a test for a specific principal component of a correlation matrix abstract in the application of principal components analysis it is common to replace an observed sample principal component vector by another vector closely resembling the sample vector but which is easier to use or interpret a useful test of hypothesis in this case is one that specifies the true ith principal component in this article we obtain an asymptotically chi squared procedure suitable for testing such a hypothesis when the principal components analysis is performed on a correlation matrix the procedure easily extends to a principal components analysis based on m estimates of scatter

antenna array pattern synthesis via convex optimization we show that a variety of antenna array pattern synthesis problems can be expressed as convex optimization problems which can be numerically solved with great efficiency by recently developed interior point methods the synthesis problems involve arrays with arbitrary geometry and element directivity constraints on far and near field patterns over narrow or broad frequency bandwidth and some important robustness constraints we show several numerical simulations for the particular problem of constraining the beampattern level of a simple array for adaptive and broadband arrays

improved estimation of eigenvalues and eigenvectors of covariance matrices using their sample estimates the problem of estimating the eigenvalues and eigenvectors of the covariance matrix associated with a multivariate stochastic process is considered the focus is on finite sample size situations whereby the number of observations is limited and comparable in magnitude to the observation dimension using tools from random matrix theory and assuming a certain eigenvalue splitting condition new estimators of the eigenvalues and eigenvectors of the covariance matrix are derived that are shown to be consistent in a more general asymptotic setting than the traditional one indeed these estimators are proven to be consistent not only when the sample size increases without bound for a fixed observation dimension but also when the observation dimension increases to infinity at the same rate as the sample size numerical evaluations indicate that the estimators have an excellent performance in small sample size scenarios where the observation dimension and the sample size are comparable in magnitude

combining information from independent sources through confidence distributions this paper develops new methodology together with related theories for combining information from independent studies through confidence distributions a formal definition of a confidence distribution and its asymptotic counterpart i e asymptotic confidence distribution are given and illustrated in the context of combining information two general combination methods are developed the first along the lines of combining p values with some notable differences in regard to optimality of bahadur type efficiency the second by multiplying and normalizing confidence densities the latter approach is inspired by the common approach of multiplying likelihood functions for combining parametric information the paper also develops adaptive combining methods with supporting asymptotic theory which should be of practical interest the key point of the adaptive development is that the methods attempt to combine only the correct information downweighting or excluding studies containing little or wrong information about the true parameter of interest the combination methodologies are illustrated in simulated and real data examples with a variety of applications

the asymptotic variance matrices of the sample correlation matrix in elliptical and normal situations and their proportionality abstract this note presents the asymptotic variance matrices of the sample correlation matrix in elliptical and normal situations and generalizes the scalar proportionality result to the multivariate case

on the removal of skewness by transformation summary we suggest empirical transformations for removing most of the skewness of an asymmetric statistic these transformations may be used as the basis for accurate confidence procedures or hypothesis tests and can be employed in conjunction with either a normal approximation or the bootstrap our approach differs from those of other researchers in that the transformation is monotone and invertible in this paper we suggest elementary but general transformations for eliminating skewness from the distribution of a studentized statistic the transformations are monotone and have simple explicit inversion formulae and so are readily applied to confidence interval problems when the transformations are used in conjunction with the bootstrap they can produce confidence intervals with particularly low levels of coverage error that confidence intervals for symmetric statistics have lower coverage error than their counterparts in the case of asymmetry may be deduced from edgeworth expansions the first term in an expansion of size n 1 2 where n is the sample size describes the error in the usual normal approximation and is due entirely to skewness if it can be eliminated for example by transformation then the normal approxima tion will be in error by only o n 1 thus our method involves transforming one statistic into another where the distribution is virtually symmetric applying the normal approximation or the bootstrap to the new statistic and then regaining the asymmetry of the original problem by inverse transformation at the same time retain ing the high coverage accuracy conferred by applying confidence interval procedures to a symmetric statistic

score confidence intervals and sample sizes for stratified comparisons of binomial proportions in a series of articles gart and nam construct the efficient score tests and confidence intervals with or without skewness correction for stratified comparisons of binomial proportions on the risk difference relative risk and odds ratio effect metrics however the stratified score methods and their properties are not well understood we rederive the efficient score tests which reveals their theoretical relationship with the contrast based score tests and provides a basis for adapting the method by using other weighting schemes the inverse variance weight is optimal for a common treatment effect in large samples we explore the behavior of the score approach in the presence of extreme outcomes when either no or all subjects in some strata are responders and provide guidance on the choice of weights in the analysis of rare events the score method is recommended for studies with a small number of moderate or large sized strata a general framework is proposed to calculate the asymptotic power and sample size for the score test in superiority noninferiority and equivalence clinical trials or case control studies we also describe a nearly exact procedure that underestimates the exact power but the degree of underestimation can be controlled to a negligible level the proposed methods are illustrated by numerical examples

