computing gaussian likelihoods and their derivatives for general linear mixed models algorithms are described for computing the gaussian likelihood or restricted likelihood corresponding to a general linear mixed model included are arbitrary covariance structures for both the random effects and errors formulas are also given for the first and second derivatives of the likelihoods thus enabling a newton raphson implementation the algorithms make heavy use of the cholesky decomposition the sweep operator and the w transformation also described are the modifications needed for variance profiling fisher scoring and mivque 0 as well as the computational order of the procedures

lattice sums for off axis electromagnetic scattering by gratings we consider both the spatial domain and spectral domain forms of the green s function appropriate in the electromagnetic diffraction of a plane wave incident at a general angle in the xy plane on a singly periodic structure or grating oriented along the x axis we equate the spatial and spectral forms of the green s function and so establish expressions from which grating lattice sums can be evaluated for oblique incidence we also obtain a set of identities among the lattice sums we use these lattice sums in an expression for the green s function which we show to be computationally fast if knowledge of this function at several points is required for small values of y

the estimation of the mean squared error of small area estimators abstract small area estimation has received considerable attention in recent years because of a growing demand for reliable small area statistics the direct survey estimators based only on the data from a given small area or small domain are likely to yield unacceptably large standard errors because of small sample size in the domain therefore alternative estimators that borrow strength from other related small areas have been proposed in the literature to improve the efficiency these estimators use models either implicitly or explicitly that connect the small areas through supplementary e g census and administrative data for example simple synthetic estimators are based on implicit modeling in this article three small area models of battese harter and fuller 1988 dempster rubin and tsutakawa 1981 and fay and herriot 1979 are investigated these models are all special cases of a general mixed linear model involving fixed and random effects and a small area mean can be expr

variance component testing in generalised linear models with random effects summary there is considerable interest in testing for overdispersion correlation and heterogeneity across groups in biomedical studies in this paper we cast the problem in the framework of generalised linear models with random effects we propose a global score test for the null hypothesis that all the variance components are zero this test is a locally asymptotically most stringent test and is robust in the special sense that the test does not require specifying the joint distribution of the random effects we also propose individual score tests and their approximations for testing the variance components separately both tests can be easily implemented using existing statistical software we illustrate these tests with an application to the study of heterogeneity of mating success across males and females in an experiment on salamander matings and evaluate their performance through simulation

systematic investigation of two dimensional static array sums we discuss general properties of doubly periodic sums over the square lattice linking phased bloch type sums in the direct lattice with displaced sums in the reciprocal lattice using the poisson summation formula we discuss cardinal points where the sums reduce to a single product of dirichlet l functions and exhibit all cardinal points for the square lattice we introduce a new method for evaluating lattice sums and illustrate this by solving low order systems of sums of integer order 2 3 4 5 for the last case the analytic expressions for the sums involve complementary l functions or alternatively l functions with complex characters

exploring the information in p values for the analysis and planning of multiple test experiments a new methodology is proposed for estimating the proportion of true null hypotheses in a large collection of tests each test concerns a single parameter delta whose value is specified by the null hypothesis we combine a parametric model for the conditional cumulative distribution function cdf of the p value given delta with a nonparametric spline model for the density g delta of delta under the alternative hypothesis the proportion of true null hypotheses and the coefficients in the spline model are estimated by penalized least squares subject to constraints that guarantee that the spline is a density the estimator is computed efficiently using quadratic programming our methodology produces an estimate g delta of the density of delta when the null is false and can address such questions as when the null is false is the parameter usually close to the null or far away this leads us to define a falsely interesting discovery rate fidr a generalization of the false discovery rate we contrast the fidr approach to efron s 2004 journal of the american statistical association 99 96 104 empirical null hypothesis technique we discuss the use of g in sample size calculations based on the expected discovery rate edr our recommended estimator of the proportion of true nulls has less bias compared to estimators based upon the marginal density of the p values at 1 in a simulation study we compare our estimators to the convex decreasing estimator of langaas lindqvist and ferkingstad 2005 journal of the royal statistical society series b 67 555 572 the most biased of our estimators is very similar in performance to the convex decreasing estimator as an illustration we analyze differences in gene expression between resistant and susceptible strains of barley

