on the existence of periodic solutions to p laplacian rayleigh differential equation with a delay abstract in this paper the authors study the existence of periodic solutions to a p laplacian rayleigh differential equation with a delay as follows p y t f y t g y t t e t where p 1 is a constant p r r p u u p 2 u f g e c r r t t t with t 0 t 0 t and e t t e t t 0 is a constant by using mawhin s continuation theorem some new results are obtained

existence and uniqueness of periodic solutions for a p laplacian duffing equation with a deviating argument abstract by means of mawhin s continuation theorem we study a class of p laplacian duffing type differential equations of the form p x t c x t g t x t x t t e t some new results on the existence and uniqueness of periodic solutions for the above equation are obtained it is significant that the growth degree with respect to the variables u v imposed on g t u v is allowed to be greater than p 1 so our results generalize and improve on the corresponding results in related papers

edge effects and efficient parameter estimation for stationary random fields summary we consider the estimation of the parameters of a stationary random field on d dimensional lattice by minimizing the classical whittle approximation to the gaussian log likelihood if the usual biased sample covariances are used the estimate is efficient only in one dimension to remove this edge effect we introduce data tapers and show that the resulting modified estimate is efficient also in two and three dimensions this avoids the use of the unbiased sample covariances which are in general not positive definite

periodic solutions for p laplacian lienard equation with a deviating argument abstract by employing mawhin s continuation theorem the existence of periodic solutions of the p laplacian lienard equation with a deviating argument p x t f x t x t g x t t e t under various assumptions are obtained

on maximum likelihood estimation of the differencing parameter of fractionally integrated noise with unknown mean abstract there are two approaches to maximum likelihood ml estimation of the parameter of fractionally integrated noise approximate frequency domain ml fox and taqqu 1986 and exact time domain ml sowell 1992b if the mean of the process is known then a clear finite sample mean squared error ranking of the estimators emerges the exact time domain estimator is superior we show in this paper however that the finite sample efficiency of approximate frequency domain ml relative to exact time domain ml rises dramatically when the mean is unknown and so must be estimated the intuition for our result is straightforward the frequency domain ml estimator is invariant to the true but unknown mean of the process while the time domain ml estimator is not feasible time domain estimation must therefore be based upon de meaned data but the long memory associated with fractional integration makes precise estimation of the mean difficult we conclude that the frequency domain estimator is an attractive and efficient alternative for situations in which large sample sizes render time domain estimation impractical

approximate likelihood for large irregularly spaced spatial data likelihood approaches for large irregularly spaced spatial datasets are often very difficult if not infeasible to implement due to computational limitations even when we can assume normality exact calculations of the likelihood for a gaussian spatial process observed at n locations requires o n3 operations we present a version of whittle s approximation to the gaussian log likelihood for spatial regular lattices with missing values and for irregularly spaced datasets this method requires o nlog2n operations and does not involve calculating determinants we present simulations and theoretical results to show the benefits and the performance of the spatial likelihood approximation method presented here for spatial irregularly spaced datasets and lattices with missing values we apply these methods to estimate the spatial structure of sea surface temperatures using satellite data with missing values

on conditional and intrinsic autoregressions summary gaussian conditional autoregressions have been widely used in spatial statistics and bayesian image analysis where they are intended to describe interactions between random variables at fixed sites in euclidean space the main appeal of these distributions is in the markovian interpretation of their full conditionals intrinsic autoregressions are limiting forms that retain the markov property despite being improper they can have advantages over the standard autoregressions both conceptually and in practice for example they often avoid difficulties in parameter estimation without apparent loss or exhibit appealing invariances as in texture analysis however on small arrays and in nonlattice applications both forms of autoregression can lead to undesirable second order characteristics either in the variables themselves or in contrasts among them this paper discusses standard and intrinsic autoregressions and describes how the problems that arise can be alleviated using dempster s 1972 algorithm or an appropriate modification the approach represents a partial synthesis of standard geostatistical and gaussian markov random field formulations some nonspatial applications are also mentioned