on the multivariate asymptotic distribution of sequential chi square statistics the multivariate asymptotic distribution of sequential chi square test statistics is investigated it is shown that a when sequential chi square statistics are calculated for nested models on the same data the statistics have an asymptotic intercorrelation which may be expressed in closed form and which is in many cases quite high and b sequential chi squaredifference tests are asymptotically independent some monte carlo evidence on the applicability of the theory is provided

covariance reducing models an alternative to spectral modelling of covariance matrices we introduce covariance reducing models for studying the sample covariance matrices of a random vector observed in different populations the models are based on reducing the sample covariance matrices to an informational core that is sufficient to characterize the variance heterogeneity among the populations they possess useful equivariance properties and provide a clear alternative to spectral models for covariance matrices

partial common principal component subspaces summary we consider the principal components analysis of g groups of m variables for those situations in which for each group the first k principal components account for most of the total variability of observations in that group if the set of these gk principal component vectors spans a space of dimension r where r is less than m then it will be possible simultaneously to reduce the dimensionality for all groups from m to r while retaining most of the within group variability methods are already available for determining if r k in which case the g groups have a common principal component subspace in this paper we develop a general procedure for determining if r s for arbitrary s this can then be used repeatedly to determine r when r k

on the treatment of correlation structures as covariance structures abstract necessary and sufficient conditions are provided for minimum discrepancy methods intended for covariance structures to retain their asymptotic properties in the analysis of correlation structures examples of correlation structures satisfying these conditions are considered and alternative discrepancy functions which are always appropriate for correlation structures under normality assumptions are discussed

raw data maximum likelihood estimation for common principal component models a state space approach the specifications of state space model for some principal component related models are described including the independent group common principal component cpc model the dependent group cpc model and principal component based multivariate analysis of variance some derivations are provided to show the equivalence of the state space approach and the existing wishart likelihood approach for each model a numeric example is used to illustrate the state space approach in addition a simulation study is conducted to evaluate the standard error estimates under the normality and nonnormality conditions in order to cope with the nonnormality conditions the robust standard errors are also computed finally other possible applications of the state space approach are discussed at the end

two generalizations of the common principal component model summary under the common principal component model the covariance matrices ti of k populations are assumed to have identical eigenvectors that is the same orthogonal matrix diagonalizes all ti simultaneously this paper modifies the common principal component model by assuming that only q out of p eigenvectors are common to all ti while the remaining p q eigenvectors are specific in each group this is called a partial common principal component model a related modification assumes that q eigenvectors of each matrix span the same subspace a problem that was first considered by krzanowski 1979 for both modifications this paper derives the normal theory maximum likelihood estimators it is shown that approximate maximum likelihood estimates can easily be computed if estimates of the ordinary common principal component model are available the methods are illustrated by numerical examples

real hyperbolic on the outside complex hyperbolic on the inside the rank one symmetric spaces of negative curvature come in three infinite families real hyperbolic space h complex hyperbolic space ch and quaternionic hyperbolic space qh the cayley plane is the remaining example aside from the obvious embeddings h ch qh the three geometries seem fairly unrelated to each other for instance h admits non arithmetic lattices in all dimensions grp while qh only admits arithmetic lattices grs see c for a related result the question of non arithmetic lattices in ch is a basic unsolved problem dm for a representation theoretic comparison of discrete subgroups in the different rank one spaces see sh in this paper we make a new connection between h and ch we construct a closed hyperbolic 3 manifold which as a diffeomorphic copy is the ideal boundary of a complex hyperbolic 4 manifold up to index 2 the isometry group of ch is pu 2 1 the group of complex projective automorphisms of the unit ball in c the ideal boundary of ch is the unit 3 sphere s a spherical cr structure on a 3 manifold is a system of coordinate charts into s whose transition functions are restrictions of elements of pu 2 1 while plenty of closed seifert fibered manifolds admit spherical cr structures kt our example gives the only known spherical cr structure on a closed hyperbolic 3 manifold

the asymptotic covariance matrix of sample correlation coefficients under general conditions abstract a general matrix expression for the asymptotic covariance matrix of correlation coefficients is derived it is applicable when the data are drawn from any distribution with finite fourth order moments the result is specialized to the cases where the data have a distribution from the elliptical class and where the sample covariance matrix has a noncentral wishart distribution

robustness and efficiency properties of scatter matrices the robustness and efficiency properties of likelihood ratio tests for functions of the population covariance matrix are studied an alternative class of tests based upon affineinvariant m estimates of scatter is proposed whenever the function of the covariance matrix is invariant under a common scale change for such inferences it is shown that a single scalar valued index of efficiency is sufficient

a class of asymptotic tests for principal component vectors in this paper the hypothesis that a set of vectors lie in the subspace spanned by a prescribed subset of the principal component vectors for a normal population is considered a class of invariant asymptotic tests based on the sample covariance matrix is derived tests in this class are shown to be consistent and their local power functions are given the arguments used in deriving the class of tests are not heavily dependent on the assumption of normality nor on the use of the sample covariance matrix the results are shown to generalize when the procedures are based on any affine invariant m estimate of scatter and when the population is elliptical