effective elastic properties of random two dimensional composites abstract consider 2d two phase random composites with circular inclusions of concentration f new analytical formulae for the effective constants are deduced up to o f 4 for macroscopically isotropic composites it is shown that the second order terms o f 2 do not depend on the location of inclusions whilst the third order terms do this implies that the previous analytical methods effective medium approximation differential scheme mori tanaka approach and so forth can be valid at most up to o f 3 for macroscopically isotropic composites first the local elastic field for a finite number n of inclusions arbitrarily located on the plane are found by a method of functional equations further the limit n yields conditionally convergent series defined by the eisenstein summation method one of the series for periodic composites is the famous lattice sum s 2 deduced by rayleigh for a conductivity problem

green s tensors and lattice sums for electrostatics and elastodynamics we present expressions for the green9s tensors in terms of lattice sums for general two dimensional arrays in both elastostatics and elastodynamics we represent the lattice sums in rapidly convergent forms by fourier transform methods we compare the static lattice sums with recently published highly accurate values for square arrays we also establish rayleigh identities for elastostatic and elastodynamic problems generalizing previous work in electrostatics and electromagnetism the rayleigh identities may be used in studies of phononic band gaps

evaluation and regularization of phase modulated eisenstein series and application to double schlomilch type sums in the study of periodic media conditionally convergent series are frequently encountered and their regularization is crucial for applications we derive an identity that regularizes two dimensional phase modulated eisenstein series for all bravais lattices yielding physically meaningful values we also obtain explicit forms for the phase modulated series in terms of holomorphic eisenstein series enabling their closed form evaluation for important high symmetry lattices results are then used to obtain representations for the related double schlomilch type sums which are also given for all bravais lattices finally we treat displaced lattices of high symmetry expressing them in terms of origin centered lattices via geometric multi set identities these identities apply to all classes of two dimensional sums allowing sums to be evaluated over each constituent of a unit cell that possesses multiple inclusions in the study of periodic media conditionally convergent series are frequently encountered and their regularization is crucial for applications we derive an identity that regularizes two dimensional phase modulated eisenstein series for all bravais lattices yielding physically meaningful values we also obtain explicit forms for the phase modulated series in terms of holomorphic eisenstein series enabling their closed form evaluation for important high symmetry lattices results are then used to obtain representations for the related double schlomilch type sums which are also given for all bravais lattices finally we treat displaced lattices of high symmetry expressing them in terms of origin centered lattices via geometric multi set identities these identities apply to all classes of two dimensional sums allowing sums to be evaluated over each constituent of a unit cell that possesses multiple inclusions

estimation in covariance components models abstract estimation techniques for linear covariance components models are developed and illustrated with special emphasis on explaining computational processes the estimation of fixed and random effects when the variances and covariances are known is presented in bayesian terms point estimates of the unknown variances and covariances are computed using the em algorithm for maximum likelihood estimation from incomplete data the techniques are illustrated with data on law schools field mice and professional football teams

asymptotic estimates for localized electromagnetic modes in doubly periodic structures with defects the paper presents analytical and numerical models describing localized electromagnetic defect modes in a doubly periodic structure involving closely located inclusions of elliptical and circular shapes two types of localized modes are considered i an axi symmetric mode for the case of transverse electric polarization with an array of metallic inclusions ii a dipole type localized mode that occurs in problems of waveguide modes confined in a defect region of an array of cylindrical fibres and propagating perpendicular to the plane of the array a thin bridge asymptotic analysis is used for case i to establish double sided bounds for the frequencies of localized modes in macro cells with thin bridges for the case ii the electric and magnetic fields independently satisfy helmholtz equations but are coupled through the boundary conditions we show that the model problem associated with localized vibration modes is the dirichlet problem for the helmholtz operator we characterize defect modes by introducing a parameter called the effective diameter we show that for circular inclusions in silica matrix the effective diameter is accurately represented by a linear function of the inclusion radius

