statistic to test changes in the size of the matrices and a modification of mantel s correlation to test for changes in the correlation structure of the matrices because the distribution of all of these measures is complex and dependent on the specifics of the particular matrices being compared we chose to use a bootstrap analysis to test for significance for females we found that following the population bottleneck va was significantly decreased and that the correlation structure of the matrix was significantly changed for males we found that following the population bottleneck va significantly increased but that the correlation structure was unchanged

an inertia formula for hermitian matrices with sparse inverses abstract let a mn be a nonsingular hermitian matrix let g be a chordal graph on vertices 1 n and suppose that g a 1 g then the inertia of a may be expressed in a simple way in terms of the inertias of those principal submatrices of a corresponding to certain readily identifiable sets of vertices of g specifically we prove that the inertia i a satisfies the identity i a c i a s m i a in which c denotes the collection of maximal cliques of g s denotes the collection of minimal vertex separators of g and m is the multiplicity of a separator

a comparison of two modified bonferroni procedures summary it is shown that hommel s 1988 modified bonferroni procedure is at least as powerful as hochberg s 1988 procedure and in general more powerful

evaluating agreement with a gold standard in method comparison studies we develop a statistical model for method comparison studies where a gold standard is present and propose a measure of agreement this measure can be interpreted as a population correlation coefficient in a constrained bivariate model an estimator of this coefficient is proposed and its statistical properties explored applications of the new methodology to data from the medical literature are presented

factorizations of operator matrices abstract this note gives explicit factorizations of a 2 2 operator matrix as a product of an upper triangular operator matrix and an involutory unitary or j unitary operator matrix a pattern is given for construction of factorizations of this kind

approximation of linear operators in the 2 norm abstract the problems of approximating linear operators in the 2 induced norm which are 1 finite dimensional unstructured and 2 infinite dimensional structured hankel have been solved the solutions of these two problems exhibit striking similarities these similarities suggest the search of a unifying framework for the approximation of linear operators in the 2 induced norm

a maximum entropy principle in the general framework of the band method abstract a maximum entropy principle is developed in the general framework of the band method known maximum entropy principles appear as corollaries and new ones are derived

comparing the variances of dependent groups recently several new attempts have been made to find a robust method for comparing the variances ofj dependent random variables however empirical studies have shown that all of these procedures can give unsatisfactory results this paper examines several new procedures that are derived heuristically one of these procedures was found to perform better than all of the robust procedures studied here and so it is recommended for general use

dualistic differential geometry of positive definite matrices and its applications to related problems abstract analysis on the set of positive definite matrices is involved in many engineering problems we develop a differential geometric theory of the set of positive definite matrices by means of a specific class of connections introduced into it consequently various dualistic aspects of the set are elucidated next using the theory we derive new results and interesting geometrical interpretations for matrix approximation positive definite matrix completion and linear matrix inequality problems

a note on extensions of band matrices with maximal and submaximal invertible blocks abstract completions or extensions of band matrices with invertible maximal and submaximal blocks are considered linear fractional descriptions for invertible minimal rank and lu ul decomposable extensions are derived self adjoint and toeplitz cases are included

extensions of band matrices with band inverses abstract let r ij be a given set of i j matrices for i j 1 n and i j m where 0 m n 1 necessary and sufficient conditions are established for the existence and uniqueness of an invertible block matrix f ij i j 1 n such that f ij r ij for i j m f admits either a left or right block triangular factorization and f 1 ij 0 for i j m the well known conditions for an invertible block matrix to admit a block triangular factorization emerge for the particular choice m n 1 the special case in which the given r ij are positive definite in an appropriate sense is explored in detail and an inequality which corresponds to burg s maximal entropy inequality in the theory of covariance extension is deduced the block toeplitz case is also studied

irreducibility and components rigid in moduli of the hilbert scheme of smooth curves denote by mathcal h d g r the hilbert scheme of smooth curves that is the union of components whose general point corresponds to a smooth irreducible and non degenerate curve of degree d and genus g in mathbb p r a component of mathcal h d g r is rigid in moduli if its image under the natural map pi mathcal h d g r dashrightarrow mathcal m g is a one point set in this note we provide a proof of the fact that mathcal h d g r has no components rigid in moduli for g 0 and r 3 from which it follows that the only smooth projective curves embedded in mathbb p 3 whose only deformations are given by projective transformations are the twisted cubic curves in case r ge 4 we also prove the non existence of a component of mathcal h d g r rigid in moduli in a certain restricted range of d g 0 and r in the course of the proofs we establish the irreducibility of mathcal h d g 3 beyond the range which has been known before