a joint optimization approach for the synthesis of large 4 d heterogeneous antenna arrays due to the inherent nonlinearity in the synthesis of large four dimensional 4 d heterogeneous arrays conventional pattern synthesis techniques are no longer suitable in this paper an effective joint optimization approach based on the combination of convex programming cp and differential evolution de algorithm is proposed by designing suitable time sequences of a 4 d heterogeneous array with uniform static amplitude excitations the original synthesis problem is simplified into two independent small problems which can be easily solved in two steps respectively in the first step the approach takes definite advantages from the convexity of the problem with respect to the static phase excitations and the switch on duration times these optimization variables become known and are fixed in the second step then the de algorithm is used to optimize the switch on time instants owing to the efficiency of the cp only 2 3 of the optimization variables need be solved in the first step while even fewer number of optimization variables are solved in the second step consequently the joint optimization of cp and de algorithm significantly improves the overall synthesis efficiency two numerical examples are presented to demonstrate the effectiveness of the proposed approach

analysis of covariance structures under elliptical distributions abstract this article examines the adjustment of normal theory methods for the analysis of covariance structures to make them applicable under the class of elliptical distributions it is shown that if the model satisfies a mild scale invariance condition and the data have an elliptical distribution the asymptotic covariance matrix of sample covariances has a structure that results in the retention of many of the asymptotic properties of normal theory methods if a scale adjustment is applied the likelihood ratio tests of fit have the usual asymptotic chi squared distributions difference tests retain their property of asymptotic independence and maximum likelihood estimators retain their relative asymptotic efficiency within the class of estimators based on the sample covariance matrix an adjustment to the asymptotic covariance matrix of normal theory maximum likelihood estimators for elliptical distributions is provided this adjustment is particularly simple in models for patterned covariance or co

linear matrix inequality formulation of spectral mask constraints with applications to fir filter design the design of a finite impulse response fir filter often involves a spectral mask that the magnitude spectrum must satisfy the mask specifies upper and lower bounds at each frequency and hence yields an infinite number of constraints in current practice spectral masks are often approximated by discretization but in this paper we derive a result that allows us to precisely enforce piecewise constant and piecewise trigonometric polynomial masks in a finite and convex manner via linear matrix inequalities while this result is theoretically satisfying in that it allows us to avoid the heuristic approximations involved in discretization techniques it is also of practical interest because it generates competitive design algorithms based on interior point methods for a diverse class of fir filtering and narrowband beamforming problems the examples we provide include the design of standard linear and nonlinear phase fir filters robust chip waveforms for wireless communications and narrowband beamformers for linear antenna arrays our main result also provides a contribution to system theory as it is an extension of the well known positive real and bounded real lemmas

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

regularized gaussian discriminant analysis through eigenvalue decomposition abstract friedman proposed a regularization technique rda of discriminant analysis in the gaussian framework rda uses two regularization parameters to design an intermediate classifier between the linear the quadratic the nearest means classifiers in this article we propose an alternative approach called edda that is based on the reparameterization of the covariance matrix k of a group gk in terms of its eigenvalue decomposition k k d k a k d k where k specifies the volume of density contours of gk the diagonal matrix of eigenvalues specifies its shape the eigenvectors specify its orientation variations on constraints concerning volumes shapes orientations k a k and d k lead to 14 discrimination models of interest for each model we derived the normal theory maximum likelihood parameter estimates our approach consists of selecting a model by minimizing the sample based estimate of future misclassification risk by cross validation numerical experiments on simulated and rea

a hierarchical eigenmodel for pooled covariance estimation summary although the covariance matrices corresponding to different populations are unlikely to be exactly equal they can still exhibit a high degree of similarity for example some pairs of variables may be positively correlated across most groups whereas the correlation between other pairs may be consistently negative in such cases much of the similarity across covariance matrices can be described by similarities in their principal axes which are the axes that are defined by the eigenvectors of the covariance matrices estimating the degree of across population eigenvector heterogeneity can be helpful for a variety of estimation tasks for example eigenvector matrices can be pooled to form a central set of principal axes and to the extent that the axes are similar covariance estimates for populations having small sample sizes can be stabilized by shrinking their principal axes towards the across population centre to this end the paper develops a hierarchical model and estimation procedure for pooling principal axes across several populations the model for the across group heterogeneity is based on a matrix valued antipodally symmetric bingham distribution that can flexibly describe notions of centre and spread for a population of orthogonal matrices

optimal array pattern synthesis using semidefinite programming we present algorithms for the solution of two problems in array pattern synthesis the first is the design of nonuniform arrays with a desired magnitude response the second is that of robust design i e design in the presence of uncertainties constraints such as power limitation can be addressed with both problems the algorithms that we present are based on semidefinite programming for which efficient software is readily available we present examples that illustrate the effectiveness of our approach

