kazhdan projections random walks and ergodic theorems in this paper we investigate generalizations of kazhdan s property t to the setting of uniformly convex banach spaces we explain the interplay between the existence of spectral gaps and that of kazhdan projections our methods employ markov operators associated to a random walk on the group for which we provide new norm estimates and convergence results this construction exhibits useful properties and flexibility and allows to reclassify the existence of kazhdan projections from a c algebraic phenomenon to a natural object associated to random walks on groups we give a number of applications of these results in particular we address several open questions we give a direct comparison of properties te and fe with lafforgue s reinforced banach property t we obtain shrinking target theorems for orbits of kazhdan groups and apply them to a question of kleinbock and margulis finally we construct non compact ghost projections for warped cones answering a question of willett and yu in this last case we conjecture that such warped cones provide new counterexamples to the coarse baum connes conjecture

accurate and efficient power calculations for 2 x m tables in unmatched case control designs the aim of this article was to derive an approximation of the distribution of pearson s statistic for 2 x m contingency tables to calculate power for case control studies accurately and efficiently in terms of computer time we first prove that rather than a non central chi square distribution pearson s statistic is asymptotically equivalent with the sum of squares of m independent normal random variables whose variances are typically different under the alternative hypothesis based on this asymptotically equivalent ae we derive a cost effective approximation ce numerical results show that ce was almost as precise as ae but computationally more efficient although somewhat less costly in terms of cpu time we show that a commonly used approximation using the non central chi square distribution can be very inaccurate and overestimate power in some scenarios and underestimate it in others simulations and exact distributions on the other hand are accurate but computationally very intensive compared to our ce the ce reached its lowest accuracy under the null hypothesis where it has a chi square distribution with m 1 degree of freedom this suggests that ce can be used to calculate power for tables where researchers would feel comfortable using pearson s chi square test

the heteroscedastic graded response model with a skewed latent trait testing statistical and substantive hypotheses related to skewed item category functions the graded response model grm samejima estimation of ability using a response pattern of graded scores psychometric monograph no 17 richmond va the psychometric society 1969 can be derived by assuming a linear regression of a continuous variable z on the trait to underlie the ordinal item scores takane de leeuw in psychometrika 52 393 408 1987 traditionally a normal distribution is specified for z implying homoscedastic error variances and a normally distributed in this paper we present the heteroscedastic grm with skewed latent trait which extends the traditional grm by incorporation of heteroscedastic error variances and a skew normal latent trait an appealing property of the extended grm is that it includes the traditional grm as a special case this enables specific tests on the normality assumption of z we show how violations of normality in z can lead to asymmetrical category response functions the ability to test this normality assumption is beneficial from both a statistical and substantive perspective in a simulation study we show the viability of the model and investigate the specificity of the effects we apply the model to a dataset on affect and a dataset on alexithymia

an autoregressive growth model for longitudinal item analysis a first order autoregressive growth model is proposed for longitudinal binary item analysis where responses to the same items are conditionally dependent across time given the latent traits specifically the item response probability for a given item at a given time depends on the latent trait as well as the response to the same item at the previous time or the lagged response an initial conditions problem arises because there is no lagged response at the initial time period we handle this problem by adapting solutions proposed for dynamic models in panel data econometrics asymptotic and finite sample power for the autoregressive parameters are investigated the consequences of ignoring local dependence and the initial conditions problem are also examined for data simulated from a first order autoregressive growth model the proposed methods are applied to longitudinal data on korean students self esteem

spectra of sparse random matrices we compute the spectral density for ensembles of sparse symmetric random matrices using replica our formulation of the replica symmetric ansatz shares the symmetries of that suggested in a seminal paper by rodgers and bray symmetry with respect to permutation of replica and rotation symmetry in the space of replica but uses a different representation in terms of superpositions of gaussians it gives rise to a pair of integral equations which can be solved by a stochastic population dynamics algorithm remarkably our representation allows us to identify pure point contributions to the spectral density related to the existence of normalizable eigenstates our approach is not restricted to matrices defined on graphs with poissonian degree distribution matrices defined on regular random graphs or on scale free graphs are easily handled we also look at matrices with row constraints such as discrete graph laplacians our approach naturally allows us to unfold the total density of states into contributions coming from vertices of different local coordinations and an example of such an unfolding is presented our results are well corroborated by numerical diagonalization studies of large finite random matrices

on the characteristic polynomial of a random unitary matrix we present a range of fluctuation and large deviations results for the logarithm of the characteristic polynomial z of a random n n unitary matrix as n first we show that evaluated at a finite set of distinct points is asymptotically a collection of i i d complex normal random variables this leads to a refinement of a recent central limit theorem due to keating and snaith and also explains the covariance structure of the eigenvalue counting function next we obtain a central limit theorem for ln z in a sobolev space of generalised functions on the unit circle in this limiting regime lower order terms which reflect the global covariance structure are no longer negligible and feature in the covariance structure of the limiting gaussian measure large deviations results for ln z a evaluated at a finite set of distinct points can be obtained for for higher order scalings we obtain large deviations results for ln z a evaluated at a single point there is a phase transition at a ln n which only applies to negative deviations of the real part reflecting a switch from global to local conspiracy

