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

smoothing of stokes discontinuities when the asymptotics of an analytic function f z are given in terms of two asymptotic forms s 1 z s 2 z alternately dominant in alternating sectors of the z plane stokes observed that the coefficients of s 1 s 2 while constant in any one sector can change from sector to sector and that to a first approximation the change appears to be discontinuous recently berry has given an interesting formal argument which shows in more detail how the change takes place and the present paper gives a rigorous treatment of berry s result

pairwise likelihood methods for inference in image models summary we consider parameter estimation for certain random set models for image data for the spatial models considered here a full likelihood based approach to this problem is often difficult since the likelihood generally cannot be computed for images of typical size this has motivated the consideration of various less computationally demanding parameter estimation techniques we discuss one method based on likelihoods for small subsets of the data consideration of the basic technique within the framework of the theory of estimating functions leads to a weighted version of the basic approach results of a simulation study are reported for some variants of the boolean model and excursion sets of random fields the methods are illustrated for some data on the spatial incidence of heather

smoothing of the stokes phenomenon using mellin barnes integrals abstract recently berry 1989 has put forward a formal theory of the smoothing of the stokes phenomenon based upon optimal truncation of asymptotic expansions and borel summation of the resulting exponentially small remainder terms for certain classes of functions defined by laplace and stieltjes transforms olver jones and boyd 1990 have obtained rigorous uniform exponentially improved asymptotic expansions which describe the smooth transition of a stokes multiplier across a stokes line in this paper we present an alternative theory for functions defined by mellin barnes integrals this approach enables uniform exponentially improved expansions to be established in a direct manner including estimates for the remainder term in the exponentially small series the theory is illustrated by application to the confluent hypergeometric functions considered by olver 1990 1991

a bayesian analysis of kriging this article is concerned with predicting for gaussian random fields in a way that appropriately deals with uncertainty in the covariance function to this end we analyze the best linear unbiased prediction procedure within a bayesian framework particular attention is paid to the treatment of parameters in the covariance structure and their effect on the quality both real and perceived of the prediction these ideas are implemented using topographical data from davis

spectral methods for nonstationary spatial processes summary we propose a nonstationary periodogram and various parametric approaches for estimating the spectral density of a nonstationary spatial process we also study the asymptotic properties of the proposed estimators via shrinking asymptotics assuming the distance between neighbouring observations tends to zero as the size of the observation region grows without bound with this type of asymptotic model we can uniquely determine the spectral density avoiding the aliasing problem we also present a new class of nonstationary processes based on a convolution of local stationary processes this model has the advantage that the model is simultaneously defined everywhere unlike moving window approaches but it retains the attractive property that locally in small regions it behaves like a stationary spatial process applications include the spatial analysis and modelling of air pollution data provided by the us environmental protection agency

gaussian quadrature for multiple orthogonal polynomials we study multiple orthogonal polynomials of type i and type ii which have orthogonality conditions with respect to r measures these polynomials are connected by their recurrence relation of order r 1 first we show a relation with the eigenvalue problem of a banded lower hessenberg matrix l n containing the recurrence coefficients as a consequence we easily find that the multiple orthogonal polynomials of type i and type ii satisfy a generalized christoffel darboux identity furthermore we explain the notion of multiple gaussian quadrature for proper multi indices which is an extension of the theory of gaussian quadrature for orthogonal polynomials and was introduced by borges in particular we show that the quadrature points and quadrature weights can be expressed in terms of the eigenvalue problem of l n

error bounds for the method of steepest descents new error bounds are found for the asymptotic expansions which arise in the method of steepest descents these bounds are found by exploiting the reformulation of the method of steepest descents given in recent work by berry howls a major difference between this and previous work is that the global properties of the integrand and not just those on the path of steepest descent play an essential role the bounds are significantly easier to compute than those in the current literature applications of the error bounds are given for asymptotic expansions arising from the pearcey integral the airy function the modified bessel function and the integral of the airy function

overlapping stokes smoothings survival of the error function and canonical catastrophe integrals we derive doubly uniform approximations for the remainder in the optimally truncated saddle point expansion for an integral containing a large parameter double uniformity means that the formulae remain valid while distant saddles responsible for the divergence of the expansion coalesce and separate as described by catastrophe theory and while the subdominant exponentials they contribute switch on and off as described by the error function smoothing of the stokes phenomenon two sorts of asymptotic singularity are thereby united in a common framework the formula for the remainder incorporates both the stokes error function and the canonical catastrophe integrals a numerical illustration is given in which the distant cluster contains two saddles the asymptotic theory gives an accurate description of the details of the fractional remainder even when this is of order exp 36