optimal design of iir digital filters with robust stability using conic quadratic programming updates in this paper minimax design of infinite impulse response iir filters with prescribed stability margin is formulated as a conic quadratic programming cqp problem cqp is known as a class of well structured convex programming problems for which efficient interior point solvers are available by considering factorized denominators the proposed formulation incorporates a set of linear constraints that are sufficient and near necessary for the iir filter to have a prescribed stability margin a second order cone condition on the magnitude of each update that ensures the validity of a key linear approximation used in the design is also included in the formulation and eliminates a line search step collectively these features lead to improved designs relative to several established methods the paper then moves on to extend the proposed design methodology to quadrantally symmetric two dimensional 2 d digital filters simulation results for both one dimensional 1 d and 2 d cases are presented to illustrate the new design algorithms and demonstrate their performance in comparison with several existing methods

asymptotics of eigenprojections of correlation matrices with some applications in principal components analysis summary the asymptotic distribution of an eigenprojection for a sample correlation matrix is obtained in particular it is shown that the rank of the asymptotic covariance matrix depends on distributional parameters in a somewhat complicated manner the results obtained in this paper can be used to determine this rank some applications of the asymptotic distribution of these eigenprojections to inferential problems involving principal components subspaces are given many statistical applications involve the analysis of a sample covariance matrix or sample correlation matrix as a result of the complexity of the exact joint distribution of the elements of these matrices most of these applications utilise the corresponding asymptotic distributions since the asymptotic distribution for the sample covariance matrix is somewhat simpler than that of the sample correlation matrix covariance matrix based analyses have received more attention in the literature unfortunately in practice analyses based on correlation matrices are more prevalent than those based on covariance matrices as a result in those situations in which correlation matrix based procedures have not yet been developed practitioners will often use procedures that have been developed for a corresponding analysis based on a sample covariance matrix this results in a procedure which at best can only be described as a crude approximation in this paper we review some of the asymptotic properties of the sample correlation matrix in particular we give the asymptotic distribution of eigenprojections computed from the sample correlation matrix the associated asymptotic covariance matrix of this distribution is obtained and its rank is determined throughout this paper we focus on the sample correlation matrix computed from the usual sample covariance matrix however it should be noted that the asymptotic results obtained can be easily generalised to the correlation matrix computed from any asymptotically normal estimate of the covariance matrix we also consider some applications which utilise eigenprojections of correlation matrices these are some inference problems in a principal components analysis of a correlation matrix in some of these situations straightforward generalisations of the covariance matrix based test statistics are not possible because of some complications associated with the rank of the asymptotic covariance matrix of the eigenprojections in these cases

pattern synthesis with multipoint accurate array response control in this paper the problem of pattern synthesis with antenna arrays is addressed and new approaches based on the recently developed accurate array response control textrm a 2 textrm rc algorithm are presented it is shown that the array weight vector obtained by the textrm a 2 textrm rc algorithm to control the normalized response at a single direction in each step belongs to a specific set thus an appropriate weight vector chosen from the intersection of weight vector sets corresponding to the desired responses at multiple directions is capable of simultaneously controlling those responses this results in the so called multipoint accurate array response control textrm ma 2 textrm rc algorithm moreover in order to avoid possible beam axis shift in pattern synthesis a modified textrm ma 2 textrm rc textrm m 2 textrm a 2 textrm rc algorithm is proposed by imposing a derivative constraint on the direction of beam axis representative numerical examples are provided to demonstrate the effectiveness of the proposed textrm ma 2 textrm rc and textrm m 2 textrm a 2 textrm rc algorithms for multipoint responses control and pattern synthesis

common principal component subspaces in two groups summary one important practical application of principal component analysis is to reduce a large number of variables say p to a smaller number m by making use of the first m principal components this technique can easily be extended to two or more groups if the subspaces spanned by the first m principal components are the same for all groups in this paper we develop an approximate procedure for testing such a hypothesis of common subspaces when two groups are involved the adequacy of the approximation is investigated by a simulation and the method is illustrated by a numerical example

the geometry of the eisenstein picard modular group the eisenstein picard modular group rm pu 2 1 mathbb z omega is defined to be the subgroup of rm pu 2 1 whose entries lie in the ring mathbb z omega where omega is a cube root of unity this group acts isometrically and properly discontinuously on bf h 2 mathbb c that is on the unit ball in mathbb c 2 with the bergman metric we construct a fundamental domain for the action of rm pu 2 1 mathbb z omega on bf h 2 mathbb c which is a 4 simplex with one ideal vertex as a consequence we elicit a presentation of the group see theorem 5 9 this seems to be the simplest fundamental domain for a finite covolume subgroup of rm pu 2 1