zeroes of zeta functions and symmetry hilbert and polya suggested that there might be a natural spectral interpretation of the zeroes of the riemann zeta function while at the time there was little evidence for this today the evidence is quite convincing firstly there are the function field analogues that is zeta functions of curves over finite fields and their generalizations for these a spectral interpretation for their zeroes exists in terms of eigenvalues of frobenius on cohomology secondly the developments both theoretical and numerical on the local spacing distributions between the high zeroes of the zeta function and its generalizations give striking evidence for such a spectral connection moreover the low lying zeroes of various families of zeta functions follow laws for the eigenvalue distributions of members of the classical groups in this paper we review these developments in order to present the material fluently we do not proceed in chronological order of discovery also in concentrating entirely on the subject matter of the title we are ignoring the standard body of important work that has been done on the zeta function and l functions 1 the montgomery odlyzko law we begin with the riemann zeta function and some phenomenology associated with it

structural modeling and psychometrika an historical perspective on growth and achievements the field of linear structural equation modeling with continuous variables is reviewed trends in psychometric theory and data analysis across the five decades of publication ofpsychometrika are discussed especially the clarification of concepts of population and sample explication of the parametric structure of models delineation of concepts of exploratory and confirmatory data analysis expansion of statistical theory in psychometrics estimation via optimization of an explicit objective function and implementation of general function minimization methods developments in the ideas of factor analysis latent variables as well as structural and causal modeling are noted some major conceptual achievements involving general covariance structure representations multiple population models and moment structures are reviewed the major statistical achievements of normal theory generalized least squares estimation elliptical and distribution free estimation and higher moment estimation are discussed computer programs that implement some of the theoretical developments are described

random matrix theory and the zeros of ζ s we study the density of the roots of the derivative of the characteristic polynomial z u z of an n n random unitary matrix with distribution given by haar measure on the unitary group based on previous random matrix theory models of the riemann zeta function s this is expected to be an accurate description for the horizontal distribution of the zeros of s to the right of the critical line we show that as n the fraction of the roots of z u z that lie in the region 1 x n 1 z 1 tends to a limit function we derive asymptotic expressions for this function in the limits x and x 0 and compare them with numerical experiments

systems of diagonal diophantine inequalities we treat systems of real diagonal forms f 1 x f 2 x f r x of degree k in s variables we give a lower bound s 0 r k which depends only on r and k such that if s s 0 r k holds then under certain conditions on the forms and for any positive real number e there is a nonzero integral simultaneous solution x z s of the system of diophantine inequalities f i x e for i i r in particular our result is one of the first to treat systems of inequalities of even degree the result is an extension of earlier work by the author on quadratic forms also a restriction in that work is removed which enables us to now treat combined systems of diophantine equations and inequalities

patterns in eigenvalues the 70th josiah willard gibbs lecture typical large unitary matrices show remarkable patterns in their eigenvalue distribution these same patterns appear in telephone encryption the zeros of riemann s zeta function a variety of physics problems and in the study of toeplitz operators this paper surveys these applications and what is currently known about the patterns

sample size determination within the scope of conditional maximum likelihood estimation with special focus on testing the rasch model this paper refers to the exponential family of probability distributions and the conditional maximum likelihood cml theory it is concerned with the determination of the sample size for three groups of tests of linear hypotheses known as the fundamental trinity of wald score and likelihood ratio tests the main practical purpose refers to the special case of tests of the class of rasch models the theoretical background is discussed and the formal framework for sample size calculations is provided given a predetermined deviation from the model to be tested and the probabilities of the errors of the first and second kinds

latent variable modeling in heterogeneous populations common applications of latent variable analysis fail to recognize that data may be obtained from several populations with different sets of parameter values this article describes the problem and gives an overview of methodology that can address heterogeneity artificial examples of mixtures are given where if the mixture is not recognized strongly distorted results occur mimic structural modeling is shown to be a useful method for detecting and describing heterogeneity that cannot be handled in regular multiple group analysis other useful methods instead take a random effects approach describing heterogeneity in terms of random parameter variation across groups these random effects models connect with emerging methodology for multilevel structural equation modeling of hierarchical data examples are drawn from educational achievement testing psychopathology and sociology of education estimation is carried out by the liscomp program

applications and generalizations of fisher hartwig asymptotics fisher hartwig asymptotics refers to the large n form of a class of toeplitz determinants with singular generating functions this class of toeplitz determinants occurs in the study of the spin spin correlations for the two dimensional ising model and the ground state density matrix of the impenetrable bose gas amongst other problems in mathematical physics we give a new application of the original fisher hartwig formula to the asymptotic decay of the ising correlations above tc while the study of the bose gas density matrix leads us to generalize the fisher hartwig formula to the asymptotic form of random matrix averages over the classical groups and the gaussian and laguerre unitary matrix ensembles another viewpoint of our generalizations is that they extend to hankel determinants the fisher hartwig asymptotic form known for toeplitz determinants

