a comparison of permutation and mixed model regression methods for the analysis of simulated data in the context of a group randomized trial our first purpose was to determine whether in the context of a group randomized trial grt with gaussian errors permutation or mixed model regression methods fare better in the presence of measurable confounding in terms of their monte carlo type i error rates and power our results indicate that given a proper randomization the type i error rate is similar for both methods whether unadjusted or adjusted even in small studies however our results also show that should the investigator face the unfortunate circumstance in which modest confounding exists in the only realization available the unadjusted analysis risks a type i error in this regard there was little to distinguish the two methods finally our results show that power is similar for the two methods and not surprisingly better for the adjusted tests our second purpose was to examine the relative performance of permutation and mixed model regression methods in the context of a grt when the normality assumptions underlying the mixed model are violated published studies have examined the impact of violation of this assumption at the member level only our findings indicate that both methods perform well when the assumption is violated so long as the icc is very small and the design is balanced at the group level however at icc 0 01 the permutation test carries the nominal type i error rate while the model based test is conservative and so less powerful binomial group and member level errors did not otherwise change the relative performance of the two methods with regard to confounding copyright 2005 john wiley sons ltd

a prediction error and stepwise regression estimation algorithm for non linear systems the identification of non linear systems based on a narmax non linear autoregressive moving average model with exogenous inputs model representation is considered and a combined stepwise regression prediction error estimation algorithm is derived the stepwise regression routine determines the model structure by detecting significant terms in the model while the prediction error algorithm provides optimal estimates of the final model parameters implementation of the algorithms is discussed in detail and several simulated examples and industrial applications are included to illustrate that parsimonious models of non linear systems can be identified using the algorithm

root properties and convergence rates of median filters median filters are a special class of ranked order filters used for smoothing signals repeated application of the filter on a quantized signal of finite length ultimately results in a sequence termed a root signal which is invariant to additional passes of the median filter in this paper the theory is developed both for determining the cardinality of the root signal space of arbitrary window width filters applied to signals with any number of quantization levels and for counting or estimating the number of passes required to produce a root for binary signals

rounding effects of quenched randomness on first order phase transitions frozen in disorder in an otherwise homogeneous system is modeled by interaction terms with random coefficients given by independent random variables with a translation invariant distribution for such systems it is proven that ind 2 dimensions there can be no first order phase transition associated with discontinuities in the thermal average of a quantity coupled to the randomized parameter discontinuities which would amount to a continuous symmetry breaking in systems which are stochastically invariant under the action of a continuous subgroup ofo n are suppressed by the randomness in dimensionsd 4 specific implications are found in the random field ising model for which we conclude that ind 2 dimensions at all h the gibbs state is unique for almost all field configurations and in the random bond potts model where the general phenomenon is manifested in the vanishing of the latent heat at the transition point the results are explained by the argument of imry and ma 1 the proofs involve the analysis of fluctuations of free energy differences which are shown using martingale techniques to be gaussian on the suitable scale

gibbs states of the hopfield model with extensively many patterns we consider the hopfield model withm n n patterns wheren is the number of neurons we show that if is sufficiently small and the temperature sufficiently low then there exist disjoint gibbs states for each of the stored patterns almost surely with respect to the distribution of the random patterns this solves a provlem left open in previous work the key new ingredient is a self averaging result on the free energy functional this result has considerable additional interest and some consequences are discussed a similar result for the free energy of the sherrington kirkpatrick model is also given

a tomography of the grem beyond the rem conjecture in a companion paper we proved that in a large class of gaussian disordered spin systems the local statistics of energy values near levels n 1 2 with 1 2 are described by a simple poisson process in this paper we address the issue as to whether this is optimal and what will happen if 1 2 we do this by analysing completely the gaussian generalised random energy models grem we show that the rem behaviour persists up to the level c n where c denotes the critical temperature we show that beyond this value the simple poisson process must be replaced by more and more complex mixed poisson point processes

