linear systems with levy fluctuations abstract we have examined the response of multivariable linear systems to levy fluctuations the characteristic function of the response is obtained explicitly as a function of time using the characteristic function the fractional moments of the process are determined the nature of the product rule obeyed by the characteristic function of a general linear dynamical system is elucidated

a generalized estimating equations approach for testing ordered group effects with repeated measurements in repeated measures studies we are often interested in comparing group effects in which groups are associated with a certain order relation we propose testing procedures for ordered group effects using the generalized estimating equations gee approach of liang and zeger 1986 biometrika 73 13 22 the order constrained gee estimators of group effects are approximated by the isotonic regression of the unconstrained gee estimators based on these constrained estimators we construct test statistics for detecting ordered group effects the limiting distributions of the test statistics are mixtures of chi square distributions a monte carlo experiment shows improved performances of the proposed tests over the usual chi square tests in detecting ordered group effects the proposed test procedures are illustrated by familial polyposis supplementation trial data

variational bayes solution of linear neural networks and its generalization performance it is well known that in unidentifiable models the bayes estimation provides much better generalization performance than the maximum likelihood ml estimation however its accurate approximation by markov chain monte carlo methods requires huge computational costs as an alternative a tractable approximation method called the variational bayes vb approach has recently been proposed and has been attracting attention its advantage over the expectation maximization em algorithm often used for realizing the ml estimation has been experimentally shown in many applications nevertheless it has not yet been theoretically shown in this letter through analysis of the simplest unidentifiable models we theoretically show some properties of the vb approach we first prove that in three layer linear neural networks the vb approach is asymptotically equivalent to a positive part james stein type shrinkage estimation then we theoretically clarify its free energy generalization error and training error comparing them with those of the ml estimation and the bayes estimation we discuss the advantage of the vb approach we also show that unlike in the bayes estimation the free energy and the generalization error are less simply related with each other and that in typical cases the vb free energy well approximates the bayes one while the vb generalization error significantly differs from the bayes one

the theory of linear models preface simple lenear regression the general linear model one sample and one factor analysis of variance multiple regression models analysis of residuals analysis of variance with two or three factors references and bibliography index

testing for ordered group effects with repeated measurements summary in the repeated measures experiment testing procedures for ordered group effects are proposed using the likelihood ratio principle the constrained maximum likelihood estimators of ordered group effects are approximated by the isotonic regression of the unconstrained maximum likelihood estimators based on these constrained maximum likelihood estimators test statistics are constructed and are used for detecting ordered group effects the limiting distributions of the proposed test statistics are shown to be mixtures of chi squared distributions a simulation study shows that the proposed tests have improved powers over the usual chi squared statistic for testing equality of group effects a real data set is analysed to illustrate the proposed statistics

on approximation of the level probabilities and associated distributions in order restricted inference summary the use of much of the distribution theory developed for order restricted inference has been limited by the lack of algorithms for the level probabilities an approximation for these which accounts for the pattern of large and small weights is developed this approximation and the equal weights approximation are examined both approximations appear to be reasonable for weight sets having a moderate amount of variability the quality of the equal weights approximation as a function of the amount of variability in the weights deteriorates more quickly for certain patterns of large and small weights than for others thus the approximation based upon the pattern of large and small weights is a significant improvement over the equal weights approximation finally siskind s 1976 approximation which can be applied if the number of parameters is not too large is discussed the chi bar squared 2 and e bar squared p2 distributions are fundamental to the theory of order restricted hypothesis tests these distributions have tail probabilities which are linear combinations of the tail probabilities of standard distributions and they depend upon the order restriction through the coefficients in these linear combinations the values of these coefficients are the probabilities that the order restricted maximum likelihood estimates of normal means assume specified numbers of distinct values which are called levels the estimates are based upon independent samples from each of the populations and depend upon the vector of relative precisions called weights of the sample means as estimates of the corresponding population means the use of these tests has been limited because these level probabilities can be virtually impossible to compute if the weights are not all equal in this paper we describe a technique for approximating the level probabilities for a linear order restriction and for unequal precisions this approximation is based upon an idea of chase 1974 and uses the pattern of large and small relative precisions it seems to be particularly good when the relative precisions have two distinct values however it seems to provide a satisfactory approximation so long as the variation in the relative precisions is not too large this is probably due to the lack of sensitivity of the level probabilities to changes in the weights the robustness was