analysis of elastic band structures for oblique incidence we propose a canonical basis that is used in the expansion of eigensolutions for a problem of oblique incidence of elastic waves on a doubly periodic array of cylindrical channels we apply a multipole method to study the spectral properties of waves in such a structure dispersion diagrams constructed on the basis of an analytical solution show the presence of a full phononic band gap when the angle of oblique incidence exceeds certain critical value explicit asymptotic formulae are presented for the effective refractive index associated with shear waves in oblique incidence

an l p theory of sparse graph convergence i limits sparse random graph models and power law distributions we introduce and develop a theory of limits for sequences of sparse graphs based on lp graphons which generalizes both the existing l1 theory of dense graph limits and its extension by bollob as and riordan to sparse graphs without dense spots in doing so we replace the no dense spots hypothesis with weaker assumptions which allow us to analyze graphs with power law degree distributions this gives the rst broadly applicable limit theory for sparse graphs with unbounded average degrees in this paper we lay the foundations of the lp theory of graphons characterize convergence and develop corresponding random graph models while we prove the equivalence of several alternative metrics in a companion paper

robust empirical bayes estimation of means from stratified samples abstract this article considers simultaneous estimation of means from several strata where each stratum contains a finite number of elements for example quite often estimates of annual incomes unemployment rates and so forth must be made simultaneously for many areas our results find application in small area estimation where very few samples are available from an individual area and an estimate of a certain area mean or simultaneous estimates of several area means can be improved by incorporating information from similar neighboring areas ghosh and meeden 1986 considered empirical bayes estimation of the finite population mean assuming a normal superpopulation model their estimators generalize automatically when one estimates a vector of stratum means this article relaxes the normality assumption and assumes instead that the posterior expectation of any stratum mean is a linear function of the sample observations this is the so called posterior linearity property as described in ericson 1

transverse conductivity and longitudinal shear of elliptic fiber composite with imperfect interface abstract the paper addresses the problem of calculating the local fields and effective transport properties and longitudinal shear stiffness of elliptic fiber composite with imperfect interface the rayleigh type representative unit cell approach has been used the micro geometry of composite is modeled by a periodic structure with a unit cell containing multiple elliptic inclusions the developed method combines the superposition principle the technique of complex potentials and certain new results in the theory of special functions an appropriate choice of the potentials provides reducing the boundary value problem to an ordinary well posed set of linear algebraic equations the exact finite form expression of the effective stiffness tensor has been obtained by analytical averaging the local gradient and flux fields the convergence of solution has been verified and the parametric study of the model has been performed the obtained accurate statistically meaningful results illustrate a substantial effect of imperfect interface on the effective behavior of composite

generalized eisenstein functions abstract in the previous works analytical exact and approximate formulas for the effective conductivity of two dimensional random composites were deduced these formulas contain the discrete convolutions of the eisenstein series e sums generalized rayleigh eisenstein mityushev sums the considered objects become the classic lattice sums for regular doubly periodic locations of the centers of inclusions in the present paper we modify the above approach and extend it to natanzon s series having applications in the elasticity theory the conditionally convergent series are investigated by the eisenstein summation method we derive a computationally effective formula for conditionally convergent series numerical simulations suggest new formulas in which some lattice sums are equal to

confidence intervals for the binomial parameter some new considerations several methods have been proposed to construct confidence intervals for the binomial parameter some recent papers introduced the mean coverage criterion to evaluate the performance of confidence intervals and suggested that exact methods because of their conservatism are less useful than asymptotic ones in these studies however exact intervals were always represented by the clopper pearson interval c p now we focus on sterne s interval which is also exact and known to be better than the c p in the two sided case introducing a computer intensive level adjustment procedure which allows constructing intervals that are exact in terms of mean coverage we demonstrate that sterne s interval performs better than the best asymptotic intervals even in the mean coverage context level adjustment improves the c p as well which with an appropriate level adjustment becomes equivalent to the mid p interval finally we show that the asymptotic behaviour of the mid p method is far poorer than is generally expected