an evaluation of constrained randomization for the design and analysis of group randomized trials with binary outcomes group randomized trials are randomized studies that allocate intact groups of individuals to different comparison arms a frequent practical limitation to adopting such research designs is that only a limited number of groups may be available and therefore simple randomization is unable to adequately balance multiple group level covariates between arms therefore covariate based constrained randomization was proposed as an allocation technique to achieve balance constrained randomization involves generating a large number of possible allocation schemes calculating a balance score that assesses covariate imbalance limiting the randomization space to a prespecified percentage of candidate allocations and randomly selecting one scheme to implement when the outcome is binary a number of statistical issues arise regarding the potential advantages of such designs in making inference in particular properties found for continuous outcomes may not directly apply and additional variations on statistical tests are available motivated by two recent trials we conduct a series of monte carlo simulations to evaluate the statistical properties of model based and randomization based tests under both simple and constrained randomization designs with varying degrees of analysis based covariate adjustment our results indicate that constrained randomization improves the power of the linearization f test the kc corrected gee t test kauermann and carroll 2001 journal of the american statistical association 96 1387 1396 and two permutation tests when the prognostic group level variables are controlled for in the analysis and the size of randomization space is reasonably small we also demonstrate that constrained randomization reduces power loss from redundant analysis based adjustment for non prognostic covariates design considerations such as the choice of the balance metric and the size of randomization space are discussed

a note on gauss hermite quadrature summary for gauss hermite quadrature we consider a systematic method for transforming the variable of integration so that the integrand is sampled in an appropriate region the effectiveness of the quadrature then depends on the ratio of the integrand to some gaussian density being a smooth function well approximated by a low order polynomial it is pointed out that in this approach order one gauss hermite quadrature becomes the laplace approximation thus the quadrature as implemented here can be thought of as a higher order laplace approximation

the random quadratic assignment problem the quadratic assignment problem qap is one of the most difficult of all com binatorial optimization problems here we use an abbreviated application of the statistical mechanics replica method to study the asymptotic behavior of instances in which the en tries of at least one of the two matrices that specify the problem are chosen from a random distributionp s urprisingly the qap has not been studied before using the replica method despite the fact that the qap was first proposed over 50 years ago and the replica method was developed over 30 years ago we find simple forms forc min andc max t he costs of the minimal and maximum solutions respectively notable features of our results are the sym metry of the results forc min andc max and their dependence onponly through its mean and standard deviation independent of the details ofp

recursive weighted median filters admitting negative weights and their optimization a recursive weighted median rwm filter structure admitting negative weights is introduced much like the sample median is analogous to the sample mean the proposed class of rwm filters is analogous to the class of infinite impulse response iir linear filters rwm filters provide advantages over linear iir filters offering near perfect stopband characteristics and robustness against noise unlike linear iir filters rwm filters are always stable under the bounded input bounded output criterion regardless of the values taken by the feedback filter weights rwm filters also offer a number of advantages over their nonrecursive counterparts including a significant reduction in computational complexity increased robustness to noise and the ability to model resonant or vibratory behavior a novel recursive decoupling adaptive optimization algorithm for the design of this class of recursive wm filters is also introduced several properties of rwm filters are presented and a number of simulations are included to illustrate the advantages of rwm filters over their nonrecursive counterparts and iir linear filters

the kac version of the sherrington kirkpatrick model at high temperatures we study the kac version of the sherrington kirkpatrick sk model of a spin glass i e a spin glass with long but finite range interaction on mathbb z d and gaussian mean zero couplings we prove that for all 1 the free energy of this model converges to that of the sk model as the range of the interaction tends to infinity moreover we prove that for all temperatures for which the infinite volume gibbs state is unique the free energy scaled by the square root of the volume converges to a gaussian with variance c where 1 is the range of the interaction moreover at least for almost all values of this variance tends to zero as goes to zero the value in the sk model we interpret our finding as a weak indication that at least at high temperatures the sk model can be seen as a reasonable asymptotic model for lattice spin glasses

small sample performance of bias corrected sandwich estimators for cluster randomized trials with binary outcomes the sandwich estimator in generalized estimating equations gee approach underestimates the true variance in small samples and consequently results in inflated type i error rates in hypothesis testing this fact limits the application of the gee in cluster randomized trials crts with few clusters under various crt scenarios with correlated binary outcomes we evaluate the small sample properties of the gee wald tests using bias corrected sandwich estimators our results suggest that the gee wald z test should be avoided in the analyses of crts with few clusters even when bias corrected sandwich estimators are used with t distribution approximation the kauermann and carroll kc correction can keep the test size to nominal levels even when the number of clusters is as low as 10 and is robust to the moderate variation of the cluster sizes however in cases with large variations in cluster sizes the fay and graubard fg correction should be used instead furthermore we derive a formula to calculate the power and minimum total number of clusters one needs using the t test and kc correction for the crts with binary outcomes the power levels as predicted by the proposed formula agree well with the empirical powers from the simulations the proposed methods are illustrated using real crt data we conclude that with appropriate control of type i error rates under small sample sizes we recommend the use of gee approach in crts with binary outcomes because of fewer assumptions and robustness to the misspecification of the covariance structure copyright 2014 john wiley sons ltd