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

moments of ratios of characteristic polynomials of a certain class of random matrices we derive a new method of calculating the mean negative moments and ratios of squared modulus of characteristic polynomials of a certain class of random matrices new results obtained with this method are presented

a note to bollen s alternative fit measure the problem of dependence of the nonnormed fit index on sample size in covariance structure analysis is discussed contrary to bollen 1986 we show that the mean of the nonnormed fit index is independent of sample size for true and almost true models whereas bollen s alternative index does depend on sample size

averages of characteristic polynomials in random matrix theory we compute averages of products and ratios of characteristic polynomials associated with orthogonal unitary and symplectic ensembles of random matrix theory the pfaffian determinantal formulae for these averages are obtained and the bulk scaling asymptotic limits are found for ensembles with gaussian weights classical results for the correlation functions of the random matrix ensembles and their bulk scaling limits are deduced from these formulae by a simple computation we employ a discrete approximation method the problem is solved for discrete analogues of random matrix ensembles originating from representation theory and then a limit transition is performed exact pfaffian determinantal formulae for the discrete averages are proven using standard tools of linear algebra no application of orthogonal or skew orthogonal polynomials is needed 2005 wiley periodicals inc

modeling λ invariants by p adic random matrices how often is the p adic invariant of an imaginary quadratic field equal to m we model this by the statistics of random p adic matrices and test these predictions numerically 2011 wiley periodicals inc

lattice point problems and values of quadratic forms for d dimensional ellipsoids e with d 5 we show that the number of lattice points in re is approximated by the volume of re as r tends to infinity up to an error of order mathcal o r d 2 for general ellipsoids and up to an error of order o rd 2 for irrational ones the estimate refines earlier bounds of the same order for dimensions d 9 as an application a conjecture of davenport and lewis about the shrinking of gaps between large consecutive values of q m m d of positive definite irrational quadratic forms q of dimension d 5 is proved finally we provide explicit bounds for errors in terms of certain minkowski minima of convex bodies related to these quadratic forms

random matrix theory the exceptional lie groups and l functions there has recently been interest in relating properties of matrices drawn at random from the classical compact groups to statistical characteristics of number theoretical l functions one example is the relationship conjectured to hold between the value distributions of the characteristic polynomials of such matrices and value distributions within families of l functions these connections are extended here to non classical groups we focus on an explicit example the exceptional lie group g2 the value distributions for characteristic polynomials associated with the 7 and 14 dimensional representations of g2 defined with respect to the uniform invariant haar measure are calculated using two of the macdonald constant term identities a one parameter family of l functions over a finite field is described whose value distribution in the limit as the size of the finite field grows is related to that of the characteristic polynomials associated with the seven dimensional representation of g2 the random matrix calculations extend to all exceptional lie groups

random matrix theory and l functions at s 1 2 recent results of katz and sarnak 8 9 suggest that the low lying zeros of families of l functions display the statistics of the eigenvalues of one of the compact groups of matrices u n o n or usp 2n we here explore the link between the value distributions of the l functions within these families at the central point s 1 2 and those of the characteristic polynomials z u of matrices u with respect to averages over so 2n and usp 2n at the corresponding point 0 using techniques previously developed for u n in 10 for any matrix size n we find exact expressions for the moments of z u 0 for each ensemble and hence calculate the asymptotic large n value distributions for z u 0 and log z u 0 the asymptotic results for the integer moments agree precisely with the few corresponding values known for l functions the value distributions suggest consequences for the non vanishing of l functions at the central point

on absolute moments of characteristic polynomials of a certain class of complex random matrices the integer moments of the spectral determinant det zi w 2 of complex random matrices w are obtained in terms of the characteristic polynomial of the positive semidefinite matrix ww for the class of matrices w au where a is a given matrix and u is random unitary this work is motivated by studies of complex eigenvalues of random matrices and potential applications of the obtained results are discussed in this context

on the generalized circle problem for a random lattice in large dimension abstract in this note we study the error term r n l x in the generalized circle problem for a ball of volume x and a random lattice l of large dimension n our main result is the following functional central limit theorem fix an arbitrary function f z r satisfying lim n f n and f n o e e e n for every e 0 then the random function t 1 2 f n r n l t f n on the interval 0 1 converges in distribution to one dimensional brownian motion as n the proof goes via convergence of moments and for the computations we develop a new version of rogers mean value formula from 18 for the individual kth moment of the variable 2 f n 1 2 r n l f n we prove convergence to the corresponding gaussian moment more generally for functions f satisfying f n o e c n for any fixed c 0 c k where c k is a constant depending on k whose optimal value we determine