periodic solutions for a kind of duffing type p laplacian equation abstract based on manasevich mawhin continuation theorem and some analysis skill some new sufficient conditions for the existence of periodic solutions for duffing type p laplacian differential equation are obtained an example is provided to demonstrate effectiveness of the proposed results

generalized levinson durbin and burg algorithms abstract a basic recursive property of orthogonal projections is presented which leads easily to a variety of different prediction algorithms examples are the classical levinson durbin and burg algorithms and a subset whittle algorithm of penm and terrell in addition some new algorithms are derived including easily applied algorithms for the recursive calculation of best h step predictors and a burg algorithm for the best subset predictor the relation to lattice algorithms is discussed

thermodynamics and statistical analysis of gaussian random fields gaussian fields are considered as gibbsian fields thermodynamic functions are calculated for them and the variational principle is proved as an application we get an approximation of log likelihood and an information theoretic interpretation of the asymptotic behaviour of the maximum likelihood estimator for gaussian markov fields

existence of periodic solutions for a lienard type p laplacian differential equation with a deviating argument by means of mawhin s continuation theorem a class of p laplacian type differential equation with a deviating argument of the form p x t f x t x t t g t x t t x e t is studied a new result related to t and the deviating argument t x is obtained it is significant that the growth degree with respect to the variable x in g t x is allowed to be greater than p 1 which could be achieved infrequently in previous papers

asymptotically optimal estimation in misspecified time series models a concept of asymptotically efficient estimation is presented whena misspecified parametric time series model is fitted to a stationary process efficiency of several minimum distance estimates is proved and the behavior ofthe gaussian maximum likelihood estimate is studied furthermore the behaviorof estimates that minimize the h step prediction error is discussed briefly the paper answers to some extent the question what happens when a misspecifiedmodel is fitted to time series data and one acts as if the model were true

the generalised autocovariance function the generalised autocovariance function is defined for a stationary stochastic process as the inverse fourier transform of the power transformation of the spectral density function depending on the value of the transformation parameter this function nests the inverse and the traditional autocovariance functions a frequency domain non parametric estimator based on the power transformation of the pooled periodogram is considered and its asymptotic distribution is derived the results are employed to construct classes of tests of the white noise hypothesis for clustering and discrimination of stochastic processes and to introduce a novel feature matching estimator of the spectrum

asymptotic equivalence of spectral density estimation and gaussian white noise we consider the statistical experiment given by a sample y 1 y n of a stationary gaussian process with an unknown smooth spectral density f asymptotic equivalence in the sense of le cam s deficiency distance to two gaussian experiments with simpler structure is established the first one is given by independent zero mean gaussians with variance approximately f i where i is a uniform grid of points in nonparametric gaussian scale regression this approximation is closely related to well known asymptotic independence results for the periodogram and corresponding inference methods the second asymptotic equivalence is to a gaussian white noise model where the drift function is the log spectral density this represents the step from a gaussian scale model to a location model and also has a counterpart in established inference methods that is log periodogram regression the problem of simple explicit equivalence maps markov kernels allowing to directly carry over inference appears in this context but is not solved here

saddlepoint approximations for short and long memory time series a frequency domain approach abstract saddlepoint techniques provide numerically accurate small sample approximations to the distribution of estimators and test statistics except for a few simple models these approximations are not available in the framework of stationary time series we contribute to fill this gap under short or long range serial dependence for gaussian and non gaussian processes we show how to derive and implement saddlepoint approximations for two relevant classes of frequency domain statistics ratio statistics and whittle s estimator we compare our new approximations to the ones obtained by the standard asymptotic theory and by two widely applied bootstrap methods the numerical exercises for whittle s estimator show that our approximations yield accuracy s improvements while preserving analytical tractability a real data example concludes the paper