stochastic analysis of the quadratic assignment problem let a and b be m m matrices the aim is to maximize q i jaijb i j over all permutations of 1 2 m we define a simple greedy algorithm that constructs a permutation that gives a large q assuming that the entries of a are i i d with a symmetric distribution and that the entries of b are independent of a we show under some weak moment conditions eq k 1m3 2 log m 1 2 where the constant k does not depend on m this contrasts with the fact that max km3 2 log m 1 2 with overwhelming probability

a theoretical analysis of the properties of median filters necessary and sufficient conditions for a signal to be invariant under a specific form of median filtering are derived these conditions state that a signal must be locally monotone to pass through a median filter unchanged it is proven that the form of successive median filtering of a signal i e the filtered output is itself again filtered eventually reduces the original signal to an invariant signal called a root signal for a signal of length l samples a maximum of frac 1 2 l 2 repeated filterings produces a root signal

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

a comparison of two bias corrected covariance estimators for generalized estimating equations summarymancl and derouen 2001 biometrics57 126 134 and kauermann and carroll 2001 jasa96 1387 1398 proposed alternative bias corrected covariance estimators for generalized estimating equations parameter estimates of regression models for marginal means the finite sample properties of these estimators are compared to those of the uncorrected sandwich estimator that underestimates variances in small samples although the formula of mancl and derouen generally overestimates variances it often leads to coverage of 95 confidence intervals near the nominal level even in some situations with as few as 10 clusters an explanation for these seemingly contradictory results is that the tendency to undercoverage resulting from the substantial variability of sandwich estimators counteracts the impact of overcorrecting the bias however these positive results do not generally hold for small cluster sizes e g 10 their estimator often results in overcoverage and the bias corrected covariance estimator of kauermann and carroll may be preferred the methods are illustrated using data from a nested cross sectional cluster intervention trial on reducing underage drinking

the sherrington kirkpatrick model of spin glasses and stochastic calculus the high temperature case we study the fluctuations of free energy energy and entropy in the high temperature regime for the sherrington kirkpatrick model of spin glasses we introduce here a new dynamical method with the help of brownian motions and continuous martingales indexed by the square root of the inverse temperature as parameter thus formulating the thermodynamic formalism in terms of random processes the well established technique of stochastic calculus leads us naturally to prove that these fluctuations are simple gaussian processes with independent increments a generalization of a result proved by aizenman lebowitz and ruelle 1

limit theorems for sums of random exponentials we study limiting distributions of exponential sums open image in new window as t n where x i are i i d random variables two cases are considered a ess sup x i 0 and b ess sup x i we assume that the function h x log p x i x case b or h x log p x i 1 x case a is regularly varying at with index 1 case b or 0 case a the appropriate growth scale of n relative to t is of the form open image in new window where the rate function h0 t is a certain asymptotic version of the function open image in new window case b or open image in new window case a we have found two critical points 1 2 below which the law of large numbers and the central limit theorem respectively break down for 0 2 under the slightly stronger condition of normalized regular variation of h we prove that the limit laws are stable with characteristic exponent 0 2 and skewness parameter 1

limit laws for power sums and norms of i i d samples we study the limit behavior of power sums and norms of i i d positive samples from the max domain of attraction of an extreme value distribution to this end we combine limit theorems for sums and for maxima and use a link between extreme value theory and the levy measures of certain infinitely divisible laws which are limit distributions of power sums in connection with the von mises representation of the gumbel max domain of attraction this new approach allows us to extend the limit results for power sums found in ben arous et al probab theory relat fields 132 579 612 2005 and bogachev j theor probab 19 849 873 2006 furthermore our findings shed a new light on the results of schlather ann probab 29 862 881 2001 and treat the gumbel case which is missing there

analysis of a cluster randomized trial with binary outcome data using a multi level model the use of multi level logistic regression models was explored for the analysis of data from a cluster randomized trial investigating whether a training programme for general practitioners reception staff could improve women s attendance at breast screening twenty six general practices were randomized with women nested within them requiring a two level model which allowed for between practice variability comparisons were made with fixed effect fe and random effects re cluster summary statistic methods ordinary logistic regression and a marginal model based on generalized estimating equations with robust variance estimates an fe summary statistic method and ordinary logistic regression considerably understated the variance of the intervention effect thus overstating its statistical significance the marginal model produced a higher statistical significance for the intervention effect compared to that obtained from the re summary statistic method and the multi level model because there was only a moderate number of practices and these had unbalanced cluster sizes reliable asymptotic properties for the robust standard errors used in the marginal model may not have been achieved while the re summary statistic method cannot handle multiple covariates easily marginal and multi level models can do so in contrast to multi level models however marginal models do not provide direct estimates of variance components but treat these as nuisance parameters estimates of the variance components were of particular interest in this example additionally parametric bootstrap methods within the multi level model framework provide confidence intervals for these variance components as well as a confidence interval for the effect of intervention which allows for the imprecision in the estimated variance components the assumption of normality of the random effects can be checked and the models extended to investigate multiple sources of variability