unbiased risk estimates for singular value thresholding and spectral estimators in an increasing number of applications it is of interest to recover an approximately low rank data matrix from noisy observations this paper develops an unbiased risk estimate holding in a gaussian model for any spectral estimator obeying some mild regularity assumptions in particular we give an unbiased risk estimate formula for singular value thresholding svt a popular estimation strategy that applies a soft thresholding rule to the singular values of the noisy observations among other things our formulas offer a principled and automated way of selecting regularization parameters in a variety of problems in particular we demonstrate the utility of the unbiased risk estimation for svt based denoising of real clinical cardiac mri series data we also give new results concerning the differentiability of certain matrix valued functions

spectral regularization algorithms for learning large incomplete matrices we use convex relaxation techniques to provide a sequence of regularized low rank solutions for large scale matrix completion problems using the nuclear norm as a regularizer we provide a simple and very efficient convex algorithm for minimizing the reconstruction error subject to a bound on the nuclear norm our algorithm soft impute iteratively replaces the missing elements with those obtained from a soft thresholded svd with warm starts this allows us to efficiently compute an entire regularization path of solutions on a grid of values of the regularization parameter the computationally intensive part of our algorithm is in computing a low rank svd of a dense matrix exploiting the problem structure we show that the task can be performed with a complexity of order linear in the matrix dimensions our semidefinite programming algorithm is readily scalable to large matrices for example soft impute takes a few hours to compute low rank approximations of a 106 x 106 incomplete matrix with 107 observed entries and fits a rank 95 approximation to the full netflix training set in 3 3 hours our methods achieve good training and test errors and exhibit superior timings when compared to other competitive state of the art techniques

a significance test for the lasso in the sparse linear regression setting we consider testing the significance of the predictor variable that enters the current lasso model in the sequence of models visited along the lasso solution path we propose a simple test statistic based on lasso fitted values called the covariance test statistic and show that when the true model is linear this statistic has an exp 1 asymptotic distribution under the null hypothesis the null being that all truly active variables are contained in the current lasso model our proof of this result for the special case of the first predictor to enter the model i e testing for a single significant predictor variable against the global null requires only weak assumptions on the predictor matrix x on the other hand our proof for a general step in the lasso path places further technical assumptions on x and the generative model but still allows for the important high dimensional case p n and does not necessarily require that the current lasso model achieves perfect recovery of the truly active variables of course for testing the significance of an additional variable between two nested linear models one typically uses the chi squared test comparing the drop in residual sum of squares rss to a formula see text distribution but when this additional variable is not fixed and has been chosen adaptively or greedily this test is no longer appropriate adaptivity makes the drop in rss stochastically much larger than formula see text under the null hypothesis our analysis explicitly accounts for adaptivity as it must since the lasso builds an adaptive sequence of linear models as the tuning parameter decreases in this analysis shrinkage plays a key role though additional variables are chosen adaptively the coefficients of lasso active variables are shrunken due to the formula see text penalty therefore the test statistic which is based on lasso fitted values is in a sense balanced by these two opposing properties adaptivity and shrinkage and its null distribution is tractable and asymptotically exp 1

improving generalised estimating equations using quadratic inference functions summary generalised estimating equations enable one to estimate regression parameters consistently in longitudinal data analysis even when the correlation structure is misspecified however under such misspecification the estimator of the regression parameter can be inefficient in this paper we introduce a method of quadratic inference functions that does not involve direct estimation of the correlation parameter and that remains optimal even if the working correlation structure is misspecified the idea is to represent the inverse of the working correlation matrix by the linear combination of basis matrices a representation that is valid for the working correlations most commonly used both asymptotic theory and simulation show that under misspecified working assumptions these estimators are more efficient than estimators from generalised estimating equations this approach also provides a chi squared inference function for testing nested models and a chi squared regression misspecification test furthermore the test statistic follows a chi squared distribution asymptotically whether or not the working correlation structure is correctly specified

testing ignorable missingness in estimating equation approaches for longitudinal data summary we address the matter of determining whether or not missing data in longitudinal studies are ignorable with regard to quasilikelihood or estimating equations approaches this involves testing for whether or not the zero mean property of estimating equations holds true chen little 1999 proposed testing for significant differences among parameter estimators calculated from sample subsets with different patterns of missing data whereas we propose a more unified generalised score type test this avoids exhaustive estimation of parameters for each missing data pattern testing instead with a single quadratic score test statistic whether or not there is a common parameter under which the means of all the pattern specific estimating equations are zero comparisons are made for the two approaches with both simulations and real data examples

significance levels in exploratory projection pursuit summary in exploratory data analysis projection pursuit methods explore the nonlinear structure of high dimensional data it is useful to have a significance test to help us decide whether apparent structure is real or just the effect of noise monte carlo methods can be helpful to achieve this but unfortunately in this case they are computationally expensive in this paper under a suitable null hypothesis we derive a theoretical approximation for the p value associated with friedman s projection pursuit index friedman 1987 the result of monte carlo simulations is compared with our analytical result some practical aspects are discussed