calibrating confidence coefficients abstract two approaches for dealing with the problem of poor coverage probabilities of certain standard confidence intervals are proposed the first is a recommendation that the actual coverage be estimated directly from the data and its value reported in addition to the nominal level this is achieved through a combination of computer simulation and density estimation the asymptotic validity of the procedure is proved for a number of common situations a classical example is the nonparametric estimation of the variance of a population using the normal theory interval here it is shown that the estimated coverage probability consistently estimates the true coverage probability if the population distribution possesses a finite sixth moment the second approach is more traditional it is a procedure for modifying an interval to yield improved coverage properties given a confidence interval its estimated coverage probability obtained in the first approach is used to alter the nominal level of the interval

on the asymptotic behavior of the sample estimates of eigenvalues and eigenvectors of covariance matrices this paper analyzes the asymptotic behavior of the sample estimators of the eigenvalues and eigenvectors of covariance matrices rather than considering traditional large sample size asymptotics in this paper the focus is on limited sample size situations whereby the number of available observations is comparable in magnitude to the observation dimension using tools from random matrix theory the asymptotic behavior of the traditional sample estimates is investigated under the assumption that both the sample size and the observation dimension tend to infinity while their quotient converges to a positive quantity assuming that an asymptotic eigenvalue splitting condition is fulfilled closed form asymptotic expressions of these estimators are derived proving inconsistency of the traditional sample estimators in these asymptotic conditions the derived expressions are shown to provide a valuable insight into the behavior of the sample estimators in the small sample size regime

asymptotic theory for common principal component analysis on donne la distribution asymptotique des estimations du maximum de vraisemblance et on utilise les resultats pour obtenir des tests d egalite des vecteurs propres et de redondance de p q composantes principales

newton algorithms for analytic rotation an implicit function approach in this paper implicit function based parameterizations for orthogonal and oblique rotation matrices are proposed the parameterizations are used to construct newton algorithms for minimizing differentiable rotation criteria applied to m factors and p variables the speed of the new algorithms is compared to that of existing algorithms and to that of newton algorithms based on alternative parameterizations several rotation criteria were examined and the algorithms were evaluated over a range of values for m initial guesses for newton algorithms were improved by subconvergence iterations of the gradient projection algorithm simulation results suggest that no one algorithm is fastest for minimizing all criteria for all values of m among competing algorithms the gradient projection algorithm alone was faster than the implicit function algorithm for minimizing a quartic criterion over oblique rotation matrices when m is large in all other conditions however the implicit function algorithms were competitive with or faster than the fastest existing algorithms the new algorithms showed the greatest advantage over other algorithms when minimizing a nonquartic component loss criterion

on the level error after bartlett adjustment of the likelihood ratio statistic summary it is well known that bartlett adjustment reduces level error of the likelihood ratio statistic from order n l to order n3 2 in the present note we show that level error of the adjusted statistic is actually order n2

cone metrics on the sphere and livne s lattices we give an explicit construction of a family of lattices in pu 1 2 originally constructed by livne following thurston we construct these lattices as the modular group of certain euclidean cone metrics on the sphere we give connections between these groups and other groups of complex hyperbolic isometries

common principal components in k groups abstract this article generalizes the method of principal components to so called common principal components as follows consider the hypothesis that the covariance matrices i for k populations are simultaneously diagonalizable that is there is an orthogonal matrix such that i is diagonal for i 1 k i derive the normal theory maximum likelihood estimates of the common component i matrices and the log likelihood ratio statistics for testing this hypothesis the solution has some favorable properties that do not depend on normality assumptions numerical examples illustrate the method applications to data reduction multiple regression and nonlinear discriminant analysis are sketched

on the bootstrap and confidence intervals we derive an explicit formula for the first term in an unconditional edgeworth type expansion of coverage probability for the nonparametric bootstrap technique applied to a very broad class of studentized statistics the class includes sample mean k sample mean sample correlation coefficient maximum likelihood estimators expressible as functions of vector means etc we suggest that the bootstrap is really an empiric one term edgeworth inversion with the bootstrap simulations implicitly estimating the first term in an edgeworth expansion this view of the bootstrap is reinforced by our discussion of the iterated bootstrap which inverts an edgeworth expansion to arbitrary order by simulating simulations

on bootstrap resampling and iteration summary we propose a single unifying approach to bootstrap resampling applicable to a very wide range of statistical problems it enables attention to be focused sharply on one or more characteristics which are of major importance in any particular problem such as coverage error or length for confidence intervals or bias for point estimation our approach leads easily and directly to a very general form of bootstrap iteration unifying and generalizing present disparate accounts of this subject it also provides simple solutions to relatively complex problems such as a suggestion by lehmann 1986 for conditionally short confidence intervals we set out a single unifying principle guiding the operation of bootstrap resampling applicable to a very wide range of statistical problems including bias reduction shrinkage hypothesis testing and confidence interval construction our principle differs from other approaches in that it focuses attention directly on a measure of quality or accuracy expressed in the form of an equation whose solution is sought a very general form of bootstrap iteration is an immediate consequence of iterating the empirical solution to this equation so as to improve accuracy when employed for bias reduction iteration of the resampling principle yields a competitor to the generalized jackknife enabling bias to be reduced to arbitrarily low levels when applied to confidence intervals it produces the techniques of hall 1986 and beran 1987 the resampling principle leads easily to solutions of new complex problems such as empirical versions of confidence intervals proposed by lehmann 1986 lehmann argued that an ideal confidence interval is one which is short when it covers the true parameter value but not necessarily otherwise the resampling principle suggests a simple empirical means of constructing such intervals section 2 describes the general principle and 3 shows how it leads naturally to bootstrap iteration there we show that in many problems of practical interest such as bias reduction and coverage error reduction in two sided confidence intervals each iteration reduces error by the factor n 1 where n is sample size in the case of confidence intervals our result sharpens one of beran 1987 who showed that coverage error is reduced by the factor n 2 in two sided intervals the main exception to our n 1 rule is coverage error of one sided intervals where error is reduced by the factor n a at each iteration our approach to bootstrap iteration serves to unify not just the philosophy of iteration for different statistical problems but also different techniques of iteration for the same