local energy statistics in disordered systems a proof of the local rem conjecture recently bauke and mertens conjectured that the local statistics of energies in random spin systems with discrete spin space should in most circumstances be the same as in the random energy model here we give necessary conditions for this hypothesis to be true which we show to be satisfied in wide classes of examples short range spin glasses and mean field spin glasses of the sk type we also show that under certain conditions the conjecture holds even if energy levels that grow moderately with the volume of the system are considered

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

power and sample size requirements for gee analyses of cluster randomized crossover trials the cluster randomized crossover design has been proposed to improve efficiency over the traditional parallel cluster randomized design which often involves a limited number of clusters in recent years the cluster randomized crossover design has been increasingly used to evaluate the effectiveness of health care policy or programs and the interest often lies in quantifying the population averaged intervention effect in this paper we consider the two treatment two period crossover design and develop sample size procedures for continuous and binary outcomes corresponding to a population averaged model estimated by generalized estimating equations accounting for both within period and interperiod correlations in particular we show that the required sample size depends on the correlation parameters through an eigenvalue of the within cluster correlation matrix for continuous outcomes and through two distinct eigenvalues of the correlation matrix for binary outcomes we demonstrate that the empirical power corresponds well with the predicted power by the proposed formulae for as few as eight clusters when outcomes are analyzed using the matrix adjusted estimating equations for the correlation parameters concurrently with a suitable bias corrected sandwich variance estimator

fluctuations of the free energy in the rem and the p spin sk models we consider the random fluctuations of the free energy in the p spin version of the sherrington kirkpatrick sk model in the high temperature regime using the martingale approach of comets and neveu as used in the standard sk model combined with truncation techniques inspired by a recent paper by talagrand on the p spin version we prove that the random corrections to the free energy are on a scale n p 2 2 only and after proper rescaling converge to a standard gaussian random variable this is shown to hold for all values of the inverse temperature smaller than a critical p we also show that p 2 in 2 as p additionally we study the formal p limit of these models the random energy model here we compute the precise limit theorem for the properly rescaled partition function at all temperatures for 2 in 2 fluctuations are found at an exponentially small scale with two distinct limit laws above and below a second critical value in 2 2 for up to that value the rescaled fluctuations are gaussian while below that there are non gaussian fluctuations driven by the poisson process of the extreme values of the random energies for larger than the critical 2 in 2 the fluctuations of the logarithm of the partition function are on a scale of 1 and are expressed in terms of the poisson process of extremes at the critical temperature the partition function divided by its expectation converges to 1 2

maintaining the validity of inference in small sample stepped wedge cluster randomized trials with binary outcomes when using generalized estimating equations stepped wedge cluster trials are an increasingly popular alternative to traditional parallel cluster randomized trials such trials often utilize a small number of clusters and numerous time intervals and these components must be considered when choosing an analysis method a generalized linear mixed model containing a random intercept and fixed time and intervention covariates is the most common analysis approach however the sole use of a random intercept applies a constant intraclass correlation coefficient structure which is an assumption that is likely to be violated given stepped wedge trials swts have multiple time intervals alternatively generalized estimating equations gee are robust to the misspecification of the working correlation structure although it has been shown that small sample adjustments to standard error estimates and the use of appropriate degrees of freedom are required to maintain the validity of inference when the number of clusters is small in this article we show using an extensive simulation study based on a motivating example and a more general design the use of gee can maintain the validity of inference in small sample swts with binary outcomes furthermore we show which combinations of bias corrections to standard error estimates and degrees of freedom work best in terms of attaining nominal type i error rates

un theoreme limite pour les covariances des spins dans le modele de sherrington kirkpatrick avec champ externe on etudie la covariance pour la mesure de gibbs des spins en deux sites dans le cas d un modele de sherrington kirkpatrick avec champ externe lorsque le nombre de sites du modele tend vers l infini une evaluation asymptotique des moments d ordre p de cette covariance permet d obtenir un theoreme limite faible avec une loi limite en general non gaussienne we study the covariance for gibbs measure of spins at two sites in the case of a sherrington kirkpatrick model with an external field when the number of sites of the model grows to infinity an asymptotic evaluation of the p moments of that covariance allows us to obtain a weak limit theorem with a generally non gaussian limit law