strictly contractive and positive completions for block matrices abstract this paper deals with the following two problems of completion of block matrices 1 let a ij i j q be given matrices find additional matrices a ij such that the block matrix a a ij n i 1 m j 1 has norm less than one 2 let b ij j i q be given matrices find additional b ij such that the completion b b ij n i j 1 is positive definite the analysis is based on a study of certain linear fractional maps this approach leads to an explicit description of all solutions by linear fractional maps of which the coefficients are given directly in terms of the given data the maximum entropy principle appears as a corollary of a general formula for the determinant of a completion special attention is paid to the toeplitz case

comparison of correlated correlations we consider a problem where k highly correlated variables are available each being a candidate for predicting a dependent variable only one of the k variables can be chosen as a predictor and the question is whether there are significant differences in the quality of the predictors we review several tests derived previously and propose a method based on the bootstrap the motivating medical problem was to predict 24 hour proteinuria by protein creatinine ratio measured at either 08 00 12 00 or 16 00 the tests which we discuss are illustrated by this example and compared using a small monte carlo study

simulated power functions on etudie par simulation la puissance de huit tests contre des alternatives comportant le meme parametre de puissance inconnu

unitary dilation approach to contractive matrix completion in this paper a new method to construct all contractive matrix completion to a block triangular matrix is given this new method uses only elementary matrix operations and has the advantages of better numerical reliability and ease in implementation on computers

schur analysis of some completion problems abstract the complete schur analysis for contractive block matrices is given together with its connections with positiveness these are used for reobtaining some results on completion problems sometimes in a more general setting these completion problems include positive completions contractive and isometric completions and the realizability of upper triangular contractions

plane answers to complex questions the theory of linear models providing a wide ranging introduction to the use of linear models in analyzing data this text presents a vector space and projections approach to the subject the topics covered include anova estimation hypothesis testing multiple comparison regression analysis and experimental design also covered are testing for lack of fit models with singular covariance matrices variance component estimation best linear prediction colinearity and variable selection

maximum entropy kernels for system identification bayesian nonparametric approaches have been recently introduced in system identification scenario where the impulse response is modeled as the realization of a zero mean gaussian process whose covariance kernel has to be estimated from data in this scheme quality of the estimates crucially depends on the parametrization of the covariance of the gaussian process a family of kernels that have been shown to be particularly effective in the system identification framework is the family of diagonal correlated dc kernels maximum entropy properties of a related family of kernels the tuned correlated tc kernels have been recently pointed out in the literature in this technical note we show that maximum entropy properties indeed extend to the whole family of dc kernels the maximum entropy interpretation can be exploited in conjunction with results on matrix completion problems in the graphical models literature to shed light on the structure of the dc kernel in particular we prove that the dc kernel admits a closed form factorization inverse and determinant these results can be exploited both to improve the numerical stability and to reduce the computational complexity associated with the computation of the dc estimator

robust estimation of dispersion matrices and principal components abstract this paper uses monte carlo methods to compare the performances of several robust procedures for estimating a correlation matrix and its principal components the estimators are formed either from separate bivariate analyses or by simultaneous manipulation of all variables by using techniques such as multivariate trimming and m estimation the m estimators stand up exceptionally well they and the multivariate trimming procedure are especially effective at estimating the principal components including a near singularity however the m estimators can break down relatively easily when the dimensionality is large and the outliers are asymmetric with missing data the element wise approach becomes more attractive

comparison of asymptotically distribution free procedures for the analysis of complete blocks abstract the friedman test or sign test when k 2 depends entirely on within block rankings in a recent paper quade 1979 attempted to provide a test with more power than the friedman test by considering a k sample extension of the wilcoxon signed ranks test this is done by taking advantage of the between block information a third way to approach the problem and still retain both the within and between block information is first to transform all the observations to ranks from 1 to bk b blocks and k treatments and then to apply the parametric f test to the ranks this approach is shown to be asymptotically distribution free under suitable conditions computer simulation results indicate that this procedure is both robust and powerful for small sample sizes