the solution path of the generalized lasso we present a path algorithm for the generalized lasso problem this problem penalizes the l1 norm of a matrix d times the coefficient vector and has a wide range of applications dictated by the choice of d our algorithm is based on solving the dual of the generalized lasso which greatly facilitates computation of the path for d i the usual lasso we draw a connection between our approach and the well known lars algorithm for an arbitrary d we derive an unbiased estimate of the degrees of freedom of the generalized lasso fit this estimate turns out to be quite intuitive in many applications

levy stable probability densities and mechanical relaxation in solid polymers early investigations by weber r and f kohlrausch maxwell and boltzmann of relaxation in viscoelastic solids are reviewed a two state model stress tensor describing strain coupling to internal conformations of a polymer chain is used to derive a linear response version of the boltzmann superposition principle for shear stress relaxation the relaxation function of kohlrausch t exp t is identical to the williams watts empirical dielectric relaxation function and in the model corresponds to the autocorrelation function of a segment s differential shape anisotropy tensor by analogy with the dielectric problem exp t is interpreted as the survival probability of a frozen segment in a swarm of hopping defects with a stable waiting time distributionat for defect motion the exponent a is the fractal dimension of a hierarchical scaling set of defect hopping times integral transforms of t needed for data analysis are evaluated the cosine and inverse laplace transforms are stable probability densities the reciprocal kernel for short time compliance is discussed

does polynomial parallel volume imply convexity for a non empty compact set a d d 2 and r 0 let a r denote the set of points whose distance from a is r at the most it is well known that the volume v d a r of a r is a polynomial of degree d in the parameter r if a is convex we pursue the reverse question and ask whether a is necessarily convex if v d a r is a polynomial in r an affirmative answer is given in dimension d 2 counterexamples are provided for d 3 a positive resolution of the question in all dimensions is obtained if the assumption of a polynomial parallel volume is strengthened to the validity of a polynomial local steiner formula

combining dependent tests with incomplete repeated measurements summary in comparing the effectiveness of two treatments repeated measurements of the same characteristic are often taken under two or more distinct conditions for each experimental subject a locally best linear combination of individual test statistics obtained under different conditions is proposed to test the equality of the two treatments the test procedure allows different patterns of missing observations for the two groups to be compared special cases such as combining wilcoxon tests t tests and 2 x 2 tables are discussed in detail with real examples

small sample adjustments in using the sandwich variance estimator in generalized estimating equations the generalized estimating equation gee approach is widely used in regression analyses with correlated response data under mild conditions the resulting regression coefficient estimator is consistent and asymptotically normal with its variance being consistently estimated by the so called sandwich estimator statistical inference is thus accomplished by using the asymptotic wald chi squared test however it has been noted in the literature that for small samples the sandwich estimator may not perform well and may lead to much inflated type i errors for the wald chi squared test here we propose using an approximate t or f test that takes account of the variability of the sandwich estimator the level of type i error of the proposed t or f test is guaranteed to be no larger than that of the wald chi squared test the satisfactory performance of the proposed new tests is confirmed in a simulation study our proposal also has some advantages when compared with other new approaches based on direct modifications of the sandwich estimator including the one that corrects the downward bias of the sandwich estimator in addition to hypothesis testing our result has a clear implication on constructing wald type confidence intervals or regions

on the distribution of the largest eigenvalue in principal components analysis let x 1 denote the square of the largest singular value of an n p matrix x all of whose entries are independent standard gaussian variates equivalently x 1 is the largest principal component variance of the covariance matrix x x or the largest eigenvalue of a p variate wishart distribution on n degrees of freedom with identity covariance consider the limit of large p and n with n p gamma ge 1 when centered by mu p sqrt n 1 sqrt p 2 and scaled by sigma p sqrt n 1 sqrt p 1 sqrt n 1 1 sqrt p 1 3 the distribution of x 1 approaches the tracey widom law of order 1 which is defined in terms of the painleve ii differential equation and can be numerically evaluated and tabulated in software simulations show the approximation to be informative for n and p as small as 5 the limit is derived via a corresponding result for complex wishart matrices using methods from random matrix theory the result suggests that some aspects of large p multivariate distribution theory may be easier to apply in practice than their fixed p counterparts