some combinatorial properties of jack symmetric functions if 1 l n then write 2 n or 1 n if p is another partition then write p c j if pi 6 for all i i e if the diagram of 1 contains the diagram of p if ipl ii 1 then write p 2 1 reverse lexicographic order if either j i or the first nonvanishing difference e pi is positive for instance writing 2 1 i for ai i it 5 5 41 32 5 311 221 2111 11111 finally write p i if ipl lal and p1 pl2 pli a1 a2 li for all i macdonald m p 63 call the partial ordering the natural ordering but we will call it the dominance ordering let x x x2 be an infinite set of indeterminates as in mi we

quadratic forms and systems of forms in many variables let f 1 ldots f r be quadratic forms with integer coefficients in n variables when n ge 9r and the variety v f 1 ldots f r is a smooth complete intersection we prove an asymptotic formula for the number of integer points in an expanding box at which these forms simultaneously vanish which in particular implies the hasse principle for v f 1 ldots f r previous work in this direction required n to grow at least quadratically with r we give a similar result for r forms of degree d conditional on an upper bound for the number of solutions to an auxiliary inequality in principle this result may apply as soon as n d2 dr in the case that d ge 3 several strategies are available to prove the necessary upper bound for the auxiliary inequality in a forthcoming paper we use these ideas to apply the circle method to nonsingular systems of forms with real coefficients

numerical computations concerning the erh this paper describes a computation which established the erh to height 10000 for all primitive dirichlet l series with conductor q 13 and to height 2500 for all q 72 all composite q 112 and other moduli the computations were based on euler maclaurin summation care was taken to obtain mathematically rigorous results the zeros were first located within 10 12 then rigorously separated using an interval arithmetic package a generalized turing criterion was used to show there were no zeros off the critical line statistics about the spacings between zeros were compiled to test the pair correlation conjecture and gue hypothesis

sign changes of the error term in weyl s law for heisenberg manifolds let r t be the error term in weyl s law for the 2l 1 dimensional heisenberg manifold hl gl in this paper several results on the sign changes and odd moments of r t are proved in particular it is proved that for some sufficiently large constant c r t changes sign in the interval t t c t for all large t moreover for a small constant c1 there exist infinitely many subintervals in t 2t of length c1 t log 5 t such that r t c1tl 1 4 holds on each of these subintervals

power of the likelihood ratio test in covariance structure analysis a procedure for computing the power of the likelihood ratio test used in the context of covariance structure analysis is derived the procedure uses statistics associated with the standard output of the computer programs commonly used and assumes that a specific alternative value of the parameter vector is specified using the noncentral chi square distribution the power of the test is approximated by the asymptotic one for a sequence of local alternatives the procedure is illustrated by an example a monte carlo experiment also shows how good the approximation is for a specific case

lattice point problems and distribution of values of quadratic forms for d dimensional irrational ellipsoids e with d 9 we show that the number of lattice points in re is approximated by the volume of re as r tends to infinity up to an error of order o rd 2 the estimate refines an earlier authors bound of order o rd 2 which holds for arbitrary ellipsoids and is optimal for rational ellipsoids as an application we prove a conjecture of davenport and lewis that the gaps between successive values say s 0 as s oo for d 9 for comparison note that sups n s s 0 for rational q x and d 5 as a corollary we derive oppenheim s conjecture for indefinite irrational quadratic forms i e the set q zd is dense in r for d 9 which was proved for d 3 by g margulis marl in 1986 using other methods finally we provide explicit bounds for errors in terms of certain characteristics of trigonometric sums

demography of cirsium vulgare in a grazing experiment the complete demographies of 16 populations of cirsium vulgare were followed in a replicated experiment the experiment was a factorial combination of two intensities of sheep grazing in each of three seasons winter grazed or ungrazed spring grazed or ungrazed and summer light or heavy grazing giving eight treatments in two blocks for 6 years from 1987 to 1992 the population sizes of c vulgare were monitored in each of the 16 paddocks after 1989 grazing in spring or winter or increased grazing in summer all increased population sizes population sizes fluctuated widely between years the effects of the grazing treatments and plant sizes on the transitions between nine life history stages were determined

on the averages of characteristic polynomials from classical groups we provide an elementary and self contained derivation of formulae for averages of products and ratios of characteristic polynomials of random matrices from classical groups using classical results due to weyl and littlewood

on the value distribution of the epstein zeta function in the critical strip we study the value distribution of the epstein zeta function e n l s for 0 infinity we prove that the random variable v n 2c e n cn has a limit distribution which we give explicitly here v n is the volume of the n dimensional unit ball more generally for any fixed epsilon 0 we determine the limit distribution of the random function c bar right arrow v n 2c e n cn c epsilon 1 4 epsilon 1 2 epsilon after compensating for the pole at c 1 2 we even obtain a limit result on the whole interval 1 4 epsilon 1 2 and as a special case we deduce the following strengthening of a result by sarnak and strombergsson concerning the height function h n l of the flat torus r n l the random variable n h n l log 4 pi gamma 1 log n has a limit distribution as n infinity which we give explicitly finally we discuss a question posed by sarnak and strombergsson as to whether there exists a lattice l subset of r n for which e n l s has no zeros in 0 infinity