a family of distributions for modelling non elliptically symmetric multivariate data summary the paucity of distributions that can be used as an aid to modelling the structure of multivariate data is a recognized limitation we present a relatively simple family of distributions for use in describing data which are not elliptically symmetric properties of the family are discussed and it is shown that multivariate pareto burr and logistic distributions are special cases in addition simulation methods for applications in monte carlo robustness studies are outlined and finally the fit to a data set on uranium exploration is compared to that obtained using the usual bivariate normal distribution

qualms about bootstrap confidence intervals abstract the percentile method and bias corrected percentile method of efron 1981 1982 are discussed when these methods are used to construct nonparametric confidence intervals for the variance of a normal distribution the coverage probabilities are substantially below the nominal level for small to moderate samples this is due to the inapplicability of assumptions underlying the methods these assumptions are difficult or impossible to check in the complicated situations for which the bootstrap is intended therefore bootstrap confidence intervals should be used with caution in complex problems

tests for precision and accuracy of multiple measuring devices we propose a multivariate linear model for analyzing measurement device data that follow grubbs s model the multivariate model generates the well known regression multiple correlation test for equality of device variances it also yields a new simultaneous test for equality of variances and biases the multivariate model readily permits testing of interesting subhypotheses involving the variances and biases we examine the power of these tests for onedimensional hypotheses compare them to tests proposed by grubbs and propose generalizations of grubbs s tests we also consider extensions of grubbs s model that allow similar regression and simultaneous tests to be performed

a nonparametric prior for simultaneous covariance estimation in the modeling of longitudinal data from several groups appropriate handling of the dependence structure is of central importance standard methods include specifying a single covariance matrix for all groups or independently estimating the covariance matrix for each group without regard to the others but when these model assumptions are incorrect these techniques can lead to biased mean effects or loss of efficiency respectively thus it is desirable to develop methods to simultaneously estimate the covariance matrix for each group that will borrow strength across groups in a way that is ultimately informed by the data in addition for several groups with covariance matrices of even medium dimension it is difficult to manually select a single best parametric model among the huge number of possibilities given by incorporating structural zeros and or commonality of individual parameters across groups in this paper we develop a family of nonparametric priors using the matrix stick breaking process of dunson et al 2008 that seeks to accomplish this task by parameterizing the covariance matrices in terms of the parameters of their modified cholesky decomposition pourahmadi 1999 we establish some theoretic properties of these priors examine their effectiveness via a simulation study and illustrate the priors using data from a longitudinal clinical trial

simple robust tests for scale differences in paired data summary the classical statistical test for assessing the difference between the two scale parameters in paired data due to pitman 1939 and morgan 1939 is not robust this paper explores robust alternatives to pitman s test using the framework of the one sample t test the asymptotic behaviour of the tests under the null hypothesis is examined in general it is easy to construct tests that are asymptotically distribution free provided the data come from a symmetric bivariate distribution versions of these tests that do not require symmetry are derived monte carlo simulation experiments are done to assess small sample behaviour and power characteristics the tests are applied to a small data set comparing cancerous and noncancerous lungs

norm preserving dilations and their applications to optimal error bounds the problem is given a b c to find d such that left begin array 20 c a c b d end array right leqq mu here we deal with hilbert space operators a b and c ar

combined maximum likelihood estimates for the equicorrelation coefficient combined maximum likelihood estimates for equicorrelation covariance matrices are considered the case of a common equicorrelation rho and possibly different standard deviations sigma 1 sigma k among k experimental groups is examined first and the estimation of rho sigma 1 sigma k is discussed second under the assumption of a common standard deviation and possibly different equicorrelations the estimation of rho 1 rho k sigma is considered in each case maximum likelihood solutions and corresponding large sample variances are presented

generalized burr pareto logistic distributions with applications to a uranium exploration data set a distribution developed previously cook and johnson 1981 has been generalizedu sing morgenstern s distribution to include more useful dependence structures the new distribution provides better fits to a uranium survey data set the effectiveness of atkinson s 1982 graphical aid for discriminating separate models is also demonstrated in comparing the usual normal model with the new distribution having normal marginal distributions

the maximum distance problem and band sequences abstract we solve the following problem for 1 j k n and j k m let a jk be a given complex number with a kj jk we wish to find linearly independent vectors x 1 x n such that x k x j a jk for j k m and such that the distance from x k to the linear span of x 1 x k 1 is maximal for 2 k n we construct and characterize all such sequences of vectors our solution leads naturally to the class of m band sequences of vectors in an inner product space we study these sequences and characterize their equivalence classes under unitary transformations