periodic solutions for a higher order p laplacian neutral functional differential equation with a deviating argument abstract in this paper a higher order p laplacian neutral functional differential equation with a deviating argument p x t c t x t n m f x t x t g t x t t e t has been studied by applying mawhin s continuation degree theorem some new criteria to guarantee the existence of periodic solutions are obtained it is interesting that the power of the variable x in function g is allowed to be greater than p 1

on the correlation matrix of the discrete fourier transform and the fast solution of large toeplitz systems for long memory time series we show that for long memory time series the toeplitz system n f x b can be solved in o nlog5 2n operations using a well known version of the preconditioned conjugate gradient method where n f is the n n covariance matrix f is the spectral density and b is a known vector solutions of such systems are needed for optimal linear prediction and interpolation we establish connections between this preconditioning method and the frequency domain analysis of time series indeed the running time of the algorithm is determined by the rate of increase in the condition number of the correlation matrix of the discrete fourier transform dft vector as the sample size tends to we derive an upper bound for this condition number the bound is of interest in its own right because it sheds some light on the widely used but heuristic approximation that the standardized dft coefficients are uncorrelated with equal variances we present applications of the preconditioning methodology to the forecasting of

valid asymptotic expansions for the maximum likelihood estimator of the parameter of a stationary gaussian strongly dependent process we establish the validity of an edgeworth expansion to the distribution of the maximum likelihood estimator of the parameter of a stationary gaussian strongly dependent process the result covers arfima type models including fractional gaussian noise the method of proof consists of three main ingredients i verification of a suitably modified version of durbin s general conditions for the validity of the edgeworth expansion to the joint density of the log likelihood derivatives ii appeal to a simple result of skovgaard to obtain from this an edgeworth expansion for the joint distribution of the log likelihood derivatives iii appeal to and extension of arguments of bhattacharya and ghosh to accomplish the passage from the result on the log likelihood derivatives to the result for the maximum likelihood estimators we develop and make extensive use of a uniform version of a theorem of dahlhaus on products of toeplitz matrices the extension of dahlhaus result is of interest in its own right a small numerical study of the efficacy of the edgeworth expansion is presented for the case of fractional gaussian noise

the debiased whittle likelihood the whittle likelihood is a widely used and computationally efficient pseudolikelihood however it is known to produce biased parameter estimates with finite sample sizes for large classes of models we propose a method for debiasing whittle estimates for second order stationary stochastic processes the debiased whittle likelihood can be computed in the same o n log n operations as the standard whittle approach we demonstrate the superior performance of our method in simulation studies and in application to a large scale oceanographic dataset where in both cases the debiased approach reduces bias by up to two orders of magnitude achieving estimates that are close to those of the exact maximum likelihood at a fraction of the computational cost we prove that the method yields estimates that are consistent at an optimal convergence rate of n 1 2 for gaussian processes and for certain classes of non gaussian or nonlinear processes this is established under weaker assumptions than in the standard theory and in particular the power spectral density is not required to be continuous in frequency we describe how the method can be readily combined with standard methods of bias reduction such as tapering and differencing to further reduce bias in parameter estimates

graphical gaussian modelling of multivariate time series with latent variable in time series analysis inference about causeeffect relationships among multiple times series is commonly based on the concept of granger causality which exploits temporal structure to achieve causal ordering of dependent variables one major problem in the application of granger causality for the identification of causal relationships is the possible presence of latent variables that affect the measured components and thus lead to so called spurious causalities in this paper we describe a new graphical approach for modelling the dependence structure of multivariate stationary time series that are affected by latent variables to this end we introduce dynamic maximal ancestral graphs dmags in which each time series is represented by a single vertex for gaussian processes this approach leads to vector autoregressive models with errors that are not independent but correlated according to the dashed edges in the graph we discuss identifiability of the parameters and show that these models can be viewed as graphical arma models that satisfy the granger causality restrictions encoded by the associated dynamic maximal ancestral graph

