a composite likelihood approach to binary spatial data abstract conventional geostatistics addresses the problem of estimation and prediction for continuous observations but in many practical applications in public health environmental remediation or ecological research the most commonly available data are in the form of counts e g number of cases or indicator variables denoting above or below threshold values also in many situations it is less expensive to obtain an imprecise categorical observation than to obtain precise measurements of the variable of interest such as a contaminant this article proposes a computationally simple method for estimation and prediction using binary or indicator data in space the proposed method is based on pairwise likelihood contributions and the large sample properties of the estimators are obtained in a straightforward manner we illustrate the methodology through application to indicator data related to gypsy moth defoliation in massachusetts

copula based geostatistical models for groundwater quality parameters 1 groundwater quality parameters exhibit considerable spatial variability geostatistical methods including the assessment of variograms are usually used to characterize this variability copulas offer an interesting opportunity to describe dependence structures for multivariate distributions bivariate empirical copulas can be used as an alternative to variograms and covariance functions for the description of the spatial variability rank correlations of these copulas express the strength of the dependence independently of the marginal distributions and thus offer an alternative to the variograms empirical copulas for four quality parameters chloride sulfate ph and nitrate obtained from a large scale groundwater quality measurement network in baden wurttemberg germany are calculated they indicate that the spatial dependence structure of the investigated parameters is not gaussian two theoretical copula based models are presented in this paper a gaussian and a non gaussian bootstrap based statistical tests using stochastic simulation of the multivariate distributions are used to investigate the appropriateness of the models according to the test results the gaussian copula is rejected for most of the parameters while the non gaussian alternative is not rejected in most cases

the smallest eigenvalues of the 1 point fixing graph abstract let s n be the symmetric group on 1 n the k point fixing graph f n k is defined to be the graph with vertex set s n and two vertices g h of f n k are joined if and only if g h 1 fixes exactly k points in this paper we determine the smallest eigenvalue for f n 1

correlated binary regression using a quadratic exponential model summary a model of quadratic exponential form is parameterized in terms of marginal means and pairwise correlations for the regression analysis of correlated binary data pseudo maximum likelihood methods using a special case termed the multiplicative model are proposed but are noted to be computationally unattractive if the blocks of correlated responses are at all large on the other hand score estimating functions for mean and correlation parameters are shown to be of a particularly simple form under the quadratic exponential family special cases of such estimating functions having attractive computational properties are identified and illustrated

the jackknife and the bootstrap for general stationary observations we extend the jackknife and the bootstrap method of estimating standard errors to the case where the observations form a general stationary sequence we do not attempt a reduction to i i d values the jackknife calculates the sample variance of replicates of the statistic obtained by omitting each block of l consecutive data once in the case of the arithmetic mean this is shown to be equivalent to a weighted covariance estimate of the spectral density of the observations at zero under appropriate conditions consistency is obtained if l l n rightarrow infty and l n n rightarrow 0 general statistics are approximated by an arithmetic mean in regular cases this approximation determines the asymptotic behavior bootstrap replicates are constructed by selecting blocks of length l randomly with replacement among the blocks of observations the procedures are illustrated by using the sunspot numbers and some simulated data

joint composite estimating functions in spatiotemporal models summary modelling of spatiotemporal processes has received considerable attention in recent statistical research however owing to the high dimensionality of the data the joint modelling of spatial and temporal processes presents a great computational challenge in both likelihood based and bayesian approaches we propose a joint composite estimating function approach to estimating spatiotemporal covariance structures this substantially reduces the computational complexity and is more efficient than existing composite likelihood methods the novelty of the proposed joint composite estimating function is rooted in the construction of three sets of estimating functions from spatial temporal and spatiotemporal cross pairs which results in overidentified estimating functions thus we form a joint inference function in a spirit that is similar to hansen s generalized method of moments we show that under practical scenarios the estimator proposed is consistent and asymptotically normal simulation studies prove that our method performs well in finite samples finally we illustrate the joint composite estimating function method by estimating the spatiotemporal dependence structure of airborne particulates pm10 in the north eastern usa over a 32 month period