mean comparison manifest variable versus latent variable mean comparisons are of great importance in the application of statistics procedures for mean comparison with manifest variables have been well studied however few rigorous studies have been conducted on mean comparisons with latent variables although the methodology has been widely used and documented this paper studies the commonly used statistics in latent variable mean modeling and compares them with parallel manifest variable statistics our results indicate that under certain conditions the likelihood ratio and wald statistics used for latent mean comparisons do not always have greater power than the hotelling t2 statistics used for manifest mean comparisons the noncentrality parameter corresponding to the t2 statistic can be much greater than those corresponding to the likelihood ratio and wald statistics which we find to be different from those provided in the literature under a fixed alternative hypothesis our results also indicate that the likelihood ratio statistic can be stochastically much greater than the corresponding wald statistic the robustness property of each statistic is also explored when the model is misspecified or when data are nonnormally distributed recommendations and advice are provided for the use of each statistic

on the recovery of a function on a circular domain we consider the problem of estimating a function f x y on the unit disk f x y x sup 2 y sup 2 spl les 1 given a discrete and noisy data recorded on a regular square grid an estimate of f x y based on a class of orthogonal and complete functions over the unit disk is proposed this class of functions has a distinctive property of being invariant to rotation of axes about the origin of coordinates yielding therefore a rotationally invariant estimate for radial functions the orthogonal set has a particularly simple form being related to the classical legendre polynomials we give the statistical accuracy analysis of the proposed estimate of f x y in the sense of the l sub 2 metric it is found that there is an inherent limitation in the precision of the estimate due to the geometric nature of a circular domain this is explained by relating the accuracy issue to the celebrated problem in the analytic number theory called the lattice points of a circle in fact the obtained bounds for the mean integrated squared error are determined by the best known result so far on the problem of lattice points within the circular domain

characteristic polynomials of random matrices number theorists have studied extensively the connections between the distribution of zeros of the riemann function and of some generalizations with the statistics of the eigenvalues of large random matrices it is interesting to compare the average moments of these functions in an interval to their counterpart in random matrices which are the expectation values of the characteristic polynomials of the matrix it turns out that these expectation values are quite interesting for instance the moments of order 2k scale for unitary invariant ensembles as the density of eigenvalues raised to the power k 2 the prefactor turns out to be a universal number i e it is independent of the specific probability distribution an equivalent behaviour and prefactor had been found as a conjecture within number theory the moments of the characteristic determinants of random matrices are computed here as limits at coinciding points of multi point correlators of determinants these correlators are in fact universal in dyson s scaling limit in which the difference between the points goes to zero the size of the matrix goes to infinity and their product remains finite

beyond the excised ensemble modelling elliptic curve l functions with random matrices the excised ensemble a random matrix model for the zeros of quadratic twist families of elliptic curve l functions was introduced by duenez et al 2012 j phys a math theor 45 115207 the excised model is motivated by a formula for central values of these l functions in a paper by kohnen and zagier 1981 invent math 64 175 98 this formula indicates that for a finite set of l functions from a family of quadratic twists the central values are all either zero or are greater than some positive cutoff the excised model imposes this same condition on the central values of characteristic polynomials of matrices from strangely the cutoff on the characteristic polynomials that results in a convincing model for the l function zeros is significantly smaller than that which we would obtain by naively transferring kohnen and zagier s cutoff to the ensemble in this current paper we investigate a modification to the excised model it lacks the simplicity of the original excised ensemble but it serves to explain the reason for the unexpectedly low cutoff in the original excised model additionally the distribution of central l values is choppier than the distribution of characteristic polynomials in the sense that it is a superposition of a series of peaks the characteristic polynomial distribution is a smooth approximation to this the excised model did not attempt to incorporate these successive peaks only the initial cutoff here we experiment with including some of the structure of the l value distribution the conclusion is that a critical feature of a good model is to associate the correct mass to the first peak of the l value distribution

moments and distribution of central l values of quadratic twists of elliptic curves we show that if one can compute a little more than a particular moment for some family of l functions then one has upper bounds of the conjectured order of magnitude for all smaller positive real moments and a one sided central limit theorem holds we illustrate our method for the family of quadratic twists of an elliptic curve obtaining sharp upper bounds for all moments below the first we also establish a one sided central limit theorem supporting a conjecture of keating and snaith our work leads to a conjecture on the distribution of the order of the tate shafarevich group for rank zero quadratic twists of an elliptic curve and establishes the upper bound part of this conjecture assuming the birch swinnerton dyer conjecture

average ranks of elliptic curves tension between data and conjecture rational points on elliptic curves are the gems of arithmetic they are to diophantine geometry what units in rings of integers are to algebraic number theory what algebraic cycles are to algebraic geometry a rational point in just the right context at one place in the theory can inhibit and control thanks to ideas of kolyvagin kol88 the existence of rational points and other mathematical structures elsewhere despite all that we know about these objects the initial mystery and excitement that drew mathematicians to this arena in the first place remains in full force today we have a network of heuristics and conjectures regarding rational points and we have massive data accumulated to exhibit instances of the phenomena generally we would expect that our data support our conjectures and if not we lose faith in our conjectures but here there is a somewhat more surprising interrelation between data and conjecture they are not exactly in open conflict one with the other but they are no great comfort to each other either we discuss various aspects of this story including recent heuristics and data that attempt to resolve this mystery we shall try to convince the reader that despite seeming discrepancy data and conjecture are in fact in harmony