dimension selection for feature selection and dimension reduction with principal and independent component analysis this letter is concerned with the problem of selecting the best or most informative dimension for dimension reduction and feature extraction in high dimensional data the dimension of the data is reduced by principal component analysis subsequent application of independent component analysis to the principal component scores determines the most nongaussian directions in the lower dimensional space a criterion for choosing the optimal dimension based on bias adjusted skewness and kurtosis is proposed this new dimension selector is applied to real data sets and compared to existing methods simulation studies for a range of densities show that the proposed method performs well and is more appropriate for nongaussian data than existing methods

path integral approach to quantum brownian motion abstract we apply the influence functional method of feynman and vernon to the study of brownian motion at arbitrary temperature by choosing a specific model for the dissipative interaction of the system of interest with its environment we are able to evaluate the influence functional in closed form and express it in terms of a few parameters such as the phenomenological viscosity coefficient we show that in the limit h 0 the results obtained from the influence functional formalism reduce to the classical fokker planck equation in the case of a simple harmonic oscillator with arbitrarily strong damping and at arbitrary temperature we obtain an explicit expression for the time evolution of the complete density matrix x x t when the system starts in a particular kind of pure state we compare our results with those of other approaches to the problem of dissipation in quantum mechanics

hotelling s theorem on the volume of tubes some illustrations in simultaneous inference and data analysis we illustrate with contemporary examples hotelling s geometric approach to simultaneous probability calculations hotelling reduces the evaluation of certain normal theory significance probabilities to finding the volume of a tube about a curve in a hypersphere and shows that this volume is often exactly given by length times cross sectional area we review hotelling s result together with some recent complements and then use the approach to set simultaneous prediction regions for some data from gait analysis to study andrews plots in multivariate data analysis and to construct significance tests for projection pursuit regression a by product is a numerical criterion for tube self overlap relevant for example to uniqueness of certain nonlinear least squares estimates

two statistics for testing for multivariate normality malkovich afifi 1973 proposed measures of multivariate skewness and kurtosis in the present paper distributional properties of the two statistics are examined a method is given for calculating approximate percentage points of their distributions for all sample sizes greater than 8 for dimensions 2 3 and 4 and in principle for all higher dimensions the percentage points are derived from appropriate transformations of the asymptotic distributions a simulation study shows the approximations to be good for sample sizes of 25 and above

random fields of multivariate test statistics with applications to shape analysis our data are random fields of multivariate gaussian observations and we fit a multivariate linear model with common design matrix at each point we are interested in detecting those points where some of the coefficients are nonzero using classical multivariate statistics evaluated at each point the problem is to find the p value of the maximum of such a random field of test statistics we approximate this by the expected euler characteristic of the excursion set our main result is a very simple method for calculating this which not only gives us the previous result of cao and worsley ann statist 27 1999 925 942 for hotelling s t 2 but also random fields of roy s maximum root maximum canonical correlations ann appl probab 9 1999 1021 1057 multilinear forms ann statist 29 2001 328 371 x 2 statist probab lett 32 1997 367 376 ann statist 25 1997 2368 2387 and x 2 scale space adv in appl probab 33 2001 773 793 the trick involves approaching the problem from the point of view of roy s union intersection principle the results are applied to a problem in shape analysis where we look for brain damage due to nonmissile trauma

two sample asymptotically distribution free tests for incomplete multivariate observations abstract asymptotically distribution free tests for equality of two multivariate distributions based on censored observations are proposed by choosing appropriate test scores we obtain the multivariate versions of the gehan and log rank tests an important application of the proposed tests is in the area in which multiple observations are collected on study subjects some of whom may have missing observations the patterns of missing observations are allowed to differ for the two groups to be compared these tests are particularly useful for longitudinal studies for example the analysis of repeated measurements and growth curves real examples are also provided for illustration

anti diagonals of symmetric and skew symmetric matrices with prescribed eigenvalues abstract the complete relationship between the kth anti diagonals and the eigenvalues of an n n k n real symmetric matrix is obtained when k is even the relationship is weak majorization the hermitian case is also studied similar results are obtained for real and complex skew symmetric matrices

tail probabilities of the maxima of multilinear forms and their applications let z be a k way array consisting of independent standard normal variables for column vectors h1 hk define a multilinear form of degree k by h 1 otimes cdots otimes h k vec z we derive formulas for upper tail probabilities of the maximum of a multilinear form with respect to the hi s under the condition that the hi s are unit vectors and of its standardized statistic obtained by dividing by the norm of z we also give formulas for the maximum of a symmetric multilinear form h 1 otimes cdots otimes h k sym z where sym z denotes the symmetrization of z with respect to indices these classes of statistics are used for testing hypotheses in the analysis of variance of multiway layout data and for testing multivariate normality in order to derive the tail probabilities we employ a geometric approach developed by hotelling weyl and sun upper and lower bounds for the tail probabilities are given by reexamining sun s results some numerical examples are given to illustrate the practical usefulness of the obtained formulas including the upper and lower bounds