sufficient conditions for the existence of periodic solutions to some second order differential equations with a deviating argument abstract by means of mawhin s continuation theorem we study some second order differential equations with a deviating argument x t f t x t x t t x t e t some new results on the existence of periodic solutions are obtained the interest is that we allow the degree with respect to the variables x 0 x 1 x 2 of f t x 0 x 1 x 2 to be greater than 1 and also the result theorem 3 2 is related to the deviating argument t meanwhile we give an example to demonstrate our result

nonparametric high resolution spectral estimation the uniform rate of convergence of the integrated relative mean square error over a with the sample sizet increasing classi t of stationary processes is studied for several estimates of the spectral density the classi t is chosen in a way such that estimates with a good uniform rate of convergence overi t may be termed high resolution spectral estimates by using this criterion several effects are explained theoretically for example the leakage effect the advantages uf using data tapers are proved and the use of local and global bandwiths are studied furthermore the behaviors of segment estimates are studied simulations are presented for the illustration of some effects

on the caratheodory john multiplier rule suppose maps an open subset u of rn into rk a u s is a subset of u and int s denotes the interior of the image s call any result with conclusion a int s an interior mapping theorem the best known example is an easy corollary of the classical implicit mapping theorem if is strongly differentiable at a u and if l a then a int u whenever l a int l u that is whenever the linear transformation l maps rn onto rk a more subtle interior mapping theorem is proved in this paper if a u c where c is a convex subset of u if is strongly differentiable at a u and if l a then a int c whenever l a int l c this convex interior mapping theorem is then applied to yield a short proof of the caratheodory john multiplier rule for minimizing a strongly differentiable function 0 subject to strongly differentiable inequality constraints 1 0 p 0 strongly differentiable equality constraints p 1 0 p q 0 a corollary of this fundamental multiplier rule is the well known karush kuhn tucker theorem the demonstration proceeds by examining interiority properties of the mapping 0 1 p q from u into rp q 1

a frequency domain bootstrap for ratio statistics in time series analysis the asymptotic properties of the bootstrap in the frequency domain based on studentized periodogram ordinates are studied it is proved that this bootstrap approximation is valid for ratio statistics such as autocorrelations by using edgeworth expansions it is shown that the bootstrap approximation even outperforms the normal approximation the results carry over to whittle estimates in a simulation study the behavior of the bootstrap is studied for empirical correlations and whittle estimates

fitting time series models to nonstationary processes a general minimum distance estimation procedure is presented for nonstationary time series models that have an evolutionary spectral representation the asymptotic properties of the estimate are derived under the assumption of possible model misspecification for autoregressive processes with time varying coefficients the estimate is compared to the least squares estimate furthermore the behavior of estimates is explained when a stationary model is fitted to a nonstationary process

autoregressive spatial spectral estimates abstract nonparametric spectral density estimates find many uses in econometrics for stationary random fields on a regular spatial lattice we propose an autoregressive nonparametric spectral density estimate that is guaranteed positive even when suitable edge effect correction is employed and is simple to compute using least squares our estimate is based on truncating a true half plane infinite autoregressive representation while also allowing the truncation length to diverge in all dimensions to avoid the potential bias due to truncation at a fixed lag length uniform consistency of the proposed estimate is established and new criteria for order selection are also suggested and studied in practical settings the asymptotic distribution of the estimate is shown to be zero mean normal and independent at fixed distinct frequencies mirroring the behaviour for time series a small monte carlo experiment examines finite sample performance technically the key to the results is the covariance structure of stationary random fields defined on regularly spaced lattices we show the covariance matrix to satisfy a generalization of the toeplitz property familiar from time series analysis

approximations for the inverse of toeplitz matrices with applications to stationary processes abstract several approximations for the inverse of symmetric toeplitz matrices are compared by a new approximation criterion the criterion requires the approximation to be uniformly good over a set of matrices which increases with the size furthermore a new approximation is introduced by applying a taper to the classical whittle approximation