weighted chi squared tests for partial common principal component subspaces summary we consider tests of the null hypothesis that g covariance matrices have a partial common principal component subspace of dimension s our approach uses a dimensionality matrix which has its rank equal to s when the hypothesis holds the test can then be based on a statistic computed from the eigenvalues of an estimate of this dimensionality matrix the asymptotic distribution of this statistic is that of a linear combination of independent one degree of freedom chi squared random variables simulation results indicate that this test yields significance levels that come closer to the nominal level than do those of a previously proposed method the procedure is also extended to a test that g correlation matrices have a partial common principal component subspace

saddlepoint approximations of the distribution of the person parameter in the two parameter logistic model large sample theory states the asymptotic normality of the maximum likelihood estimator of the person parameter in the two parameter logistic 2pl model in short tests however the assumption of normality can be grossly wrong as a consequence intended coverage rates may be exceeded and confidence intervals are revealed to be overly conservative methods belonging to the higher order theory more specifically saddlepoint approximations are a convenient way to deal with small sample problems confidence bounds obtained by these means hold the approximate confidence level for a broad range of the person parameter moreover an approximation to the exact distribution permits to compute median unbiased estimates mue that are as likely to overestimate as to underestimate the true person parameter additionally in small samples these mue are less mean biased than the often used maximum likelihood estimator

nonparametric statistical methods for cost effectiveness analyses two measures often used in a cost effectiveness analysis are the incremental cost effectiveness ratio icer and the net health benefit nhb inferences on these two quantities are often hindered by highly skewed cost data in this article we derive the edgeworth expansions for the studentized t statistics for the two measures and show how they could be used to guide inferences in particular we use the expansions to study the theoretical performance of existing confidence intervals based on normal theory and to derive new confidence intervals for the icer and the nhb we conduct a simulation study to compare our new intervals with several existing methods the methods evaluated include taylor s interval fieller s interval the bootstrap percentile interval and the bootstrap bias corrected acceleration interval we found that our new intervals give good coverage accuracy and are narrower compared to the current recommended intervals

semiparametric partial common principal component analysis for covariance matrices we consider the problem of jointly modeling multiple covariance matrices by partial common principal component analysis pcpca which assumes a proportion of eigenvectors to be shared across covariance matrices and the rest to be individual specific this paper proposes consistent estimators of the shared eigenvectors in the pcpca as the number of matrices or the number of samples to estimate each matrix goes to infinity we prove such asymptotic results without making any assumptions on the ranks of eigenvalues that are associated with the shared eigenvectors when the number of samples goes to infinity our results do not require the data to be gaussian distributed furthermore this paper introduces a sequential testing procedure to identify the number of shared eigenvectors in the pcpca in simulation studies our method shows higher accuracy in estimating the shared eigenvectors than competing methods applied to a motor task functional magnetic resonance imaging data set our estimator identifies meaningful brain networks that are consistent with current scientific understandings of motor networks during a motor paradigm

prepivoting to reduce level error of confidence sets summary approximate confidence sets for a parameter 0 may be obtained by referring a function of 0 and of the sample to an estimated quantile of that function s sampling distribution we call this function the root of the confidence set either asymptotic theory or bootstrap methods can be used to estimate the desired quantile when the root is not a pivot in the sense of classical statistics the actual level of the approximate confidence set may differ substantially from the intended level prepivoting is the transformation of a confidence set root by its estimated bootstrap cumulative distribution function prepivoting can be iterated bootstrap confidence sets generated from a root prepivoted one or more times have smaller error in level than do confidence sets based on the original root the first prepivoting is nearly equivalent to studentizing when that operation is appropriate further iterations of prepivoting make higher order corrections automatically