semiparametric regression of multidimensional genetic pathway data least squares kernel machines and linear mixed models we consider a semiparametric regression model that relates a normal outcome to covariates and a genetic pathway where the covariate effects are modeled parametrically and the pathway effect of multiple gene expressions is modeled parametrically or nonparametrically using least squares kernel machines lskms this unified framework allows a flexible function for the joint effect of multiple genes within a pathway by specifying a kernel function and allows for the possibility that each gene expression effect might be nonlinear and the genes within the same pathway are likely to interact with each other in a complicated way this semiparametric model also makes it possible to test for the overall genetic pathway effect we show that the lskm semiparametric regression can be formulated using a linear mixed model estimation and inference hence can proceed within the linear mixed model framework using standard mixed model software both the regression coefficients of the covariate effects and the lskm estimator of the genetic pathway effect can be obtained using the best linear unbiased predictor in the corresponding linear mixed model formulation the smoothing parameter and the kernel parameter can be estimated as variance components using restricted maximum likelihood a score test is developed to test for the genetic pathway effect model variable selection within the lskm framework is discussed the methods are illustrated using a prostate cancer data set and evaluated using simulations

a renormalization method for the evaluation of lattice sums a number of problems in solid state physics and materials science can be resolved by the evaluation of certain lattice sums sums over all sites of an infinite perfect lattice of some potential energy function one classical example the calculation of lattice sums of circular and spherical harmonics dates back to the last century to lord rayleigh s work on computing the effective conductivity of a simple composite while lord rayleigh presented an efficient asymptotic method for two dimensional problems he resorted to direct evaluation of the lattice sums in the three dimensional case more recent methods based on ewald summation have been developed by nijboer and de wette schmidt and lee and others in this article a fast method for evaluating lattice sums which is based on a new renormalization identity is described

eigenvalue problems for doubly periodic elastic structures and phononic band gaps we consider the problem of elastic waves propagating in a two dimensional array of circular cavities taking rigorous account of coupling between shear and dilational waves a technique originally due to rayleigh is derived that involves an elegant identity between the singular and non singular components of the stress fields in the array this leads to an infinite linear system which can be truncated and solved in order to determine the complete structure of the propagating modes of particular interest is the possibility of exhibiting phononic band gaps i e domains of frequency for which all propagating vibration in the material is suppressed

asymptotic properties of maximum likelihood estimators and likelihood ratio tests under nonstandard conditions abstract large sample properties of the likelihood function when the true parameter value may be on the boundary of the parameter space are described specifically the asymptotic distribution of maximum likelihood estimators and likelihood ratio statistics are derived these results generalize the work of moran 1971 chant 1974 and chernoff 1954 some of chant s results are shown to be incorrect the approach used in deriving these results follows from comments made by moran and chant the problem is shown to be asymptotically equivalent to the problem of estimating the restricted mean of a multivariate gaussian distribution from a sample of size 1 in this representation the gaussian random variable corresponds to the limit of the normalized score statistic and the estimate of the mean corresponds to the limit of the normalized maximum likelihood estimator thus the limiting distribution of the maximum likelihood estimator is the same as the distribution of the projection of the gaussian random v

parametric empirical bayes inference theory and applications abstract this article reviews the state of multiparameter shrinkage estimators with emphasis on the empirical bayes viewpoint particularly in the case of parametric prior distributions some successful applications of major importance are considered recent results concerning estimates of error and confidence intervals are described and illustrated with data

static bloch sums for the square array we consider sums arising in doubly quasiperiodic green s functions for the laplace equation over the square array the sums are represented as fourier series and it is shown that the coefficients in the series can be obtained as polynomials we give expressions from which the first six array sums can be evaluated efficiently and accurate to better than one part in 107 over most of the brillouin zone