the use of subseries values for estimating the variance of a general statistic from a stationary sequence let z i infty i infty be a strictly stationary infty mixing sequence without specifying the dependence model giving rise to z i and without specifying the marginal distribution of z i we address the question of variance estimation for a general statistic t n t n z 1 z n for estimating var t n from just the available data z 1 z n we propose computing subseries values t m z i 1 z i 2 z i m 0 leq i i m leq n these subseries values are used as replicates to model the sampling variability of t n in particular we use adjacent nonoverlapping subseries of length m m n with m rightarrow infty and m n n rigtharrow 0 our variance estimator is just the usual sample variance computed amongst these subseries values after appropriate standardization this estimator is shown to be consistent under mild integrability conditions we present optimal i e minimum m s e choices of m n for the special case where t n overset z n and z i is a normal ar 1 sequence a simulation study is conducted showing that those same choices of m n are effective when t n is a robust estimator of location and z i is subject to contamination

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

nonparametric estimation of the moments of a general statistic computed from spatial data abstract a statistic s is computed on spatially indexed data x i i d where d is a finite subset of the integer lattice z2 we propose a simple nonparametric method for estimating the moments e g variance skewness of s d using only the observed data at hand the method uses replicates of s computed on smaller subsets of d no explicit knowledge of the underlying spatial dependence mechanism is needed and the marginal distribution of x i may also be unknown the shape of d can be quite irregular and s is allowed to be a general statistic the proposed estimator is shown to be consistent and asymptotically normal under mild conditions on s d and mixing conditions on the strength of spatial dependence as a numerical example the estimator is used in assessing the geographic clumping of cancer deaths in the united states

geostatistical interpolation using copulas 1 in many applications of geostatistical methods the dependence structure of the investigated parameter is described solely with the variogram or covariance functions which are susceptible to measurement anomalies and implies the assumption of gaussian dependence moreover the kriging variance respects only observation density data geometry and the variogram model to address these problems we borrow the idea from copulas to depict the dependence structure without the influence of the marginal distribution the methodology and basic hypotheses for application of copulas as geostatistical methods are discussed and the gaussian copula as well as a non gaussian copula are used in this paper copula parameters are estimated using a division of the observations into multipoint subsets and a subsequent maximization of the corresponding likelihood function the interpolation is carried out with two different copulas where the expected and median values are calculated from the copulas conditioned with the nearby observations the full conditional copulas provide the estimation distributions for the unobserved locations and can be used to define confidence intervals which depend on both the observation geometry and values observations of a large scale groundwater quality measurement network in baden wurttemberg are used to demonstrate the methodology five groundwater quality parameters chloride nitrate ph sulfate and dissolved oxygen are investigated all five parameters show non gaussian dependence the copula based interpolation results of the five parameters are compared to the results of conventional ordinary and indicator kriging different statistical measures including mean squared error relative differences and probability scores are used to compare cross validation and split sampling results of the interpolation methods the non gaussian copulas give better results than the geostatistical interpolations validation of the confidence intervals shows that they are more realistic than the estimation variances obtained by ordinary kriging

spectrum of cayley graphs on the symmetric group generated by transpositions abstract for an integer n 2 let x n be the cayley graph on the symmetric group s n generated by the set of transpositions 12 13 1 n it is shown that the spectrum of x n contains all integers from n 1 to n 1 except 0 if n 2 or n 3

analysis and numerical simulation of positive and dead core solutions of singular two point boundary value problems we investigate the solvability of the dirichlet boundary value problem u t lg u t l 0 u 0 1 u 1 1 where l is a nonnegative parameter we discuss the existence of multiple positive solutions and show that for certain values of l there also exist solutions that vanish on a subinterval r 1 r 0 1 the so called dead core solutions in order to illustrate the theoretical findings we present computational results for g u 1 u computed using the collocation method implemented in bvpsuite a new version of the standard matlab code sbvp1 0

on the eigenvalues of certain cayley graphs and arrangement graphs abstract in this paper we show that the eigenvalues of certain classes of cayley graphs are integers the n k r arrangement graph a n k r is a graph with all the k permutations of an n element set as vertices where two k permutations are adjacent if they differ in exactly r positions we establish a relation between the eigenvalues of the arrangement graphs and the eigenvalues of certain cayley graphs as a result the conjecture on integrality of eigenvalues of a n k 1 follows