stieltjes transforms and the stokes phenomenon recently berry olver and jones have found uniform asymptotic expansions for the exponentially small remainder terms that result when asymptotic expansions are optimally truncated these uniform expansions describe the rapid change in the behaviour of the remainders as a stokes line is crossed we show how such uniform expansions may be found when a function can be expressed as a stieltjes transform such an approach has the following advantages the uniform expansion is calculated directly non uniform expansions away from the stokes line are readily found and explicit error bounds may be established we illustrate the method by application to the modified bessel function k v z

on optimal truncation of divergent series solutions of nonlinear differential systems we prove that for divergent series solutions of nonlinear or linear differential systems near a generic irregular singularity the common prescription of summation to the least term is if properly interpreted meaningful and correct and we extend this method to trans series solutions in every direction in the complex plane at the singularity stokes9 directions not excepted there exists a non empty set of solutions whose difference from the optimally i e near the least term truncated asymptotic series is of the same exponentially small order of magnitude as the least term of the series there is a family of generalized borel summation formulae b which commute with the usual algebraic and analytic operations addition multiplication differentiation etc we show that there is exactly one of them b 0 such that for any formal series solution f b 0 f differs from the optimal truncation of f by at most the order of the least term of f we show in addition that the berry smoothing phenomenon is universal within this class of differential systems whenever the terms beyond all orders change in crossing a stokes line these terms vary smoothly on the berry scale x x 1 2 and the transition is always given by the error function under the same conditions we show that dingle9 rule of signs for stokes9 transitions holds

smoothing of the stokes phenomenon for high order differential equations we extend the class of functions for which the smooth transition of a stokes multiplier across a stokes line can be rigorously established to functions satisfying a certain differential equation of arbitrary order n the equation chosen admits solutions of hypergeometric function type which in the case n 2 are related to the parabolic cylinder functions in general the solutions of this equation involve compound asymptotic expansions valid in certain sectors of the complex z plane with more than one dominant and subdominant series the functional form of the stokes multipliers expressed in terms of an appropriately scaled variable describing transition across a stokes line is found to obey the error function smoothing law derived by berry

the asymptotic behaviour of pearcey s integral for complex variables a deeper understanding of the rich structure of the canonical form of the oscillatory integral describing the cusp diffraction catastrophe generally known as pearcey s integral p prime x y can be obtained by considering its analytic continuation to arbitrary complex variables x and y a new integral representation for p prime x y is given in the form of a contour integral involving a weber parabolic cylinder function whose order is the variable of integration it is shown how the asymptotics of p prime x y may be obtained from this representation for complex x and y when either x or y rightarrow infty without reference to the usual stationary points of the integrand for the case x rightarrow infty y finite the full asymptotic expansion of p prime x y is derived and its asymptotic character is found to be either exponentially large or algebraic in certain sectors of the x plane the case y rightarrow infty x finite is complicated by the presence of exponentially small subdominant terms in certain sectors of the y plane and only the first terms in the expansion are given the asymptotic behaviour of p prime x y on the caustic y 2 frac 2 3 x 3 0 is also obtained from the new representation and is shown to agree with recent results of d kaminski the various properties of the weber parabolic cylinder function required in this paper are collected together in the appendix

bias correction in generalised linear mixed models with a single component of dispersion summary general expressions are derived for the asymptotic biases in three approximate estimators of regression coefficients and variance component for small values of the variance component in generalised linear mixed models with canonical link function and a single source of extraneous variation the estimators involve first and second order laplace expansions of the integrated likelihood and a related procedure known as penalised quasilikelihood numerical studies of a series of matched pairs of binary outcomes show that the first order estimators of the variance component are seriously biased easily computed correction factors produce satisfactory estimators of small variance components comparable to those obtained with a second order laplace expansion and markedly improve the asymptotic performance for larger values for a series of matched pairs of binomial observations the variance correction factors rapidly approach one as the binomial denominators increase these results greatly extend the range of parameter values for which the approximate estimation procedures have satisfactory asymptotic properties