inference in adaptive regression via the kac rice formula we derive an exact p value for testing a global null hypothesis in a general adaptive regression setting our approach uses the kac rice formula as described in random fields and geometry 2007 springer new york applied to the problem of maximizing a gaussian process the resulting test statistic has a known distribution in finite samples assuming gaussian errors we examine this test statistic in the case of the lasso group lasso principal components and matrix completion problems for the lasso problem our test relates closely to the recently proposed covariance test of lockhart et al ann statist 2004 42 413 468 in a few specific settings our proposed tests will be less powerful than other previously known and well established tests however it should be noted that the real strength of our proposal here is its generality we provide a framework for constructing valid tests across a wide class of regularized regression problems and as far as we can tell such a unified view was not possible before this work

exploratory projection pursuit abstract a new projection pursuit algorithm for exploring multivariate data is presented that has both statistical and computational advantages over previous methods a number of practical issues concerning its application are addressed a connection to multivariate density estimation is established and its properties are investigated through simulation studies and application to real data the goal of exploratory projection pursuit is to use the data to find low one two or three dimensional projections that provide the most revealing views of the full dimensional data with these views the human gift for pattern recognition can be applied to help discover effects that may not have been anticipated in advance since linear effects are directly captured by the covariance structure of the variable pairs which are straightforward to estimate the emphasis here is on the discovery of nonlinear effects such as clustering or other general nonlinear associations among the variables although arbitrary

a general framework for estimation and inference from clusters of features abstractapplied statistical problems often come with prespecified groupings to predictors it is natural to test for the presence of simultaneous group wide signal for groups in isolation or for multiple groups together current tests for the presence of such signals include the classical f test or a t test on unsupervised group prototypes either group centroids or first principal components in this article we propose test statistics that aim for power improvements over these classical approaches in particular we first create group prototypes with reference to the response and then test with likelihood ratio statistics incorporating only these prototypes we propose a model called the prototype model which naturally models this two step procedure furthermore we introduce an inferential schema detailing the unique considerations for different combinations of prototype formation and univariate multivariate testing models the prototype model also suggests new applications to estimation and pr

linearly coupled quantum oscillators with levy stable noise abstract the influence of an external noise on two linearly coupled quantum oscillators is investigated the noise is described by the probability measure corresponding to a symmetric levy stable distribution in calculations the path integral technique is used this paper is a continuation of investigations started by bose datta and feinsilver

nonparametric analysis of random utility models abstract this paper considers two sided tests for the parameter of an endogenous variable in an instrumental variable iv model with heteroskedastic and autocorrelated errors we develop the finite sample theory of weighted average power wap tests with normal errors and a known long run variance a typical weight choice yields tests with power near zero for parts of the parameter space we propose two boundary conditions that yield two sided tests with hac errors the locally unbiased lu condition is related to the power around the null hypothesis and is a weaker requirement than unbiasedness the strongly unbiased su condition is more restrictive than lu but the associated wap tests are easier to implement the wap su tests can outperform several tests under the weak iv asymptotics as they make use of the data beyond the anderson rubin and score statistics

a covariance estimator for gee with improved small sample properties summary in this paper we propose an alternative covariance estimator to the robust covariance estimator of generalized estimating equations gee hypothesis tests using the robust covariance estimator can have inflated size when the number of independent clusters is small resampling methods such as the jackknife and bootstrap have been suggested for covariance estimation when the number of clusters is small a drawback of the resampling methods when the response is binary is that the methods can break down when the number of subjects is small due to zero or near zero cell counts caused by resampling we propose a bias corrected covariance estimator that avoids this problem in a small simulation study we compare the bias corrected covariance estimator to the robust and jackknife covariance estimators for binary responses for situations involving 10 40 subjects with equal and unequal cluster sizes of 16 64 observations the bias corrected covariance estimator gave tests with sizes close to the nominal level even when the number of subjects was 10 and cluster sizes were unequal whereas the robust and jackknife covariance estimators gave tests with sizes that could be 2 3 times the nominal level the methods are illustrated using data from a randomized clinical trial on treatment for bone loss in subjects with periodontal disease

analysis of incomplete multivariate data from repeated measurement experiments this paper analyses two sets of data that consist of repeated measurements with missing data the missing observations always occur at the end of the series of repeated measurements the score test for multivariate normal data is used to compare treatment groups if the original data are not multivariate normal they are replaced by expected normal scores