empirical bayes estimation in finite population sampling abstract empirical bayes methods are becoming increasingly popular in statistics robbins 1955 introduced the method in the context of nonparametric estimation of a completely unspecified prior distribution subsequently the method has been explored very successfully in a series of articles by efron and morris 1973 1975 1977 in a parametric framework in the efron morris setup a family of parametric distributions is used as possible priors but only when one or more of the parameters of the family of prior distributions is estimated from the data morris 1983 listed a number of areas where empirical bayes methods are used one of the main features of empirical bayes analysis is to borrow strength from the ensemble that is use information from similar sources in constructing estimators and predictors in addition to the most directly available source of information there are some situations in finite population sampling where such methods might be suitable for instance in many repetitive surve

asymptotic analysis of high contrast phononic crystals and a criterion for the band gap opening we investigate the band gap structure of the frequency spectrum for elastic waves in a high contrast two component periodic elastic medium we consider two dimensional phononic crystals consisting of a background medium which is perforated by an array of holes periodic along each of the two orthogonal coordinate axes in this paper we establish a full asymptotic formula for dispersion relations of phononic band structures as the contrast of the shear modulus and that of the density become large the main ingredients are integral equation formulations of the solutions to the harmonic oscillatory linear elastic equation and several theorems concerning the characteristic values of meromorphic operator valued functions in the complex plane such as the generalized rouche s theorem we establish a connection between the band structures and the dirichlet eigenvalue problem on the elementary hole we also provide a criterion for exhibiting gaps in the band structure which shows that smaller the density of the matrix is the wider the band gap is provided that the criterion is fulfilled this phenomenon was reported by economou and sigalas j acoust soc am 95 1734 1740 1994 who observed that periodic elastic composites whose matrix has lower density and higher shear modulus compared to those of inclusions yield better open gaps our analysis in this paper agrees with this experimental finding

linear score tests for variance components in linear mixed models and applications to genetic association studies summary following the rapid development of genome scale genotyping technologies genetic association mapping has become a popular tool to detect genomic regions responsible for certain disease phenotypes especially in early phase pharmacogenomic studies with limited sample size in response to such applications a good association test needs to be 1 applicable to a wide range of possible genetic models including but not limited to the presence of gene by environment or gene by gene interactions and non linearity of a group of marker effects 2 accurate in small samples fast to compute on the genomic scale and amenable to large scale multiple testing corrections and 3 reasonably powerful to locate causal genomic regions the kernel machine method represented in linear mixed models provides a viable solution by transforming the problem into testing the nullity of variance components in this study we consider score based tests by choosing a statistic linear in the score function when the model under the null hypothesis has only one error variance parameter our test is exact in finite samples when the null model has more than one variance parameter we develop a new moment based approximation that performs well in simulations through simulations and analysis of real data we demonstrate that the new test possesses most of the aforementioned characteristics especially when compared to existing quadratic score tests or restricted likelihood ratio tests

effect of crack orientation statistics on effective stiffness of mircocracked solid this paper addresses the problem of calculating effective elastic properties of a solid containing multiple cracks with prescribed orientation statistics to do so the representative unit cell approach has been used the microgeometry of a cracked solid is modeled by a periodic structure with a unit cell containing multiple cracks a sufficient number is taken to account for the microstructure statistics the developed method combines the superposition principle the technique of complex potentials and certain new results in the theory of special functions a proper choice of potentials provides reducing the boundary value problem to an ordinary well posed set of linear algebraic equations the exact finite form expression of the effective stiffness tensor has been obtained by analytical averaging the strain and stress fields the convergence study has been performed the statistically meaningful results obtained show dependence of the effective elastic stiffness on angular scattering of cracks comparison has been made with the selected simple micromechanical models namely non interaction approximation differential scheme and modified differential scheme it is found that among these models the differential scheme provides the best fit of the numerical data