on insensitivity of the chi square model test to nonlinear misspecification in structural equation models in this paper we show that for some structural equation models sem the classical chi square goodness of fit test is unable to detect the presence of nonlinear terms in the model as an example we consider a regression model with latent variables and interactions terms not only the model test has zero power against that type of misspecifications but even the theoretical chi square distribution of the test is not distorted when severe interaction term misspecification is present in the postulated model we explain this phenomenon by exploiting results on asymptotic robustness in structural equation models the importance of this paper is to warn against the conclusion that if a proposed linear model fits the data well according to the chi quare goodness of fit test then the underlying model is linear indeed it will be shown that the underlying model may in fact be severely nonlinear in addition the present paper shows that such insensitivity to nonlinear terms is only a particular instance of a more general problem namely the incapacity of the classical chi square goodness of fit test to detect deviations from zero correlation among exogenous regressors either being them observable or latent when the structural part of the model is just saturated

derivative moments for characteristic polynomials from the cue we calculate joint moments of the characteristic polynomial of a random unitary matrix from the circular unitary ensemble and its derivative in the case that the power in the moments is an odd positive integer the calculations are carried out for finite matrix size and in the limit as the size of the matrices goes to infinity the latter asymptotic calculation allows us to prove a long standing conjecture from random matrix theory

the maximum size of l functions we conjecture the true rate of growth of the maximum size of the riemann zeta function and other l functions we support our conjecture using arguments from random matrix theory conjectures for moments of l functions and also by assuming a random model for the primes

linear statistics of point processes via orthogonal polynomials for arbitrary 0 we use the orthogonal polynomials techniques developed in killip and nenciu in arxiv math 0508113v1 2005 killip and nenciu in int math res not 50 2665 2701 2004 to study certain linear statistics associated with the circular and jacobi ensembles we identify the distribution of these statistics then prove a joint central limit theorem in the circular case similar statements have been proved using different methods by a number of authors in the jacobi case these results are new

autocorrelation of random matrix polynomials we calculate the autocorrelation functions or shifted moments of the characteristic polynomials of matrices drawn uniformly with respect to haar measure from the groups u n o 2n and usp 2n in each case the result can be expressed in three equivalent forms as a determinant sum and hence in terms of symmetric polynomials as a combinatorial sum and as a multiple contour integral these formulae are analogous to those previously obtained for the gaussian ensembles of random matrix theory but in this case are identities for any size of matrix rather than large matrix asymptotic approximations they also mirror exactly the autocorrelation formulae conjectured to hold for l functions in a companion paper this then provides further evidence in support of the connection between random matrix theory and the theory of l functions

moment testing for interaction terms in structural equation modeling starting with kenny and judd psychol bull 96 201 210 1984 several methods have been introduced for analyzing models with interaction terms in all these methods more information from the data than just means and covariances is required in this paper we also use more than just first and second order moments however we are aiming to adding just a selection of the third order moments the key issue in this paper is to develop theoretical results that will allow practitioners to evaluate the strength of different third order moments in assessing interaction terms of the model to select the third order moments we propose to be guided by the power of the goodness of fit test of a model with no interactions which varies with each selection of third order moments a theorem is presented that relates the power of the usual goodness of fit test of the model with the power of a moment test for the significance of third order moments the latter has the advantage that it can be computed without fitting a model the main conclusion is that the selection of third order moments can be based on the power of a moment test thus assessing the relevance in the analysis of different sets of third order moments can be computationally simple the paper gives an illustration of the method and argues for the need of refraining from adding into the analysis an excess of higher order moments

on the distribution of spacings between zeros of the zeta function etude numerique de la distribution des espacements des zeros de la fonction zeta de riemann observation de certains phenomenes inatendus

sample size determination for rasch model tests this paper is concerned with supplementing statistical tests for the rasch model so that additionally to the probability of the error of the first kind type i probability the probability of the error of the second kind type ii probability can be controlled at a predetermined level by basing the test on the appropriate number of observations an approach to determining a practically meaningful extent of model deviation is proposed and the approximate distribution of the wald test is derived under the extent of model deviation of interest

developments in random matrix theory in this introduction to the journal of physics a special issue on random matrix theory we give a review of the main historical developments in random matrix theory a short summary of the papers that appear in this special issue is also given

singularity dominated strong fluctuations for some random matrix averages the circular and jacobi ensembles of random matrices have their eigenvalue support on the unit circle of the complex plane and the interval 0 1 of the real line respectively the averaged value of the modulus of the corresponding characteristic polynomial raised to the power 2 diverges for 2 1 at points approaching the eigenvalue support using the theory of generalized hypergeometric functions based on jack polynomials the functional form of the leading asymptotic behaviour is established rigorously in the circular ensemble case this confirms a conjecture of berry and keating