stochastic analysis for the recursive median filter process vector probability measure functions density function for recursively median filtered signals are found when the underlying input binary sequences are either independent identically distributed i i d or markov chains the results are parametric in the window size of the filter and in the probability distribution of the input sequence using statistical threshold decomposition the same results are found for discrete alphabet random sequences that are either i i d or markov chains some examples illustrating the efficacy of the recursive median filter relative to the nonrecursive implementation are presented in particular the breakdown probabilities are tabulated for both recursive and nonrecursive median filters

asymptotic properties of the quadratic assignment problem for the general quadratic assignment problem as well as for a planar version of this problem we extend earlier work by burkard and fincke to prove that the ratio of the maximal to the minimal solution value converges to 1 almost surely in fact any solution value can almost surely be written asymptotically as a simple explicitly given function of the problem size theoretical analysis and computational experiments reveal the convergence to be relatively fast

nonrecursive and recursive stack filters and their filtering behavior it is shown that stack filters that are based on symmetric threshold functions and preserve median filter roots make all inputs converge to roots or to cycles of period two this is an important result since these filters have useful roots the median filter roots and they are time symmetric i e time reversal of an input sequence of such a filter is equivalent to time reversal of the output sequence of the filter in order to construct stack filters without cycles the recursive stack filter which is an extension of the recursive median filter is introduced it is shown that a recursive stack filter has the same roots as the corresponding nonrecursive stack filter also given a nonrecursive filter from the class mentioned above the corresponding recursive filter will make every input signal of finite converge to a root in a finite number of passes

an asymptotical study of combinatorial optimization problems by means of statistical mechanics the analogy between combinatorial optimization and statistical mechanics has proven to be a fruitful object of study simulated annealing a metaheuristic for combinatorial optimization problems is based on this analogy in this paper we show how a statistical mechanics formalism can be utilized to analyze the asymptotic behavior of combinatorial optimization problems with sum objective function and provide an alternative proof for the following result under a certain combinatorial condition and some natural probabilistic assumptions on the coefficients of the problem the ratio between the optimal solution and an arbitrary feasible solution tends to one almost surely as the size of the problem tends to infinity so that the problem of optimization becomes trivial in some sense whereas this result can also be proven by purely probabilistic techniques the above approach allows one to understand why the assumed combinatorial condition is essential for such a type of asymptotic behavior

nonlinear mean filters in image processing the use of nonlinear means in image processing is introduced the properties of these means in the presence of different types of noise are investigated it is shown that nonlinear filters based on these means behave well for both additive and impulse noise their performance in the presence of signal dependent noise is satisfactory they preserve the edges better than linear filters and they reject the noise better than median filters

quenched large deviation principle for the overlap of a p spins system in this note we prove a quenched large deviation result for the overlap of a p spins interaction system at high temperature the rate function of the large deviation principle is proved to be deterministic and some of its basic properties are studied our result is based on a pure state result for a multidimensional p spins system combined with a careful application of the gartner ellis theorem

glauber dynamics of the random energy model i metastable motion on the extreme states we investigate the long time behavior of the glauber dynamics for the random energy model below the critical temperature we give very precise estimates on the motion of the process to and between the states of extremal energies we show that when disregarding time the consecutive steps of the process on these states are governed by a markov chain that jumps uniformly on all possible states the mean times of these jumps are also computed very precisely and are seen to be asymptotically independent of the terminal point a first indicator of aging is the observation that the mean time of arrival in the set of states that have waiting times of order t is itself of order t the estimates proven in this paper will furnish crucial input for a follow up paper where aging is analysed in full detail

on the existence of thermodynamics for the random energy model derrida s random energy model is considered almost sure andlp convergence of the free energy at any inverse temperature are proven rigorous upper and lower bounds to the finite size corrections to the free energy are given

complex random energy model zeros and fluctuations the partition function of the random energy model at inverse temperature beta is a sum of random exponentials mathcal z n beta sum k 1 n exp beta sqrt n x k where x 1 x 2 ldots are independent real standard normal random variables random energies and n log n we study the large n limit of the partition function viewed as an analytic function of the complex variable beta we identify the asymptotic structure of complex zeros of the partition function confirming and extending predictions made in the theoretical physics literature we prove limit theorems for the random partition function at complex beta both on the logarithmic scale and on the level of limiting distributions our results cover also the case of the sums of independent identically distributed random exponentials with any given correlations between the real and imaginary parts of the random exponent