maximum likelihood estimation of stationary univariate fractionally integrated time series models abstract to estimate the parameters of a stationary univariate fractionally integrated time series the unconditional exact likelihood function is derived this allows the simultaneous estimation of all the parameters of the model by exact maximum likelihood issues involved in obtaining maximum likelihood estimates are discussed particular attention is given to efficient procedures to evaluate the likelihood function obtaining starting values and the small sample properties of the estimators limitations of previous estimation procedures are also discussed

bias of autoregressive spectral estimators abstract bias of the least squares estimator of the log of the spectral density of an autoregression attenuates the peaks of the estimator under the assumption of an autoregressive generating process of known finite order we obtain an expression for the order 1 t bias where t is the sample length of the observed series this approximation is a sum of several simple functions of the unknown coefficients when the spectral density has sharp peaks one of these functions dominates the bias the attenuation from this dominant component can be substantial when the spectral peak is well defined and several examples illustrate this effect since the integral of the order 1 t bias components that are frequency dependent is 0 unbiased estimation of entropy to this order is possible for autoregressive processes these bias expressions extend to autoregressive models in which the mean is a polynomial function of time similar results obtain for the log of the yule walker spectral estimator for which the order

pseudolinear estimation of fractionally integrated arma arfima models with automotive application a novel pseudolinear method for the estimation of fractionally integrated arma arfima models that are capable of representing combined long and short term dependency is introduced the method is based on the relationship of the ar ma parameters and the coefficients of the fractional power operator binomial series expansion with the model s inverse function these lead to the formulation of a special form regression problem that can be decomposed into a univariate nonlinear and a multivariate linear regression and may be thus tackled via a special pseudolinear procedure this decomposition in turn leads to the elimination of the need for initial guess parameter values drastic simplification in the detection and handling of potential local extrema problems as well as computational simplicity the method s strong consistency is established whereas its performance characteristics are assessed via monte carlo experiments and comparisons with the maximum likelihood method the pseudolinear method is also used for the arfima modeling and prediction of power consumption in an experimental automobile fully active suspension system the consumption of which is shown to exhibit long term dependency a comparison with arma arima type modeling is also made and the obtained arfima models are shown to achieve improved predictive performance at a drastically reduced parametric complexity

on the existence of periodic solutions for p laplacian generalized lienard equation abstract by employing mawhin s continuation theorem the existence of periodic solutions of the p laplacian generalized lienard equation with deviating argument p x t f t x t x t t g x t t e t under various assumptions are obtained

homeomorphisms and generalized derivatives when d is open in r f d rk is said to be lipschitzian provided each point in d has a neighborhood where f is lipschitz continuous for such maps the derivative f p may not exist everywhere in d but one can at least assign to each p a certain collection f p of linear transformations from r into r k called the generalized derivative for the definition see section 4 reading clarke 3 or pourciau 20 j one finds this notion of differentiability enjoys many nice properties but only the following extension of the inverse function theorem needs to be singled out here supposing f has values in r if af p is invertible that is if each linear transformation in af p is invertible at every p then f is a local homeomorphism of course local homeomorphisms are not generally homeomorphisms and in this paper we seek additional conditions on the generalized derivative and the boundary behavior of the map forcing f to be a homeomorphism in mathematical economics theorems of this type are crucial for establishing the uniqueness of competitive equilibrium points in abstract economies for such applications theorems that require the image off to satisfy a difficult to verify condition like simple connectedness are not very useful involving boundary restrictions and covering spaces our results at first are topological but then in section 4 we blend these topological ideas with properties of the generalized derivative to obtain analytic conditions when the map is lipschitzian throughout we suppose d is a domain with compact closure k in euclidean n space r b is the boundary k d f is a continuous map from k into r and f is the restriction f d our path will take us through covering spaces we call f d m a covering if each point of fz has an open neighborhood v whose preimage f v is the pairwise disjoint union of open sets each of which f maps homeomorphically onto v when f is a covering it is a local homeomorphism yet not all local homeomorphisms are coverings

small sample effects in time series analysis a new asymptotic theory and a new estimate on introduit un modele mathematique qui couvre les effets des petits echantillons dans l approximation de whittle 1953 on montre que l estimation du maximum de vraisemblance exacte est toujours optimale dans ce modele