l infinity norm design of linear phase robust broadband beamformers using constrained optimization a new method for the design of linear phase robust far field broadband beamformers using constrained optimization is proposed in the method the maximum passband ripple and minimum stopband attenuation are ensured to be within prescribed levels while at the same time maintaining a good linear phase characteristic at a prescribed group delay in the passband since the beamformer is intended primarily for small sized microphone arrays where the microphone spacing is small relative to the wavelength at low frequencies the beamformer can become highly sensitive to spatial white noise and array imperfections if a direct minimization of the error is performed therefore to limit the sensitivity of the beamformer the optimization is carried out by constraining a sensitivity parameter namely the white noise gain wng to be above prescribed levels across the frequency band two novel design variants have been developed the first variant is formulated as a convex optimization problem where the maximum error in the passband is minimized while the second variant is formulated as an iterative optimization problem and has the advantage of significantly improving the linear phase characteristics of the beamformer under any prescribed group delay or linear array configuration in the second variant the passband group delay deviation is minimized while ensuring that the maximum passband ripple and stopband attenuation are within prescribed levels to reduce the computational effort in carrying out the optimization a nonuniform variable sampling approach over the frequency and angular dimensions is used to compute the required parameters experiment results show that beamformers designed using the proposed methods have much smaller passband group delay deviation for similar passband ripple and stopband attenuation than a modified version of an existing method

proportionality of covariance matrices on considere l inference pour un modele statistique specifiant la proportionnalite entre deux distributions de wishart independantes

principal component models for correlation matrices summary distributional theory regarding principal components is less well developed for correlation matrices than it is for covariance matrices the intent of this paper is to reduce this disparity methods are proposed that enable investigators to fit and to make inferences about flexible principal components models for correlation matrices the models allow arbitrary eigenvalue multiplicities and allow the distinct eigenvalues to be modelled parametrically or nonparametrically local parameterisations and implicit functions are used to construct full rank unconstrained parameterisations first order asymptotic distributions are obtained directly from the theory of estimating functions second order accurate distributions for making inferences under normality are obtained directly from likelihood theory simulation studies show that the bartlett correction is effective in controlling the size of the tests and that first order approximations to nonnull distributions are reasonably accurate the methods are illustrated on a dataset

estimation of common linear functional relationships in k data sets summary this paper studies a functional type linear measurement error model in which k data sets of observations are assumed to have in common a set of linear restrictions on their respective unobservable constant vectors each data set consists of replicated measurements of each one of several generally unequal constant vectors with normal error assumed independent for each measurement made the error covariance matrices are unspecified and are equal for measurements within a data set but in general unequal among the data sets this work generalises the results of anderson 1951 1984 who derived maximum likelihood estimators for the constant vectors and error covariance matrix of a single such data set the model recalls the common space analysis described by flury 1987 1988 pp 131 4 but differs by the presence of a fixed effects subspace maximum likelihood estimation of the model parameters requires special numerical methods an example from electroencephalography is used to introduce and illustrate the method

asymptotic inference for eigenvectors abstract asymptotic procedures are given for testing certain hypotheses concerning eigenvectors and for constructing confidence regions for eigenvectors these asymptotic procedures are derived under fairly general conditions on the estimates of the matrix whose eigenvectors are of interest applications of the general results to principal components analysis and canonical variate analysis are given author

a matrix derivation of the asymptotic covariance matrix of sample correlation coefficients abstract the asymptotic covariance matrix of the sample correlation matrix is derived in matrix form as an application of some new matrix theory in multivariate statistics

unbiased estimation of fourth order matrix moments we formulate browne s 1984 unbiased estimator for the elements of the matrix of fourth order moments in terms of matrices we show that this matrix is indeed an unbiased estimator without using the theory of cumulants and k statistics

on symmetric bootstrap confidence intervals on montre que dans des circonstances generales les intervalles de confiance du bootstrap de percentile t symetrique ont une erreur de recouvrement o n 2 ou n denote la taille d echantillon

unusual properties of bootstrap confidence intervals in regression problems we show that the percentile t method and one of the two percentile methods have unusually good performance when employed to construct bootstrap confidence intervals in a regression setting in the case of slope parameters percentile t produces two sided intervals with coverage errorn 2 and one sided intervals with coverage errorn 3 2 wheren is sample size the errors are onlyn 1 in most other problems one of the percentile methods produces critical points which are third order correct for efron s 11 relatively complex accelerated bias corrected points

elliptical multivariate analysis abstract in this paper we investigate the interpretation and the asymptotic properties of the well known statistics derived in multivariate analysis when the parent distribution of the observations is not necessary normal but belongs to the wider class of elliptical distributions including the normal our conclusion is that those statistics keep their geometric interpretation and consistency and that for most of those statistics the asymptotic distribution of t is n 0 1 k var n where varn is the covariance matrix under the normality assumption and where k is the joint kurtosis parameter for statistics of the x2 type used in asymptotic test criteria we find similar expressions it follows that all multivariate software may be utilized in the elliptical context under the provision that k is estimated in addition

modelling covariance structure in bivariate marginal models for longitudinal data summary it can be more challenging to efficiently model the covariance matrices for multivariate longitudinal data than for the univariate case due to the correlations arising between multiple responses the positive definiteness constraint and the high dimensionality are further obstacles in covariance modelling in this paper we develop a data based method by which the parametersinthecovariancematricesarereplacedbyunconstrainedandinterpretableparameterswith reduced dimensions the maximum likelihood estimators for the mean and covariance parameters are shown to be consistent and asymptotically normally distributed simulations and real data analysis show that the new approach performs very well even when modelling bivariate nonstationary dependence structures