the retrieval phase of the hopfield model a rigorous analysis of the overlap distribution standard large deviation estimates or the use of the hubbard stratonovich transformation reduce the analysis of the distribution of the overlap parameters essentially to that of an explicitly known random function n on m in this article we present a rather careful study of the structure of the minima of this random function related to the retrieval of the stored patterns we denote by m the modulus of the spontaneous magnetization in the curie weiss model and by the ratio between the number of the stored patterns and the system size we show that there exist strictly positive numbers 0 a c such that 1 if a m 2 then the absolute minima of are located within small balls around the points m e where e denotes the th unit vector while 2 if c m 2 at least a local minimum surrounded by extensive energy barriers exists near these points the random location of these minima is given within precise bounds these are used to prove sharp estimates on the support of the gibbs measures

critical weight statistics of the random energy model and of the directed polymer on the cayley tree we consider the critical point of two mean field disordered models i the random energy model rem introduced by derrida as a mean field spin glass model of n spins and ii the directed polymer of length n on a cayley tree dpct with random bond energies both models are known to exhibit a freezing transition between a high temperature phase where the entropy is extensive and a low temperature phase of finite entropy where the weight statistics coincides with the weight statistics of l evy sums with index ensuremath mu t t c l1 in this paper we study the weight statistics at criticality via the entropy s ensuremath ensuremath sum w i phantom rule 0 2em 0ex mathrm ln phantom rule 0 2em 0ex w i and the generalized moments y k ensuremath sum w i k where the w i are the boltzmann weights of the 2 n configurations in the rem we find that the critical weight statistics is governed by the finite size exponent ensuremath nu 2 the entropy scales as overline s n t c ensuremath sim n 1 2 the typical values e overline mathrm ln phantom rule 0 2em 0ex y k decay as n ensuremath k 2 and the disorder averaged values overline y k are governed by rare events and decay as n ensuremath 1 2 for any kg1 for the dpct we find that the entropy scales similarly as overline s n t c ensuremath sim n 1 2 whereas another exponent ensuremath nu ensuremath 1 governs the y k statistics the typical values e overline mathrm ln phantom rule 0 2em 0ex y k decay as n ensuremath k and the disorder averaged values overline y k decay as n ensuremath 1 for any kg1 as a consequence the asymptotic probability distribution overline ensuremath pi n ensuremath infty q of the overlap q in addition to the delta function ensuremath delta q which bears the whole normalization contains an isolated point at q 1 as a memory of the delta peak 1 ensuremath t t c ensuremath delta q ensuremath 1 of the low temperature phase tl t c the associated value overline ensuremath pi n ensuremath infty q 1 is finite for the dpct and diverges as overline ensuremath pi n ensuremath infty q 1 ensuremath sim n 1 2 for the rem

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

a survey for the quadratic assignment problem the quadratic assignment problem qap one of the most difficult problems in the np hard class models many real life problems in several areas such as facilities location parallel and distributed computing and combinatorial data analysis combinatorial optimization problems such as the traveling salesman problem maximal clique and graph partitioning can be formulated as a qap in this paper we present some of the most important qap formulations and classify them according to their mathematical sources we also present a discussion on the theoretical resources used to define lower bounds for exact and heuristic algorithms we then give a detailed discussion of the progress made in both exact and heuristic solution methods including those formulated according to metaheuristic strategies finally we analyze the contributions brought about by the study of different approaches

posterior agreement for large parameter rich optimization problems abstract most real world combinatorial optimization problems are affected by noise in the input data thus behaving in the high noise limit like large disordered particle systems e g spin glasses or random networks due to uncertainty in the input optimization of such disordered instances should infer stable posterior distributions of solutions conditioned on the noisy input instance the maximum entropy principle states that the most stable distribution given the noise influence is defined by the gibbs distribution and it is characterized by the free energy in this paper we first provide rigorous asymptotics of the difficult problem to compute the free energy for two combinatorial optimization problems namely the sparse minimum bisection problem smbp and lawler s quadratic assignment problem lqap we prove that both problems exhibit phase transitions equivalent to the discontinuous behavior of derrida s random energy model rem furthermore the derived free energy asymptotics lead to a theoretical justification of a recently introduced concept 3 of gibbs posterior agreement that measures stability of the gibbs distributions when the cost function fluctuates due to randomness in the input this relatively new stability concept may potentially provide a new method to select robust solutions for a large class of optimization problems