convergent sequences of dense graphs ii multiway cuts and statistical physics we consider sequences of graphs gn and dene various notions of convergence related to these sequences including left convergence dened in terms of the densities of homomorphisms from small graphs into gn and right convergence dened in terms of the densities of homomorphisms from gn into small graphs we show that right convergence is equivalent to left convergence both for simple graphs gn and for graphs gn with nontrivial nodeweights and edgeweights other equivalent conditions for convergence are given in terms of fundamental notions from combinatorics such as maximum cuts and szemer edi partitions and fundamental notions from statistical physics like energies and free energies we thereby relate local and global properties of graph sequences quantitative forms of these results express the relationships among dierent measures of similarity of large graphs

delta method and bootstrap in linear mixed models to estimate a proportion when no event is observed application to intralesional resection in bone tumor surgery resecting bone tumors requires good cutting accuracy to reduce the occurrence of local recurrence this issue is considerably reduced with a navigated technology the estimation of extreme proportions is challenging especially with small or moderate sample sizes when no success is observed the commonly used binomial proportion confidence interval is not suitable while the rule of three provides a simple solution unfortunately these approaches are unable to differentiate between different unobserved events different delta methods and bootstrap procedures are compared in univariate and linear mixed models with simulations and real data by assuming the normality the delta method on the z score and parametric bootstrap provide similar results but the delta method requires the estimation of the covariance matrix of the estimates in mixed models the observed fisher information matrix with unbounded variance components should be preferred the parametric bootstrap easier to apply outperforms the delta method for larger sample sizes but it may be time costly copyright 2016 john wiley sons ltd

oblique propagation of electromagnetic and elastic waves for an array of cylindrical fibres this paper presents analysis of electromagnetic and elastodynamic waves propagating through a doubly periodic array of cylindrical channels in oblique incidence a new method based on a multipole scattering approach has been proposed to reduce these spectral problems for partial differential equations to certain algebraic problems of the rayleigh type we obtain a formulation in terms of an eigenvalue problem that enables us to construct the high order dispersion curves and to study both photonic and phononic band gap structures in oblique incidence we also address the question of a singular perturbation induced by the conical incidence and discuss some effective properties for ferromagnetic photonic crystal fibres in the long wavelength limit

a score test against one sided alternatives abstract a score type statistic t s is introduced for testing h 0 against k 0 and more general one sided hypotheses when nuisance parameters may be present is a vector parameter the main advantages of t s are that it requires estimation of the model only under the null hypothesis and that it is asymptotically equivalent to the likelihood ratio statistic these are precisely the reasons for the popularity of the score tests for testing against two sided alternatives in this sense t s preserves the main attractive features of the classical two sided score test the theoretical results are presented in a general framework where the likelihood based score function is replaced by an estimating function so that the test is applicable even if the exact population distribution is unknown computation of t s is simplified by the fact that it can be computed easily once the corresponding two sided statistic has been computed the relevance and simplicity of t s are illustrated by discussing

comparisons of alternative predictors under the balanced one way random model abstract prediction of an arbitrary linear combination of the random effects of a balanced one way random model is investigated alternative two stage predictors are compared on the basis of their conditional on the random effects and unconditional bias and mean squared errors when the true value of the ratio of expected mean squares is known there exists a best linear unbiased predictor blup when the true value is unknown a two stage predictor obtained from the blup by replacing the true value with an estimated value can be used when the ratio of expected mean squares is estimated by maximum likelihood bayesian methods or various related methods a two stage predictor is obtained whose properties compare favorably with for example those of the least squares predictor and the positive part james stein predictor

functions of bounded variation the derivative of the one dimensional maximal function and applications to inequalities we prove that if f i r r is of bounded variation then the uncentered maximal function mf is absolutely continuous and its derivative satisfies the sharp inequality dmf l 1 i df i this allows us to obtain under less regularity versions of classical inequalities involving derivatives