analysis of semiparametric regression models for repeated outcomes in the presence of missing data abstract we propose a class of inverse probability of censoring weighted estimators for the parameters of models for the dependence of the mean of a vector of correlated response variables on a vector of explanatory variables in the presence of missing response data the proposed estimators do not require full specification of the likelihood they can be viewed as an extension of generalized estimating equations estimators that allow for the data to be missing at random but not missing completely at random these estimators can be used to correct for dependent censoring and nonrandom noncompliance in randomized clinical trials studying the effect of a treatment on the evolution over time of the mean of a response variable the likelihood based parametric g computation algorithm estimator may also be used to attempt to correct for dependent censoring and nonrandom noncompliance but because of possible model misspecification the parametric g computation algorithm estimator in contrast with the proposed w

lower bounds for the spread of a matrix a characterization of the spread of a normal matrix is used to derive several simple lower bounds for the spread comparisons are then made with several known bounds

using historical controls in the analysis of developmental toxicity data historical control data can often aid in the interpretation of the results from laboratory dose response experiments for example formal statistical methods for using historical data are well established for carcinogenicity studies in that case a score test derived from a beta binomial model yields a simple modification of the standard cochran armitage test for trend however this test cannot be used for developmental toxicity studies since it does not allow for the additional correlation structure induced by the presence of litter effects in this paper quasi likelihood methods are used to incorporate historical control information into a trend test for the types of correlated binary outcomes that typically arise in developmental toxicity studies the proposed test is asymptotically equivalent to the beta binomial score test in the special case when all the litter sizes are equal to 1 malformation data from a series of developmental toxicity studies illustrate the results

reml estimation with exact covariance in the logistic mixed model residual maximum likelihood reml estimation is adapted to certain logistic mixed models for which representation of the unconditional mean as a linear function of the fixed effects is possible only the first two moments of the unconditional distribution need be evaluated and except for the form of the covariance the maximization algorithm carries over directly from linear models the exact unconditional covariance is computed from the logistic normal mixture for input into the algorithm residual log likelihood plots provide a means of inference for the dispersion components the taylor series approximation to this covariance besides exhibiting insufficient accuracy fails to be positive definite for large values of the dispersion components as a consequence reml loglikelihood plots based on this approximate covariance attribute misleadingly high precision to the dispersion component estimates the method is presented in the context of a salamander mating experiment in which random effects corresponding to male and female animals occur in a crossed design analyses of these data by several methods are compared

comparison of methods for the analysis of longitudinal interval count data longitudinal studies are often concerned with estimating the recurrence rate of a non fatal event in many cases only the total number of events occurring during successive time intervals is known we compared a mixed poisson gamma regression method proposed by thall and a quasi likelihood method proposed by zeger and liang for the analysis of such data in the case where the mean was correctly specified using simulation techniques with large samples both methods produced similar standard errors in most situations except in the case of time dependent covariates with non poisson gamma data where they were seriously underestimated by the thall method a simple method for discriminating between the variance forms of the two methods is described the findings are applied to the analyses of clinical trials of non melanoma skin cancer and familial polyposis this study extends the findings of breslow concerning variance misspecification in overdispersed poisson and quasi likelihood models to the longitudinal setting

on levy or stable distributions and the williams watts model of dielectric relaxation this paper is concerned with the levy or stable distribution function defined by the fourier transform q alpha left z right frac 1 2 pi int infty infty exp left izu left u right alpha right du with 0 alpha leqslant 2 when 2 it becomes the gauss distribution function and when 1 the cauchy distribution when 2 the distribution has a long inverse power tail q alpha left z right sim frac gamma left 1 alpha right sin tfrac 1 2 pi alpha pi left z right 1 alpha in the regime of small if logz 1 the distribution is mimicked by a log normal distribution we have derived rapidly converging algorithms for the numerical calculation ofq z for various in the range 0 1 the functionq z appears naturally in the williams watts model of dielectric relaxation in that model one expresses the normalized dielectric parameter as in n left omega right equiv in n left omega right i in n left omega right int 0 infty e i omega t left d phi left t right mathord left vphantom d phi left t right dt right kern nulldelimiterspace dt right dt with phi left t right exp left t mathord left vphantom t tau right kern nulldelimiterspace tau right alpha it has been found empirically by various authors that observed dielectric parameters of a wide variety of materials of a broad range of frequencies are fitted remarkably accurately by using this form of t e n is shown to be directly related toq z it is also shown that if the williams watts exponential is expressed as a weighted average of exponential relaxation functions exp left t mathord left vphantom t tau right kern nulldelimiterspace tau right alpha int 0 infty g left lambda alpha right e lambda t dt the weight functiong is expressible as a stable distribution some suggestions are made about physical models that might lead to the williams watts form of t