aberrant crypt foci and semiparametric modeling of correlated binary data motivated by the spatial modeling of aberrant crypt foci acf in colon carcinogenesis we consider binary data with probabilities modeled as the sum of a nonparametric mean plus a latent gaussian spatial process that accounts for short range dependencies the mean is modeled in a general way using regression splines the mean function can be viewed as a fixed effect and is estimated with a penalty for regularization with the latent process viewed as another random effect the model becomes a generalized linear mixed model in our motivating data set and other applications the sample size is too large to easily accommodate maximum likelihood or restricted maximum likelihood estimation reml so pairwise likelihood a special case of composite likelihood is used instead we develop an asymptotic theory for models that are sufficiently general to be used in a wide variety of applications including but not limited to the problem that motivated this work the splines have penalty parameters that must converge to zero asymptotically we derive theory for this along with a data driven method for selecting the penalty parameter a method that is shown in simulations to improve greatly upon standard devices such as likelihood crossvalidation finally we apply the methods to the data from our experiment acf we discover an unexpected location for peak formation of acf

exponentially improved asymptotics for the gamma function by expressing the error term in truncation of the asymptotic expansion in terms of a mellin barnes integral we obtain an exponentially improved asymptotic expansion for z as z in arg z we describe the need for such an expansion arising in applications the result is compared with formal results of dingle obtained via borel summation and rigorous methods of olver using laplace integrals for a different class of functions

penalized spline estimation for partially linear single index models single index models are potentially important tools for multivariate nonparametric regression they generalize linear regression by replacing the linear combination 0tx with a nonparametric component 0 0tx where 0 is an unknown univariate link function by reducing the dimensionality from that of a general covariate vector x to a univariate index 0tx single index models avoid the so called curse of dimensionality we propose penalized spline p spline estimation of 0 in partially linear single index models where the mean function has the form 0 0tx 0tz the p spline approach offers a number of advantages over other fitting methods for single index models all parameters in the p spline single index model can be estimated simultaneously by penalized nonlinear least squares as a direct least squares fitting method our approach is rapid and computationally stable standard nonlinear least squares software can be used moreover joint inference for 0 0 and 0 is possible by

approximate inference in generalized linear mixed models statistical approaches to overdispersion correlated errors shrinkage estimation and smoothing of regression relationships may be encompassed within the framework of the generalized linear mixed model glmm given an unobserved vector of random effects observations are assumed to be conditionally independent with means that depend on the linear predictor through a specified link function and conditional variances that are specified by a variance function known prior weights and a scale factor the random effects are assumed to be normally distributed with mean zero and dispersion matrix depending on unknown variance components for problems involving time series spatial aggregation and smoothing the dispersion may be specified in terms of a rank deficient inverse covariance matrix approximation of the marginal quasi likelihood using laplace s method leads eventually to estimating equations based on penalized quasilikelihood or pql for the mean parameters and pseudo likelihood for the variances im

multiple orthogonal polynomials of mixed type gauss borel factorization and the multi component 2d toda hierarchy multiple orthogonality is considered in the realm of a gauss borel factorization problem for a semi infinite moment matrix perfect combinations of weights and a finite borel measure are constructed in terms of m nikishin systems these perfect combinations ensure that the problem of mixed multiple orthogonality has a unique solution that can be obtained from the solution of a gauss borel factorization problem for a semi infinite matrix which plays the role of a moment matrix this leads to sequences of multiple orthogonal polynomials their duals and second kind functions it also gives the corresponding linear forms that are bi orthogonal to the dual linear forms expressions for these objects in terms of determinants from the moment matrix are given recursion relations are found which imply a multi diagonal jacobi type matrix with snake shape and results like the abc theorem or the christoffel darboux formula are re derived in this context using the factorization problem and the generalized hankel symmetry of the moment matrix the connection between this description of multiple orthogonality and the multi component 2d toda hierarchy which can be also understood and studied through a gauss borel factorization problem is discussed deformations of the weights natural for m nikishin systems are considered and the correspondence with solutions to the integrable hierarchy represented as a collection of lax equations is explored corresponding lax and zakharov shabat matrices as well as wave functions and their adjoints are determined the construction of discrete flows is discussed in terms of miwa transformations which involve darboux transformations for the multiple orthogonality conditions the bilinear equations are derived and the tau function representation of the multiple orthogonality is given

exponentially small expansions in the asymptotics of the wright function we consider exponentially small expansions present in the asymptotics of the generalised hypergeometric function or wright function email protected q z for large z that have not been considered in the existing theory our interest is principally with those functions of this class that possess either a finite algebraic expansion or no such expansion and with parameter values that produce exponentially small expansions in the neighbourhood of the negative real z axis numerical examples are presented to demonstrate the presence of these exponentially small expansions