the geometry of the gauss picard modular group we give a construction of a fundamental domain for rm pu 2 1 mathbb z i that is the group of holomorphic isometries of complex hyperbolic space with coefficients in the gaussian ring of integers mathbb z i we obtain from that construction a presentation of that lattice and relate it in particular to lattices constructed by mostow

generators of the sister of euclidean picard modular groups the sister of eisenstein picard modular group was described in parker duke math j 94 3 433 464 1998 in fact one can define the sister groups of all picard modular groups in particular we find the generators of the sister of euclidean picard modular groups by using a geometric method

robust beamforming with magnitude response constraints using iterative second order cone programming the problem of robust beamforming for antenna arrays with arbitrary geometry and magnitude response constraints is one of considerable importance due to the presence of the non convex magnitude response constraints conventional convex optimization techniques cannot be applied directly a new approach based on iteratively linearizing the non convex constraints is then proposed to reformulate the non convex problem to a series of convex subproblems each of which can be optimally solved using second order cone programming socp moreover in order to obtain a more robust beamformer against array imperfections the proposed method is further extended by optimizing its worst case performance using again socp different from some conventional methods which are restricted to linear arrays the proposed method is applicable to arbitrary array geometries since the weight vector rather than its autocorrelation sequence is used as the variable simulation results show that the performance of the proposed method is comparable to the optimal solution previously proposed for uniform linear arrays and it also gives satisfactory results under different array specifications and geometries tested

new non arithmetic complex hyperbolic lattices we produce a family of new non arithmetic lattices in mathrm pu 2 1 all previously known examples were commensurable with lattices constructed by picard mostow and deligne mostow and fell into nine commensurability classes our groups produce five new distinct commensurability classes most of the techniques are completely general and provide efficient geometric and computational tools for constructing fundamental domains for discrete groups acting on the complex hyperbolic plane

power analyses for correlations from clustered study designs power analysis constitutes an important component of modern clinical trials and research studies although a variety of methods and software packages are available almost all of them are focused on regression models with little attention paid to correlation analysis however the latter is arguably a simpler and more appropriate approach for modelling concurrent events especially in psychosocial research in this paper we discuss power and sample size estimation for correlation analysis arising from clustered study designs our approach is based on the asymptotic distribution of correlated pearson type estimates although this asymptotic distribution is easy to use in data analysis the presence of a large number of parameters creates a major problem for power analysis due to the lack of real data to estimate them by introducing a surrogacy type assumption we show that all nuisance parameters can be eliminated making it possible to perform power analysis based only on the parameters of interest simulation results suggest that power and sample size estimates obtained under the proposed approach are robust to this assumption copyright 2005 john wiley sons ltd

some tests for common principal component subspaces in several groups summary an approximate test based on sample eigenprojections is obtained for testing the hypothesis that the subspaces spanned by the first m principal components of several different covariance matrices are identical this procedure can fairly easily be extended to an analogous test of hypothesis concerning correlation matrices extensions to a robust principal components analysis are also considered in recent years principal components analysis has been generalized from the typical single population setting to that of several populations this paper focuses on an extension that may be useful in those situations in which one needs a variance preserving simultaneous reduction of dimensionality in several groups this sort of reduction will be possible when the subspace spanned by the most important principal component vectors of a particular group is the same as that of any other group formally suppose the same p variables are being measured on objects in g different groups with the covariance

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

regularized generalized canonical correlation analysis for multiblock or multigroup data analysis this paper presents an overview of methods for the analysis of data structured in blocks of variables or in groups of individuals more specifically regularized generalized canonical correlation analysis rgcca which is a unifying approach for multiblock data analysis is extended to be also a unifying tool for multigroup data analysis the versatility and usefulness of our approach is illustrated on two real datasets

pattern synthesis of narrowband conformal arrays using iterative second order cone programming a new design method is proposed for the power or shaped beam pattern synthesis problem of narrowband conformal arrays where only the magnitude response is specified the proposed method iteratively linearizes the non convex power pattern function to obtain a convex subproblem in the design variables which can be solved optimally using second order cone programming socp in addition a wide variety of magnitude constraints such as non convex lower bound magnitude constraints can be incorporated an efficient technique for determining a reasonably good initial guess to the problem is also proposed to further improve the reliability of the method computer simulations show that the initial guesses so obtained converge to satisfactory solutions while satisfying various prescribed magnitude constraints design results show that the performance of the proposed method is comparable to the optimal solution previously obtained for uniform linear arrays with isotropic elements moreover we show by means of examples that the proposed method is also applicable to general non convex power pattern synthesis problems involving arbitrary array geometries arbitrary polarization characteristics and mutual coupling effect