a note on the efficiency of sandwich covariance matrix estimation the sandwich estimator also known as robust covariance matrix estimator heteroscedasticity consistent covariance matrix estimate or empirical covariance matrix estimator has achieved increasing use in the econometric literature as well as with the growing popularity of generalized estimating equations its virtue is that it provides consistent estimates of the covariance matrix for parameter estimates even when the fitted parametric model fails to hold or is not even specified surprisingly though there has been little discussion of properties of the sandwich method other than consistency we investigate the sandwich estimator in quasi likelihood models asymptotically and in the linear case analytically we show that under certain circumstances when the quasi likelihood model is correct the sandwich estimate is often far more variable than the usual parametric variance estimate the increased variance is a fixed feature of the method and the price that one pays to obtain consistency even when the p

on a third order phase transition the asymptotic behaviour of random variables of the general form ln sum limits i 1 kappa n exp n 1 p beta zeta i with independent identically distributed random variables i is studied this generalizes the random energy model of derrida in the limitn there occurs a particular kind of phase transition which does not incorporate a bifurcation phenomenon or symmetry breaking the hypergeometric character of the problem see definitions of sect 4 its function and its entropy function are discussed

temperature dependence of the gibbs state in the random energy model we consider the problem of temperature dependence of the gibbs states in two spin glass models derrida s random energy model and its analogue where the random variables in the hamiltonian are replaced by independent standard brownian motions for both of them we compute in the thermodynamic limit the overlap distribution ni 1 i i n 1 1 of two spin configurations under the product of two gibbs measures which are taken at temperatures t t respectively if t t are fixed then at low temperature phase the results are different for these models for the first one this distribution is d0 0 d1 1 with random weights d0 d1 while for the second one it is 0 we compute consequently the overlap distribution for the second model whenever t t 0 at different speeds as n

small sample adjustments for wald type tests using sandwich estimators the sandwich estimator of variance may be used to create robust wald type tests from estimating equations that are sums of k independent or approximately independent terms for example for repeated measures data on k individuals each term relates to a different individual these tests applied to a parameter may have greater than nominal size if k is small or more generally if the parameter to be tested is essentially estimated from a small number of terms in the estimating equation we offer some practical modifications to these robust wald type tests which asymptotically approach the usual robust wald type tests we show that one of these modifications provides exact coverage for a simple case and examine by simulation the modifications applied to the generalized estimating equations of liang and zeger 1986 conditional logistic regression and the cox proportional hazard model

average performance of greedy heuristics for the integer knapsack problem abstract this paper derives a lower bound on the average performance of a total value greedy heuristic for the integer knapsack problem this heuristic selects items in order of their maximum possible contribution to the solution value at each stage we show that as for the worst case bound the average performance bound for the total value heuristic dominates the corresponding bound for the density ordered greedy heuristic

a recursive variant of mallows filter class mallows 1980 nonlinear fir filter class includes most of the nonlinear digital filters in wide use today this filter class is defined in terms of five axioms and this paper modifies two of these axioms to obtain a class of symmetric recursive nonlinear filters one of the main points considered in this paper is the consequence of mallows location invariance axiom a2 and general prescriptions are given for both mallows original fir filter class and its recursive extension this representation is also extremely useful in discussing the root sequences of both the recursive and nonrecursive structures since recursive filters can exhibit various forms of instability a number of stability results are presented for this class of filters finally some simple examples are considered that illustrate the potential utility of the results presented

high temperature regime for a multidimensional sherrington kirkpatrick model of spin glass comets and neveu have initiated in 5 a method to prove convergence of the partition function of disordered systems to a log normal random variable in the high temperature regime by means of stochastic calculus we generalize their approach to a multidimensional sherrington kirkpatrick model with an application to the heisenberg model of uniform spins on a sphere of d see 9 the main tool that we use is a truncation of the partition function outside a small neighbourhood of the typical energy path

phase transition in the jarzynski estimator of free energy differences the transition between a regime in which thermodynamic relations apply only to ensembles of small systems coupled to a large environment and a regime in which they can be used to characterize individual macroscopic systems is analyzed in terms of the change in behavior of the jarzynski estimator of equilibrium free energy differences from nonequilibrium work measurements given a fixed number of measurements the jarzynski estimator is unbiased for sufficiently small systems in these systems the directionality of time is poorly defined and the configurations that dominate the empirical average but which are in fact typical of the reverse process are sufficiently well sampled as the system size increases the arrow of time becomes better defined the dominant atypical fluctuations become rare and eventually cannot be sampled with the limited resources that are available asymptotically only typical work values are measured the jarzynski estimator becomes maximally biased and approaches the exponential of minus the average work which is the result that is expected from standard macroscopic thermodynamics in the proper scaling limit this regime change has been recently described in terms of a phase transition in variants of the random energy model in this paper this correspondence is further demonstrated in two examples of physical interest the sudden compression of an ideal gas and adiabatic quasistatic volume changes in a dilute real gas