asymptotics of the generalized exponential integral and error bounds in the uniform asymptotic smoothing of its stokes discontinuities uniform asymptotic approximations are derived for the generalized exponential integral e p z where p is real and z complex both the cases p rightarrow infty and z rightarrow infty are considered for the case p rightarrow infty an expansion in inverse powers of p is derived which involves elementary functions and readily computed coefficients and is uniformly valid for pi delta leq arg z leq pi delta where delta is an arbitrary small positive constant an approximation for large p involving the complementary error function is also derived which is valid in an unbounded z domain which contains the negative real axis the case z rightarrow infty is then considered and uniform asymptotic approximations are derived which involve the complementary error function in the first approximation and the parabolic cylinder function in an expansion both approximations are valid for values of p satisfying 0 leq p leq z a where a is bounded uniformly for pi delta leq arg z leq 3 pi delta these are examples of the so called stokes smoothing theory which was initiated by berry the novelty of the new stokes smoothing approximations is that they include explicit and realistic error bounds as do all the other approximations in the present investigation

a note on pseudolikelihood constructed from marginal densities for likelihood based inference involving distributions in which high dimensional dependencies are present it may be useful to use approximate likelihoods based for example on the univariate or bivariate marginal distributions the asymptotic properties of formal maximum likelihood estimators in such cases are outlined in particular applications in which only a single q x i vector of observations is observed are examined conditions under which consistent estimators of parameters result from the approximate likelihood using only pairwise joint distributions are studied some examples are analysed in detail

exponentially improved asymptotic solutions of ordinary differential equations ii irregular singularities of rank one re expansions are found for the optimal remainder terms in the well known asymptotic series solutions of homogeneous linear differential equations of the second order in the neighbourhood of an irregular singularity of rank one the re expansions are in terms of generalized exponential integrals and have greater regions of validity than the original expansions as well being considerably more accurate and providing a smooth interpretation of the stokes phenomenon they are also of strikingly simple form in addition explicit asymptotic expansions for the higher coefficients of the original asymptotic solutions are obtained

waves near stokes lines the large k asymptotics of d 2 u z d z 2 k 2 r 2 z u z are studied near a stokes line z z 0 r d z real where z 0 is a zero of r 2 z of any order on which there is greatest disparity between the dominant and subdominant exponential waves in the phase integral wkb approximations the aim is to establish precisely how the multiplier b of the subdominant wave varies across the stokes line although b always has a total change proportional to i times the multiplier of the dominant wave the stokes phenomenon the form of the change depends on the convention used to define the two waves the optimal convention for which the variation is maximally compact and smooth is to define them by the phase integral approximation truncated at its least term whose order is proportional to k and therefore large asymptotics of asymptotics then the variation of b is proportional to the error function of the natural stokes crossing variable im k re this result is obtained without resumming divergent series thereby avoiding asymptotics of asymptotics of asymptotics an application is given to the birth of exponentially weak reflected waves in media with smoothly varying refractive index

maximizing generalized linear mixed model likelihoods with an automated monte carlo em algorithm summary two new implementations of the em algorithm are proposed for maximum likelihood fitting of generalized linear mixed models both methods use random independent and identically distributed sampling to construct monte carlo approximations at the e step one approach involves generating random samples from the exact conditional distribution of the random effects given the data by rejection sampling using the marginal distribution as a candidate the second method uses a multivariate t importance sampling approximation in many applications the two methods are complementary rejection sampling is more efficient when sample sizes are small whereas importance sampling is better with larger sample sizes monte carlo approximation using random samples allows the monte carlo error at each iteration to be assessed by using standard central limit theory combined with taylor series methods specifically we construct a sandwich variance estimate for the maximizer at each approximate e step this suggests a rule for automatically increasing the monte carlo sample size after iterations in which the true em step is swamped by monte carlo error in contrast techniques for assessing monte carlo error have not been developed for use with alternative implementations of monte carlo em algorithms utilizing markov chain monte carlo e step approximations three different data sets including the infamous salamander data of mccullagh and nelder are used to illustrate the techniques and to compare them with the alternatives the results show that the methods proposed can be considerably more efficient than those based on markov chain monte carlo algorithms however the methods proposed may break down when the intractable integrals in the likelihood function are of high dimension

uniform asymptotic smoothing of stokes s discontinuities across a stokes line where one exponential in an asymptotic expansion maximally dominates another the multiplier of the small exponential changes rapidly if the expansion is truncated near its least term the change is not discontinuous but smooth and moreover universal in form in terms of the singulant f the difference between the larger and smaller exponents and real on the stokes line the change in the multiplier is the error function pi frac 1 2 int sigma infty dt exp t 2 text where sigma imf 2ref frac 1 2 the derivation requires control of exponentially small terms in the dominant series this is achieved with dingle s method of borel summation of late terms starting with the least term in numerical illustrations the multiplier is extracted from dawson s integral erfi and the airy function of the second kind bi the small exponential emerges in the predicted universal manner from the dominant one which can be 10 10 times larger