positive and dead core solutions of singular bvps for third order differential equations abstract the paper discusses the existence of positive and dead core solutions of the singular differential equation u f t u u u satisfying the boundary conditions u 0 a u t a min u t t 0 t 0 here is a nonnegative parameter a is a positive constant and the caratheodory function f t x y z is singular at the value 0 of its space variable y

estimating space and space time covariance functions for large data sets a weighted composite likelihood approach in this article we propose two methods for estimating space and space time covariance functions from a gaussian random field based on the composite likelihood idea the first method relies on the maximization of a weighted version of the composite likelihood function while the second one is based on the solution of a weighted composite score equation this last scheme is quite general and could be applied to any kind of composite likelihood an information criterion for model selection based on the first estimation method is also introduced the methods are useful for practitioners looking for a good balance between computational complexity and statistical efficiency the effectiveness of the methods is illustrated through examples simulation experiments and by analyzing a dataset on ozone measurements

positive and dead core solutions of singular dirichlet boundary value problems with φ laplacian the paper discusses the existence of positive solutions dead core solutions and pseudodead core solutions of the singular dirichlet boundary value problem f u email protected f t u u h t u u u 0 u t a here l is the positive parameter a 0 f is singular at the value 0 of its first phase variable and h may be singular at the value 0 of its second phase variable

variance estimation for statistics computed from spatial lattice data summary a statistic s is computed on spatially indexed data x i e d where d is a finite subset of the integer lattice 22 we propose a method for estimating the variance and other moments of s by using only the data the set d may be irregularly shaped the statistic s may be arbitrarily complicated and no distributional assumptions marginal or joint are necessary the method uses the statistic computed on overlapping subshapes of d as replicates of s the estimator is simply the sample variance of the standardized replicates we demonstrate c2 consistency of the estimator under mild conditions on s and the strength of spatial dependence a rate of convergence is also given giving guidance into the appropriate choice of subshape size and the estimator is illustrated in two data examples

a generalized estimating equations approach for spatially correlated binary data applications to the analysis of neuroimaging data this paper proposes a generalized estimating equations approach for the analysis of spatially correlated binary data when there are large numbers of spatially correlated observations on a moderate number of subjects this approach is useful when the scientific focus is on modeling the marginal mean structure proper modeling of the spatial correlation structure is shown to provide large efficiency gains along with precise standard error estimates for inference on mean structure parameters generalized estimating equations for estimating the parameters of both the mean and spatial correlation structure are proposed the use of semivariogram models for parameterizing the correlation structure is discussed and estimation of the sample semivariogram is proposed as a technique for choosing parametric models and starting values for generalized estimating equations estimation the methodology is illustrated with neuroimaging data collected as part of the national institute of neurological disorders and stroke ninds stroke data bank a simulation study demonstrates the importance of accurate modeling of the spatial correlation structure in data with large numbers of spatially correlated observations such as those found in neuroimaging studies

efficient pairwise composite likelihood estimation for spatial clustered data spatial clustered data refer to high dimensional correlated measurements collected from units or subjects that are spatially clustered such data arise frequently from studies in social and health sciences we propose a unified modeling framework termed as geocopula to characterize both large scale variation and small scale variation for various data types including continuous data binary data and count data as special cases to overcome challenges in the estimation and inference for the model parameters we propose an efficient composite likelihood approach in that the estimation efficiency is resulted from a construction of over identified joint composite estimating equations consequently the statistical theory for the proposed estimation is developed by extending the classical theory of the generalized method of moments a clear advantage of the proposed estimation method is the computation feasibility we conduct several simulation studies to assess the performance of the proposed models and estimation methods for both gaussian and binary spatial clustered data results show a clear improvement on estimation efficiency over the conventional composite likelihood method an illustrative data example is included to motivate and demonstrate the proposed method

maximum likelihood estimation for probit linear mixed models with correlated random effects the probit normal model for binary data mcculloch 1994 journal of the american statistical association 89 330 335 is extended to allow correlated random effects to obtain maximum likelihood estimates we use the em algorithm with its m step greatly simplified under the assumption of a probit link and its e step made feasible by gibbs sampling standard errors are calculated by inverting a monte carlo approximation of the information matrix rather than via the sem algorithm a method is also suggested that accounts for the monte carlo variation explicitly as an illustration we present a new analysis of the famous salamander mating data unlike previous analyses we find it necessary to introduce different variance components for different species of animals finally we consider models with correlated errors as well as correlated random effects