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

analysis of nonadditive multiway classifications abstract this article considers the problems of testing additivity and estimating 2 in unreplicated multiway classifications to model nonadditivity and jointly estimate 2 the interaction parameter space must be restricted otherwise the model is saturated the parameterization we use is a multiway extension of the two way multiplicative interaction model of mandel 1971 and johnson and graybill 1972a for example in a three way classification we model interaction as ijk 1i 2j 3k this structure is a special case of the k mode principal components model which has received considerable attention in the psychometric literature kapteyn neudecker and wansbeek 1986 we construct an exact test of 0 and propose an estimator of 2 that can be used when interaction has been detected our test is an approximation to the likelihood ratio test lrt of ho 0 the proposed test has essentially the same power as the lrt but is easier to compute and the exact null distribution of the test

tests of independence treatment heterogeneity dose related trend with exchangeable binary data abstract existing methods of testing for treatment effects for clustered binary data include the beta binomial quasilikelihood gee procedures all of these methods revolve around the mean response and the second order correlation however these two parameters alone do not fully determine the effect of treatment this article develops nonparametric likelihood ratio procedures to test for independence heterogeneity dose related trend in dose response studies involving exchangeable binary data the hypotheses of independence heterogeneity trend are expressed in terms of joint probabilities of similar responses among cluster mates constrained maximum likelihood estimates of these probabilities are computed and used to construct test statistics unlike the test statistics for independence and heterogeneity the asymptotic distribution of the likelihood ratio test for trend is not exactly a chi square however an upper bound of its p value is obtained by using a chi squared distribution a set of clustere

asymptotic distribution of test statistics in the analysis of moment structures under inequality constraints summary the analysis of moment structural models has become an important tool of investigation in behavioural educational and economic studies the chi squared largesample test is routinely employed to assess the goodness of fit of the model considered however in order to invoke the standard asymptotic distribution theory certain regularity conditions have to be met here we consider the case where the population value of the parameter vector is a boundary point of the feasible region we show that in this case the asymptotic distribution of test statistic is a mixture of chi squared distributions the problem of finding the corresponding weights is discussed

tests based on t statistics for iv regression with weak instruments abstract this paper considers tests of the parameter of an endogenous variable in an instrumental variables regression model the focus is on one sided conditional t tests theoretical and numerical work shows that the conditional 2sls and fuller t tests perform well even when instruments are weakly correlated with the endogenous variable when the population f statistic is as small as two their power is reasonably close to the power envelopes for similar and non similar tests which are invariant to rotation transformations of the instruments this finding is surprising considering the bad performance of two sided conditional t tests found in andrews et al 2007 we show these tests have bad power because the conditional null distributions of t statistics are asymmetric when instruments are weak taking this asymmetry into account we propose two sided tests based on t statistics these novel tests are approximately unbiased and can perform as well as the conditional likelihood ratio clr test

williams watts dielectric relaxation a fractal time stochastic process dielectric relaxation in amorphous materials is treated in a defect diffusion model where relaxation occurs when a mobile defect such as a vacancy reaches a frozen in dipole the random motion of the defect is assumed to be governed by a fractal time stochastic process where the mean duration between defect movements is infinite when there are many more defects than dipoles the williams watts decaying fractional exponential relaxation law is derived the argument of the exponential is related to the number of distinct sites visited by the random walk of the defect for the same reaction dynamics but with more traps than walkers an algebraically decaying relaxation is found

weights of bar x 2 distribution for smooth or piecewise smooth cone alternatives we study the problem of testing a simple null hypothesis about the multivariate normal mean vector against smooth or piecewise smooth cone alternatives we show that the mixture weights of the x 2 distribution of the likelihood ratio test can be characterized as mixed volumes of the cone and its dual the weights can be calculated by integration involving the second fundamental form on the boundary of the cone we illustrate our technique by examples involving a spherical cone and a piecewise smooth cone

antidiagonals of matrices in a unitary similarity orbit abstract let mn be the algebra of n n complex matrices for a aij mn the kth antidiagonal of a is the vector ak 1 ak 1 2 a1 k t and u a u au u is unitary is the unitary similarity orbit of a we study the sets 1 d k a the collection of all kth antidiagonals of matrices in u a 2 i k a the collection of z k i 1 v i c such that v v 1 v k t d k a 3 p k a the collection of z k i 1 v i c such that v v 1 v k t d k a particular attention is paid to the relation between the geometrical properties of the sets and the algebraic properties of the matrix a when k 1 all three sets reduce to w a the numerical range of a which has been studied extensively we also consider the following related concepts 1 d k a max x x d k a where x x x 1 2 2 s k a max z z i k a 3 p k a max z z p k a