bayesian prediction of spatial count data using generalized linear mixed models summary spatial weed count data are modeled and predicted using a generalized linear mixed model combined with a bayesian approach and markov chain monte carlo informative priors for a data set with sparse sampling are elicited using a previously collected data set with extensive sampling furthermore we demonstrate that so called langevin hastings updates are useful for efficient simulation of the posterior distributions and we discuss computational issues concerning prediction

stokes phenomenon and matched asymptotic expansions this paper describes the use of matched asymptotic expansions to illuminate the description of functions exhibiting stokes phenomenon in particular the approach highlights the way in which the local structure and the possibility of finding stokes multipliers explicitly depend on the behaviour of the coefficients of the relevant asymptotic expansions

hyperasymptotic solutions of higher order linear differential equations with a singularity of rank one a sequence of re expansions is developed for the remainder terms in the well known poincare series expansions of the solutions of homogeneous linear differential equations of higher order in the neighbourhood of an irregular singularity of rank one these re expansions are a series whose terms are a product of stokes multipliers coefficients of the original poincare series expansions and certain multiple integrals the so called hyperterminants each step of the process reduces the estimate of the error term by an exponentially small factor the method of this paper is based on the borel laplace transform which makes it applicable to other problems at the end of the paper the method is applied to integrals with saddles also a powerful new method is presented to compute the stokes multipliers a numerical example is included

interlacing properties of zeros of multiple orthogonal polynomials abstract it is well known that the zeros of orthogonal polynomials interlace in this paper we study the case of multiple orthogonal polynomials we recall known results and some recursion relations for multiple orthogonal polynomials our main result gives a sufficient condition based on the coefficients in the recurrence relations for the interlacing of the zeros of neighboring multiple orthogonal polynomials we give several examples illustrating our result

infinitely many stokes smoothings in the gamma function the stokes lines for gamma z are the positive and negative imaginary axes where all terms in the divergent asymptotic expansion for ln gamma z have the same phase on crossing these lines from the right to the left half plane infinitely many subdominant exponentials appear rather than the usual one the exponentials increase in magnitude towards the negative real axis anti stokes line where they add to produce the poles of gamma z corresponding to each small exponential is a separate component asymptotic series in the expansion for ln gamma z if each is truncated near its least term its exponential switches on smoothly across the stokes lines according to the universal error function law by appropriate subtractions from ln gamma z the switching on of successively smaller exponentials can be revealed the procedure is illustrated by numerical computations

a composite likelihood approach in fitting spatial point process models we propose a new likelihood based approach in fitting spatial point process models a composite likelihood is first formed by adding some pairwise composite likelihood functions that are defined in terms of the second order intensity function of the underlying process and then used for estimating the unknown parameters the estimation procedure is computationally simple and yields consistent and asymptotically normal estimators under some mild conditions we demonstrate through a simulation study and applications to two real data examples that the proposed approach may lead to improved estimations compared with the commonly used minimum contrast estimation approach

a jacobi type christoffel darboux formula for multiple orthogonal polynomials of mixed type abstract an alternative expression for the christoffel darboux formula for multiple orthogonal polynomials of mixed type is derived from the lu factorization of the moment matrix of a given measure and two sets of weights we use the action of the generalized jacobi matrix j also responsible for the recurrence relations on the linear forms and their duals to obtain the result

bias correction in generalized linear mixed models with multiple components of dispersion abstract general formulas are derived for the asymptotic bias in regression coefficients and variance components estimated by penalized quasi likelihood pql in generalized linear mixed models with canonical link function and multiple sets of independent random effects easily computed correction matrices result in variance component estimates that have satisfactory asymptotic behavior for small values of the variance components and significantly reduce bias for larger values both first order and second order correction procedures are developed for regression coefficients estimated by pql the methods are illustrated through an analysis of an experiment on salamander matings involving crossed male and female random effects and their properties are evaluated in a simulation study

mixed type multiple orthogonal polynomials perfectness and interlacing properties of zeros abstract the zeros of orthogonal polynomials on the real line are simple lie in the interior of the convex hull of the support of the measure and the zeros of consecutive orthogonal polynomials interlace we study a large class of mixed type multiple orthogonal polynomials for which similar properties hold