on the stochastic calculus method for spins systems in this note we show how to generalize the stochastic calculus method introduced by comets and neveu comm math phys 166 1995 549 564 for two models of spin glasses namely the sk model with external field and the perceptron model this method allows to derive quite easily some fluctuation results for the free energy in those two cases

optimal permutation tests for the analysis of group randomized trials two facts complicate the comparison of interventions in group randomized trials grt s a family of clinical trials in which each member of a particular group receives the same treatment assignment first individual outcomes within each group are often correlated second the number of groups in a grt is often not sufficient to make asymptotic approximations possible therefore the tests used with methods such as generalized estimating equations gee and penalized quasi likelihood pql originally developed for longitudinal studies may not be valid as an alternative a class of permutation tests is derived to maximize the power of testing for an intervention effect in a grt while maintaining a nominal test size the test uses a statistic that is a weighted sum of residuals with the weights based on the group sizes and the variability of each individual outcome through simulation we demonstrate the importance of weights to a permutation test s power and compare the power of permutation tests ge

random assignment problems abstract analysis of random instances of optimization problems provides valuable insights into the behavior and properties of problem s solutions feasible region and optimal values especially in large scale cases a class of problems that have been studied extensively in the literature using the methods of probabilistic analysis is represented by the assignment problems and many important problems in operations research and computer science can be formulated as assignment problems this paper presents an overview of the recent results and developments in the area of probabilistic assignment problems including the linear and multidimensional assignment problems quadratic assignment problem etc

glauber dynamics of the random energy model ii aging below the critical temperature we investigate the long time behavior of the glauber dynamics for the ran dom energy model below the critical temperature we establish that for a suitably chosen timescale that diverges with the size of the system one can prove that a natural auto correlation function exhibits aging moreover we show that the long time asymptotics of this function coincide with those of the so called rem like trap model proposed by bouchaud and dean our results rely on very precise estimates on the distribution of transition times of the process between different states of extremely low energy

on root structures of median and median type filters in this correspondence we analyze the root structures of the standard median sm filters the recursive median rm filters and the fir median hybrid fmh filters it is shown that these filters have many oscillatory infinitely long root signals when a section of an oscillatory root is present in a signal the filter s noise attenuation of the filter is not as good as predicted by statistical measures we also suggest methods to prevent the possible harmful effect caused by the existence of these roots

quadratic assignment problems abstract this paper surveys quadratic assignment problems qap at first several applications of this problem class are described and mathematical formulations of qaps are given then some exact solution methods and good heuristics are outlined their computational behaviour is illustrated by numerical results further recent results on the asymptotic probabilistic behaviour of qaps are outlined

the geometry of the gibbs measure of pure spherical spin glasses we analyze the statics for pure p spin spherical spin glass models with p ge 3 at low enough temperature with f n beta denoting the free energy we compute the logarithmic second order term of nf n beta and prove that for an appropriate centering c n beta nf n beta c n beta is a tight sequence we further establish the absence of temperature chaos those results follow from the following geometric picture we prove for the gibbs measure of interest by itself asymptotically the measure splits into infinitesimal spherical bands centered at deep minima playing the role of pure states for the pure models the latter makes precise the picture of many valleys separated by high mountains and significant parts of the tap analysis from the physics literature

design and analysis considerations for cohort stepped wedge cluster randomized trials with a decay correlation structure a stepped wedge cluster randomized trial is a type of longitudinal cluster design that sequentially switches clusters to intervention over time until all clusters are treated while the traditional posttest only parallel design requires adjustment for a single intraclass correlation coefficient the stepped wedge design allows multiple outcome measurements from the same cluster and so additional correlation parameters are necessary to characterize the within cluster correlation structure although a number of studies have differentiated between the concepts of within period and between period correlations only a few studies have allowed the between period correlation to decay over time in this article we consider the proportional decay correlation structure for a cohort stepped wedge design and provide a matrix adjusted quasi least squares approach to accurately estimate the correlation parameters along with the marginal intervention effect we further develop the sample size and power procedures accounting for the correlation decay and investigate the accuracy of the power procedure with continuous outcomes in a simulation study we show that the empirical power agrees well with the prediction even with as few as nine clusters when data are analyzed with matrix adjusted quasi least squares concurrently with a suitable bias corrected sandwich variance two trial examples are provided to illustrate the new sample size procedure