asymptotic normality for a general statistic from a stationary sequence on obtient un ensemble de conditions necessaires et suffisantes pour la normalite asymptotique conjointe de t n et une valeur de sous serie t m m n

cayley graph on symmetric group generated by elements fixing k points abstract let s n be the symmetric group on n 1 n the k point fixing graph f n k is defined to be the graph with vertex set s n and two vertices g h of f n k are joined if and only if g h 1 fixes exactly k points in this paper we derive a recurrence formula for the eigenvalues of f n k then we apply our result to determine the sign of the eigenvalues of f n 1

the spectrum of eigenvalues for certain subgraphs of the k point fixing graph abstract let s n be the symmetric group on n points the k point fixing graph f n k is defined to be the graph with vertex set s n and two vertices g h of f n k are joined if and only if g h 1 fixes exactly k points in this paper we give a recurrence formula for the eigenvalues of a class of regular subgraphs of f n k by using this recurrence formula we will determine the smallest eigenvalues for this class of regular subgraphs of f n 1 for sufficiently large n

window subsampling of estimating functions with application to regression models abstract we propose a subsampling method for estimating the asymptotic standard error of a statistic n that is the solution to an estimating equation 1 n n j 1 uj y j xj 0 where the data yj may be temporally or spatially correlated and the estimating function may depend on covariates xj a key statistic that we consider in detail is a generalized linear model regression coefficient computed under the assumption of independence the availability of a consistent variance estimator allows semiparametric regression approaches for clustered and longitudinal data to be used with time series and spatial data the methods that we develop extend the subsampling ideas of carlstein sherman and garcia soidan and hall to estimating functions our approach provides an attractive alternative to the jackknife method of lele particularly for large datasets because we do not require parameter reestimation

decay of correlations under dobrushin s uniqueness condition and its applications an estimate on the correlation of functionals of gibbs fields satisfying dobrushin s uniqueness condition is given as a consequence a result of gross saying that the truncated pair correlation function decays in the same weighted summability sense as the potential can be extended to the whole dobrushin uniqueness region applications to the central limit theorem and the second derivative of the pressure are also given

maximum likelihood estimation of models for residual covariance in spatial regression we describe the maximum likelihood method for fitting the linear model when residuals are correlated and when the covariance among the residuals is determined by a parametric model containing unknown parameters observations are assumed to be gaussian we give conditions which ensure consistency and asymptotic normality of the estimators our main concern is with the analysis of spatial data and in this context we describe some simulation experiments to assess the small sample behaviour of estimators we also discuss an application of the spectral approximation to the likelihood for processes on a lattice

constrained monte carlo maximum likelihood for dependent data maximum likelihood estimates mles in autologistic models and other exponential family models for dependent data can be calculated with markov chain monte carlo methods the metropolis algorithm or the gibbs sampler which simulate ergodic markov chains having equilibrium distributions in the model from one realization of such a markov chain a monte carlo approximant to the whole likelihood function can be constructed the parameter value if any maximizing this function approximates the mle

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

large sample confidence regions based on subsamples under minimal assumptions in this article the construction of confidence regions by approximating the sampling distribution of some statistic is studied the true sampling distribution is estimated by an appropriate normalization of the values of the statistic computed over subsamples of the data in the i i d context the method has been studied by wu in regular situations where the statistic is asymptotically normal the goal of the present work is to prove the method yields asymptotically valid confidence regions under minimal conditions essentially all that is required is that the statistic suitably normalized possesses a limit distribution under the true model unlike the bootstrap the convergence to the limit distribution need not be uniform in any sense the method is readily adapted to parameters of stationary time series or more generally homogeneous random fields for example an immediate application is the construction of a confidence interval for the spectral density function of a homogeneous random field

on the central limit theorem for stationary mixing random fields a simple proof of a central limit theorem for stationary random fields under mixing conditions is given generalizing some results obtained by more complicated methods e g bernstein s method