how should we assess the fit of rasch type models approximating the power of goodness of fit statistics in categorical data analysis we investigate the performance of three statistics r 1 r 2 glas in psychometrika 53 525 546 1988 and m 2 maydeu olivares joe in j am stat assoc 100 1009 1020 2005 psychometrika 71 713 732 2006 to assess the overall fit of a one parameter logistic model 1pl estimated by marginal maximum likelihood ml r 1 and r 2 were specifically designed to target specific assumptions of rasch models whereas m 2 is a general purpose test statistic we report asymptotic power rates under some interesting violations of model assumptions different item discrimination presence of guessing and multidimensionality as well as empirical rejection rates for correctly specified models and some misspecified models all three statistics were found to be more powerful than pearson s x 2 against two and three parameter logistic alternatives 2pl and 3pl and against multidimensional 1pl models the results suggest that there is no clear advantage in using goodness of fit statistics specifically designed for rasch type models to test these models when marginal ml estimation is used

moments of the logarithmic derivative of characteristic polynomials from so n and usp 2n we study moments of the logarithmic derivative of characteristic polynomials of orthogonal and symplectic random matrices in particular we compute the asymptotics for large matrix size n of these moments evaluated at points that are approaching 1 this follows the work of bailey et al j math phys 60 8 083509 2019 where they computed these asymptotics in the case of unitary random matrices

moments of the riemann zeta function assuming the riemann hypothesis we obtain an upper bound for the moments of the riemann zeta function on the critical line our bound is nearly as sharp as the conjectured asymptotic formulae for these moments the method extends to moments in other families of l functions

effect size power and sample size determination for structured means modeling and mimic approaches to between groups hypothesis testing of means on a single latent construct while effect size estimates post hoc power estimates and a priori sample size determination are becoming a routine part of univariate analyses involving measured variables e g anova such measures and methods have not been articulated for analyses involving latent means the current article presents standardized effect size measures for latent mean differences inferred from both structured means modeling and mimic approaches to hypothesis testing about differences among means on a single latent construct these measures are then related to post hoc power analysis a priori sample size determination and a relevant measure of construct reliability

moments of the characteristic polynomial in the three ensembles of random matrices moments of the characteristic polynomial of a random matrix taken from any of the three ensembles orthogonal unitary or symplectic are given either as a determinant or a pfaffian or as a sum of determinants for gaussian ensembles on comparing two expressions of the same moment one obtains two remarkable identities one between an n n determinant and an m m determinant and another between the pfaffian of a 2n 2n anti symmetric matrix and a sum of m m determinants

the zeta function on the critical line numerical evidence for moments and random matrix theory models results of extensive computations of moments of the riemann zeta function on the critical line are presented calculated values are compared with predictions motivated by random matrix theory the results can help in deciding between those and competing predictions it is shown that for high moments and at large heights the variability of moment values over adjacent intervals is substantial even when those intervals are long as long as a block containing 109 zeros near zero number 1023 more than anything else the variability illustrates the limits of what one can learn about the zeta function from numerical evidence it is shown the rate of decline of extreme values of the moments is modelled relatively well by power laws also some long range correlations in the values of the second moment as well as asymptotic oscillations in the values of the shifted fourth moment are found the computations described here relied on several representations of the zeta function the numerical comparison of their effectiveness that is presented is of independent interest for future large scale computations

random matrices and l functions in recent years there has been a growing interest in connections between the statistical properties of number theoretical l functions and random matrix theory we review the history of these connections some of the major achievements and a number of applications

integral moments of l functions we give a new heuristic for all of the main terms in the integral moments of various families of primitive l functions the results agree with previous conjectures for the leading order terms our conjectures also have an almost identical form to exact expressions for the corresponding moments of the characteristic polynomials of either unitary orthogonal or symplectic matrices where the moments are defined by the appropriate group averages this lends support to the idea that arithmetical l functions have a spectral interpretation and that their value distributions can be modelled using random matrix theory numerical examples show good agreement with our conjectures

symmetry transitions in random matrix theory l functions we compute the moments of the characteristic polynomials of random orthogonal and symplectic matrices defined by averages with respect to haar measure on so 2n and usp 2n to leading order as n on the unit circle as functions of the angle measured from one of the two symmetry points in the eigenvalue spectrum exp pm i theta n 1 le n le n our results extend previous formulae that relate just to the symmetry points i e to 0 local spectral statistics are expected to converge to those of random unitary matrices in the limit as n when is fixed and to show a transition from the orthogonal or symplectic to the unitary forms on the scale of the mean eigenvalue spacing if y n they become functions of y in the limit when n we verify that this is true for the spectral two point correlation function but show that it is not true for the moments of the characteristic polynomials for which the leading order asymptotic approximation is a function of rather than y symmetry points therefore influence the moments asymptotically far away on the scale of the mean eigenvalue spacing we also investigate the moments of the logarithms of the characteristic polynomials in the same context the moments of the characteristic polynomials of random matrices are conjectured to be related to the moments of families of l functions previously moments at the symmetry point 0 have been related to the moments of families of l functions evaluated at the centre of the critical strip our results motivate general conjectures for the moments of orthogonal and symplectic families of l functions evaluated at a fixed height t up the critical line these conjectures suggest that the symmetry of the non trivial zeros of the l functions influences the moments asymptotically far on the scale of the mean zero spacing from the centre of the critical strip we verify that the second moments of real quadratic dirichlet l functions and a family of automorphic l functions are consistent with our conjectures