on the degrees of freedom of reduced rank estimators in multivariate regression we study the effective degrees of freedom of a general class of reduced rank estimators for multivariate regression in the framework of stein s unbiased risk estimation a finite sample exact unbiased estimator is derived that admits a closed form expression in terms of the thresholded singular values of the least squares solution and hence is readily computable the results continue to hold in the high dimensional setting where both the predictor and the response dimensions may be larger than the sample size the derived analytical form facilitates the investigation of theoretical properties and provides new insights into the empirical behaviour of the degrees of freedom in particular we examine the differences and connections between the proposed estimator and a commonly used naive estimator the use of the proposed estimator leads to efficient and accurate prediction risk estimation and model selection as demonstrated by simulation studies and a data example

tests with correct size when instruments can be arbitrarily weak abstract this paper applies classical exponential family statistical theory to develop a unifying framework for testing structural parameters in the simultaneous equations model under the assumption of normal errors with known reduced form variance matrix the results can be divided into the limited information and full information categories in the limited information model it is possible to characterize the entire class of similar tests in a model with only one endogenous explanatory variable in the full information framework this paper proposes a family of similar tests for subsets of endogenous variables coefficients for both limited and full information models there exist power upper bounds for unbiased tests when the model is just identified the anderson rubin score and pseudo conditional likelihood ratio tests are optimal when the model is over identified the pseudo conditional likelihood ratio test has power close to the power envelope when identification is strong

selective inference with unknown variance via the square root lasso summarythere has been much recent work on inference after model selection in situations where the noise level is known however the error variance is rarely known in practice and its estimation is difficult in high dimensional settings in this work we propose using the square root lasso also known as the scaled lasso to perform inference for selected coefficients and the noise level simultaneously the square root lasso has the property that the choice of a reasonable tuning parameter does not depend on the noise level in the data we provide valid p values and confidence intervals for coefficients after variable selection and estimates for the model specific variance our estimators perform better in simulations than other estimators of the noise variance these results make inference after model selection significantly more applicable

testing for order restricted hypotheses in longitudinal data an inherent ordering we propose a quadratic score test qst based on a quadratic inference function for detecting an order in treatment effects for correlated data the quadratic inference function is similar to the negative of a log likelihood and it provides test statistics in the spirit of a x2 test for testing nested hypotheses as well as for assessing the goodness of fit of model assumptions under the null hypothesis of no order restriction it is shown that the qst statistic has a wald type asymptotic representation and that the asymptotic distribution of the qst statistic is a weighted x2 distribution furthermore an asymptotic distribution of the qst statistic under an arbitrary convex cone alternative is provided the performance of the qst is investigated through monte carlo simulation experiments analysis of the polyposis data demonstrates that the qst outperforms the wald test when data are highly correlated with a small sample size and there is a significant amount of missing data with a small number of clusters the proposed test statistic accommodates both time dependent and time independent covariates in a model

multiple testing of local maxima for detection of peaks in random fields a topological multiple testing scheme is presented for detecting peaks in images under stationary ergodic gaussian noise where tests are performed at local maxima of the smoothed observed signals the procedure generalizes the one dimensional scheme of 31 to euclidean domains of arbitrary dimension two methods are developed according to two different ways of computing p values i using the exact distribution of the height of local maxima available explicitly when the noise field is isotropic 9 10 ii using an approximation to the overshoot distribution of local maxima above a pre threshold applicable when the exact distribution is unknown such as when the stationary noise field is non isotropic 9 the algorithms combined with the benjamini hochberg procedure for thresholding p values provide asymptotic strong control of the false discovery rate fdr and power consistency with specific rates as the search space and signal strength get large the optimal smoothing bandwidth and optimal pre threshold are obtained to achieve maximum power simulations show that fdr levels are maintained in non asymptotic conditions the methods are illustrated in the analysis of functional magnetic resonance images of the brain

longitudinal data analysis using generalized linear models summary this paper proposes an extension of generalized linear models to the analysis of longitudinal data we introduce a class of estimating equations that give consistent estimates of the regression parameters and of their variance under mild assumptions about the time dependence the estimating equations are derived without specifying the joint distribution of a subject s observations yet they reduce to the score equations for multivariate gaussian outcomes asymptotic theory is presented for the general class of estimators specific cases in which we assume independence m dependence and exchangeable correlation structures from each subject are discussed efficiency of the proposed estimators in two simple situations is considered the approach is closely related to quasi likelih ood some key ironh estimating equation generalized linear model longitudinal data quasi likelihood repeated measures