large deviations of subgraph counts for sparse erdős renyi graphs abstract for any fixed simple graph h v e and any fixed u 0 we establish the leading order of the exponential rate function for the probability that the number of copies of h in the erd s renyi graph g n p exceeds its expectation by a factor 1 u assuming n h p 1 with h 1 2 where 1 is the maximum degree of h this improves on a previous result of chatterjee and the second author who obtained h c e for a constant c 0 moreover for the case of cycle counts we can take as large as 1 2 we additionally obtain the sharp upper tail for schatten norms of the adjacency matrix as well as the sharp lower tail for counts of graphs for which sidorenko s conjecture holds as a key step we establish quantitative versions of szemeredi s regularity lemma and the counting lemma suitable for the analysis of random graphs in the large deviations regime

variance components testing in the longitudinal mixed effects model this article discusses the asymptotic behavior of likelihood ratio tests for nonzero variance components in the longitudinal mixed effects linear model described by laird and ware 1982 biometrics 38 963 974 our discussion of the large sample behavior of likelihood ratio tests for nonzero variance components is based on the results for nonstandard testing situations by self and liang 1987 journal of the american statistical association 82 605 610

confidence prediction and tolerance in linear mixed models the literature about prediction interval pi and tolerance interval ti in linear mixed models is usually developed for specific designs which is a main limitation to their use this paper proposes to reformulate the two sided pi to be generalizable under a wide variety of designs one random factor nested and crossed designs for multiple random factors and balanced or unbalanced designs this new methodology is based on the hessian matrix namely the inverse of observed fisher information matrix and is built with a cell mean model the degrees of freedom for the total variance are calculated with the generalized satterthwaite method and compared to the kenward roger s degrees of freedom for fixed effects construction of two sided tis are also detailed with one random factor and two nested and two crossed random variables an extensive simulation study is carried out to compare the widths and coverage probabilities of confidence intervals ci pis and tis to their nominal levels it shows excellent coverage whatever the design and the sample size are finally these cis pis and tis are applied to two real data sets one from orthopedic surgery study intralesional resection risk and the other from assay validation study during vaccine development

local convergence for permutations and local limits for uniform rho avoiding permutations with rho 3 we set up a new notion of local convergence for permutations and we prove a characterization in terms of proportions of consecutive pattern occurrences we also characterize random limiting objects for this new topology introducing a notion of shift invariant property corresponding to the notion of unimodularity for random graphs we then study two models in the framework of random pattern avoiding permutations we compute the local limits of uniform rho avoiding permutations for rho 3 when the size of the permutations tends to infinity the core part of the argument is the description of the asymptotics of the number of consecutive occurrences of any given pattern for this result we use bijections between rho avoiding permutations and rooted ordered trees local limit results for galton watson trees the second moment method and singularity analysis

testing against a high dimensional alternative summary as the dimensionality of the alternative hypothesis increases the power of classical tests tends to diminish quite rapidly this is especially true for high dimensional data in which there are more parameters than observations we discuss a score test on a hyperparameter in an empirical bayesian model as an alternative to classical tests it gives a general test statistic which can be used to test a point null hypothesis against a high dimensional alternative even when the number of parameters exceeds the number of samples this test will be shown to have optimal power on average in a neighbourhood of the null hypothesis which makes it a proper generalization of the locally most powerful test to multiple dimensions to illustrate this new locally most powerful test we investigate the case of testing the global null hypothesis in a linear regression model in more detail the score test is shown to have significantly more power than the f test whenever under the alternative the large variance principal components of the design matrix explain substantially more of the variance of the outcome than do the small variance principal components the score test is also useful for detecting sparse alternatives in truly high dimensional data where its power is comparable with the test based on the maximum absolute t statistic

exact tests for variance components in unbalanced mixed linear models a method for deriving exact tests for variance components in some unbalanced mixed linear models is presented the derivation is based on a new kind of preliminary orthogonal transformation and a subsequent resampling procedure the resulting tests are based on mutually independent sums of squares which under the null hypothesis are distributed as scalar multiples of chi square variates for balanced data the tests are identical to the traditional f tests