random matrix theory and the derivative of the riemann zeta function random matrix theory is used to model the asymptotics of the discrete moments of the derivative of the riemann zeta function s evaluated at the complex zeros i n we also discuss the probability distribution of ln 1 2 i n proving the central limit theorem for the corresponding random matrix distribution and analysing its large deviations

alternative test criteria in covariance structure analysis a unified approach in the context of covariance structure analysis a unified approach to the asymptotic theory of alternative test criteria for testing parametric restrictions is provided the discussion develops within a general framework that distinguishes whether or not the fitting function is asymptotically optimal and allows the null and alternative hypothesis to be only approximations of the true model also the equivalent of the information matrix and the asymptotic covariance matrix of the vector of summary statistics are allowed to be singular when the fitting function is not asymptotically optimal test statistics which have asymptotically a chi square distribution are developed as a natural generalization of more classical ones issues relevant for power analysis and the asymptotic theory of a testing related statistic are also investigated

on nonequivalence of several procedures of structural equation modeling the normal theory based maximum likelihood procedure is widely used in structural equation modeling three alternatives are the normal theory based generalized least squares the normal theory based iteratively reweighted least squares and the asymptotically distribution free procedure when data are normally distributed and the model structure is correctly specified the four procedures are asymptotically equivalent however this equivalence is often used when models are not correctly specified this short paper clarifies conditions under which these procedures are not asymptotically equivalent analytical results indicate that when a model is not correct two factors contribute to the nonequivalence of the different procedures one is that the estimated covariance matrices by different procedures are different the other is that they use different scales to measure the distance between the sample covariance matrix and the estimated covariance matrix the results are illustrated using real as well as simulated data implication of the results to model fit indices is also discussed using the comparative fit index as an example

on the mean values of dirichlet l functions we study the 2kth power moment of dirichlet l functions l s at the centre of the critical strip s 2 where the average is over all primitive characters mod q we extend to this case the hybrid euler hadamard product results of gonek hughes and keating for the riemann zeta function this allows us to recover conjectures for the moments based on random matrix models incorporating the arithmetical terms in a natural way

lower bounds for moments of l functions the moments of central values of families of l functions have recently attracted much attention and with the work of keating and snaith 2000 commun math phys 214 57 89 and 91 110 there are now precise conjectures for their limiting values we develop a simple method to establish lower bounds of the conjectured order of magnitude for several such families of l functions as an example we work out the case of the family of all dirichlet l functions to a prime modulus

random matrix theory and ζ 1 2 it we study the characteristic polynomials z u of matrices u in the circular unitary ensemble cue of random matrix theory exact expressions for any matrix size n are derived for the moments of z and z z and from these we obtain the asymptotics of the value distributions and cumulants of the real and imaginary parts of log z as n in the limit we show that these two distributions are independent and gaussian costin and lebowitz 15 previously found the gaussian limit distribution for im log z using a different approach and our result for the cumulants proves a conjecture made by them in this case we also calculate the leading order n asymptotics of the moments of z and z z these cue results are then compared with what is known about the riemann zeta function s on its critical line re s 1 2 assuming the riemann hypothesis equating the mean density of the non trivial zeros of the zeta function at a height t up the critical line with the mean density of the matrix eigenvalues gives a connection between n and t invoking this connection our cue results coincide with a theorem of selberg for the value distribution of log 1 2 it in the limit t they are also in close agreement with numerical data computed by odlyzko 29 for large but finite t this leads us to a conjecture for the moments of 1 2 it finally we generalize our random matrix results to the circular orthogonal coe and circular symplectic cse ensembles

mixed moments of characteristic polynomials of random unitary matrices following the work of conrey rubinstein and snaith commun math phys 267 611 2006 and forrester and witte j phys a math gen 39 8983 2006 we examine a mixed moment of the characteristic polynomial and its derivative for matrices from the unitary group u n also known as the cue and relate the moment to the solution of a painleve differential equation we also calculate a simple form for the asymptotic behavior of moments of logarithmic derivatives of these characteristic polynomials evaluated near the unit circle following the work of conrey rubinstein and snaith commun math phys 267 611 2006 and forrester and witte j phys a math gen 39 8983 2006 we examine a mixed moment of the characteristic polynomial and its derivative for matrices from the unitary group u n also known as the cue and relate the moment to the solution of a painleve differential equation we also calculate a simple form for the asymptotic behavior of moments of logarithmic derivatives of these characteristic polynomials evaluated near the unit circle