polynomial tapered two stage least squares method in nonlinear regression nonlinear models play an important role in various scientific disciplines and engineering the parameter estimation of these models should be efficient to make better decisions ordinary least squares ols method is used for estimating the parameters of nonlinear regression models when all regression assumptions are satisfied if there is a problem with these assumptions ols fails to give efficient results this paper examines the efficiency of parameter estimation under the problem of autocorrelated errors some methods have been proposed in order to overcome the problem and obtain efficient parameter estimates especially for autoregressive ar processes one of the most commonly used method is two stage least squares 2sls this method is based on generalized least squares in this paper a novel approach is proposed for 2sls method by evaluating a polynomial tapering procedure on autocorrelated errors this new method is called tapered two stage least squares t2sls the finite sample properties and improvements of t2sls are explored by means of some real life examples and a monte carlo simulation study both numerical and experimental results reveal that t2sls can give more efficient parameter estimates especially in small samples under the autocorrelation problem when compared to ols and 2sls

periodic solutions for a kind of high order p laplacian differential equation with sign changing coefficient ahead of the non linear term abstract in this paper by using the theory of fourier series bernoulli number theory and the continuation theorem of coincidence degree theory we study a kind of high order p laplacian differential equation as follows p y m t m f y t y t h y t t g y t t e t some new results on the existence of periodic solutions are obtained the interesting thing is that the coefficient t is allowed to change sign but the methods used to estimate a priori bounds of periodic solutions are different from the corresponding ones used in the past

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

new results on the existence of periodic solutions to a p laplacian differential equation with a deviating argument by means of mawhin s continuation theorem a kind of p laplacian differential equation with a deviating argument as follows p x t f t x t x t t x t e t is studied some new results on the existence of periodic solutions are obtained the main results theorems 3 2 and 3 3 are all related to the deviating argument t meanwhile the degrees with respect to the variables x0 x1 of f t x0 x1 x2 are allowed to be grater than p 1 which is different from the corresponding conditions of known literature

periodic solutions for a fourth order rayleigh type p laplacian delay equation abstract by applying mawhin s continuation theory and some new inequalities the existence of periodic solutions for a fourth order rayleigh type p laplacian delay equation is obtained moreover an example is given to illustrate the results

spectral density estimation for random fields via periodic embeddings we introduce methods for estimating the spectral density of a random field on a formula see text dimensional lattice from incomplete gridded data data are iteratively imputed onto an expanded lattice according to a model with a periodic covariance function the imputations are convenient computationally in that circulant embedding and preconditioned conjugate gradient methods can produce imputations in formula see text time and formula see text memory however these so called periodic imputations are motivated mainly by their ability to produce accurate spectral density estimates in addition we introduce a parametric filtering method that is designed to reduce periodogram smoothing bias the paper contains theoretical results on properties of the imputed data periodogram and numerical and simulation studies comparing the performance of the proposed methods to existing approaches in a number of scenarios we present an application to a gridded satellite surface temperature dataset with missing values

asymptotic theory for nonparametric regression with spatial data nonparametric regression with spatial or spatio temporal data is considered the conditional mean of a dependent variable given explanatory ones is a nonparametric function while the conditional covariance reflects spatial correlation conditional heteroscedasticity is also allowed as well as non identically distributed observations instead of mixing conditions a possibly non stationary linear process is assumed for disturbances allowing for long range as well as short range dependence while decay in dependence in explanatory variables is described using a measure based on the departure of the joint density from the product of marginal densities a basic triangular array setting is employed with the aim of covering various patterns of spatial observation sufficient conditions are established for consistency and asymptotic normality of kernel regression estimates when the cross sectional dependence is sufficiently mild the asymptotic variance in the central limit theorem is the same as when observations are independent otherwise the rate of convergence is slower we discuss the application of our conditions to spatial autoregressive models and models defined on a regular lattice

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