exact tests for variance components ofversten 1993 presented methods for obtaining exact f tests of variance components in three unbalanced mixed linear models he developed methods for models with one random factor models with nested classifications and models with interaction between two random factors in section 2 we review and extend wald s test which is a generalization of one of ofversten s methods ofversten s other method is a generalization of khuri 1987 1990 and khuri and littell 1987 it does not apply to the one random factor model in section 3 we present a unified treatment of ofversten s second method this treats all of ofversten s examples and gives a formal justification to ofversten s claim that the method extends quite generally section 3 also gives a formal condition for when the method does not require resampling from the residuals the unified treatment is conceptually simpler than ofversten s treatment in that it uses a single orthonormal basis rather than a series of matrix decompositions that change slightly from example to example there do not seem to be any particular computational advantages to the unified treatment the unified treatment also appears to generalize ofversten s method in that it applies in some cases where wald s test also applies section 4 compares the two tests when both apply all of the arguments are given in terms of vector space ideas cf christensen 1987

testing against a high dimensional alternative in the generalized linear model asymptotic type i error control summary testing a low dimensional null hypothesis against a high dimensional alternative in a generalized linear model may lead to a test statistic that is a quadratic form in the residuals under the null model using asymptotic arguments we show that the distribution of such a test statistic can be approximated by a ratio of quadratic forms in normal variables for which algorithms are readily available for generalized linear models the asymptotic distribution shows good control of type i error for moderate to small samples even when the number of covariates in the model far exceeds the sample size

approximations for standard errors of estimators of fixed and random effects in mixed linear models abstract best linear unbiased estimators of the fixed and random effects of mixed linear models are available when the true values of the variance ratios are known if the true values are replaced by estimated values the mean squared errors of the estimators of the fixed and random effects increase in size the magnitude of this increase is investigated and a general approximation is proposed the performance of this approximation is investigated in the context of a the estimation of the effects of the balanced one way random model and b the estimation of treatment contrasts for balanced incomplete block designs

order restricted score tests for homogeneity in generalised linear and nonlinear mixed models lin 1997 building on work by solomon cox 1992 and lin breslow 1996 introduced a score test for homogeneity in the generalised linear mixed model in this paper we propose an improvement to lin s test that exploits the fact that covariance parameters associated with random effects in the model are constrained so as to form a positive semidefinite covariance matrix therefore an omnibus score test will have power against alternatives that are known never to occur our improvement works by concentrating the score test into directions given by the positive semidefiniteness constraint we apply the same sort of modification to the bias corrected version of her score test in the generalised linear mixed model context and in a nonlinear mixed model context to a score test analogous to hers we show how to obtain the limiting distribution of our proposed test statistic we also argue via simulations and local alternative power calculations that the restricted test has better power than the unrestricted one

two sided confidence intervals for the single proportion comparison of seven methods simple interval estimate methods for proportions exhibit poor coverage and can produce evidently inappropriate intervals criteria appropriate to the evaluation of various proposed methods include closeness of the achieved coverage probability to its nominal value whether intervals are located too close to or too distant from the middle of the scale expected interval width avoidance of aberrations such as limits outside 0 1 or zero width intervals and ease of use whether by tables software or formulae seven methods for the single proportion are evaluated on 96 000 parameter space points intervals based on tail areas and the simpler score methods are recommended for use in each case methods are available that aim to align either the minimum or the mean coverage with the nominal 1 alpha

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

sparse regular random graphs spectral density and eigenvectors we examine the empirical distribution of the eigenvalues and the eigenvectors of adjacency matrices of sparse regular random graphs we find that when the degree sequence of the graph slowly increases to infinity with the number of vertices the empirical spectral distribution converges to the semicircle law moreover we prove concentration estimates on the number of eigenvalues over progressively smaller intervals we also show that with high probability all the eigenvectors are delocalized

bounds for effective parameters of heterogeneous media by analytic continuation we give a mathematical formulation of a method for obtaining bounds on effective parameters developed by d bergman and g w milton this method in contrast to others used before does not rely on a variational principle but exploits the properties of the effective parameter as an analytic function of the component parameters the method is at present restricted to two component media