inconsistent estimation and asymptotically equal interpolations in model based geostatistics it is shown that in model based geostatistics not all parameters in the matern class can be estimated consistently if data are observed in an increasing density in a fixed domain regardless of the estimation methods used nevertheless one quantity can be estimated consistently by the maximum likelihood method and this quantity is more important to spatial interpolation the results are established by using the properties of equivalence and orthogonality of probability measures some sufficient conditions are provided for both gaussian and non gaussian equivalent measures and necessary conditions are provided for gaussian equivalent measures two simulation studies are presented that show that the fixed domain asymptotic properties can explain some finite sample behavior of both interpolation and estimation when the sample size is moderately large

nonparametric estimation of probability density functions for irregularly observed spatial data nonparametric estimation of probability density functions both marginal and joint densities is a very useful tool in statistics the kernel method is popular and applicable to dependent data including time series and spatial data but at least for the joint density one has had to assume that data are observed at regular time intervals or on a regular grid in space though this is not very restrictive in the time series case it often is in the spatial case in fact to a large degree it has precluded applications of nonparametric methods to spatial data because such data often are irregularly positioned over space in this article we propose nonparametric kernel estimators for both the marginal and in particular the joint probability density functions for nongridded spatial data large sample distributions of the proposed estimators are established under mild conditions and a new framework of expanding domain infill asymptotics is suggested to overcome the shortcomings of spatial asymptotics in the

hierarchical spatial process models for multiple traits in large genetic trials this article expands upon recent interest in bayesian hierarchical models in quantitative genetics by developing spatial process models for inference on additive and dominance genetic variance within the context of large spatially referenced trial datasets of multiple traits of interest direct application of such multivariate models to large spatial datasets is often computationally infeasible because of cubic order matrix algorithms involved in estimation the situation is even worse in markov chain monte carlo mcmc contexts where such computations are performed for several thousand iterations here we discuss approaches that help obviate these hurdles without sacrificing the richness in modeling for genetic effects we demonstrate how an initial spectral decomposition of the relationship matrices negates the expensive matrix inversions required in previously proposed mcmc methods for spatial effects we discuss a multivariate predictive process that reduces the computational burden by projecting th

stochastic switching for partially observable dynamics and optimal asset allocation abstractin industrial applications optimal control problems frequently appear in the context of decision making under incomplete information in such framework decisions must be adapted dynamically to account for possible regime changes of the underlying dynamics using stochastic filtering theory markovian evolution can be modelled in terms of latent variables which naturally leads to high dimensional state space making practical solutions to these control problems notoriously challenging in our approach we utilise a specific structure of this problem class to present a solution in terms of simple reliable and fast algorithms the algorithms presented in this paper have already been implemented in an r package

asymptotic theory of generalized information criterion for geostatistical regression model selection information criteria such as akaike s information criterion and bayesian information criterion are often applied in model selection however their asymptotic behaviors for selecting geostatistical regression models have not been well studied particularly under the fixed domain asymptotic framework with more and more data observed in a bounded fixed region in this article we study the generalized information criterion gic for selecting geostatistical regression models under a more general mixed domain asymptotic framework via uniform convergence developments of some statistics we establish the selection consistency and the asymptotic loss efficiency of gic under some regularity conditions regardless of whether the covariance model is correctly or wrongly specified we further provide specific examples with different types of explanatory variables that satisfy the conditions for example in some situations gic is selection consistent even when some spatial covariance parameters cannot be estimated consistently on the other hand gic fails to select the true polynomial order consistently under the fixed domain asymptotic framework moreover the growth rate of the domain and the degree of smoothness of candidate regressors in space are shown to play key roles for model selection

covariance tapering for likelihood based estimation in large spatial data sets maximum likelihood is an attractive method of estimating covariance parameters in spatial models based on gaussian processes but calculating the likelihood can be computationally infeasible for large data sets requiring o n3 calculations for a data set with n observations this article proposes the method of covariance tapering to approximate the likelihood in this setting in this approach covariance matrixes are tapered or multiplied element wise by a sparse correlation matrix the resulting matrixes can then be manipulated using efficient sparse matrix algorithms we propose two approximations to the gaussian likelihood using tapering one of these approximations simply replaces the model covariance with a tapered version whereas the other is motivated by the theory of unbiased estimating equations focusing on the particular case of the matern class of covariance functions we give conditions under which estimators maximizing the tapering approximations are like the maximum likelihood estimat

a comparison of generalized cross validation and modified maximum likelihood for estimating the parameters of a stochastic process wahba compared the performance of generalized cross validation gcv and modified maximum likelihood mml procedures for choosing the smoothing parameter of a smoothing spline this work makes a more careful study of the two procedures when the stochastic model motivating the modified maximum likelihood estimate is correct it is shown that in the case of the linear smoothing spline with equally spaced observations both estimates are asymptotically normal with the gcv estimate having twice the asymptotic variance of the mml estimate the impact of using these estimates on the subsequent predictions is also calculated

the stability of special symmetric solutions of x t αf x t 1 with small amplitudes we consider the differential delay equation f x t f x t 1 with odd f c 3 r f 0 0 f 0 0 f 0 0 f 0 0 a theorem of kaplan and yorke 4 implies that 2f 0 is a bifurcation point where a certain class of periodic solutions bifurcates from the trivial zero solutions

maximum likelihood estimation of parameters under a spatial sampling scheme we study in detail asymptotic properties of maximum likelihood estimators of parameters when observations are taken from a two dimensional gaussian random field with a multiplicative ornstein uhlenbeck covariance function under the complete lattice sampling plan it is shown that the maximum likelihood estimators are strongly consistent and asymptotically normal the asymptotic normality here is normalized by the fourth root of the sample size and is obtained through higher order expansions of the likelihood score equations extensions of these results to higher dimensional processes are also obtained showing that the convergence rate becomes better as the dimension gets higher

optimal stopping under partial observation near value iteration we propose a new approximate value iteration method namely near value iteration nvi to solve continuous state optimal stopping problems under partial observation which in general cannot be solved analytically and also pose a great challenge to numerical solutions nvi is motivated by the expression of the value function as the supremum over an uncountable set of linear functions in the belief state after a smart manipulation of the operations in the updating equation for the value function we reduce the set to only two functions at every time step so as to achieve significant computational savings nvi yields a value function approximation bounded by the tightest lower and upper bounds that can be achieved by existing algorithms in the same class so the nvi approximation is closer to the true value function than at least one of these bounds we demonstrate the effectiveness of our approach on an example of pricing american options under stochastic volatility

bayesian nonparametric covariance regression capturing predictor dependent correlations amongst the elements of a multivariate response vector is fundamental to numerous applied domains including neuroscience epidemiology and finance although there is a rich literature on methods for allowing the variance in a univariate regression model to vary with predictors relatively little has been done in the multivariate case as a motivating example we consider the google flu trends data set which provides indirect measurements of influenza incidence at a large set of locations over time our predictor to accurately characterize temporally evolving influenza incidence across regions it is important to develop statistical methods for a time varying covariance matrix importantly the locations provide a redundant set of measurements and do not yield a sparse nor static spatial dependence structure we propose to reduce dimensionality and induce a flexible bayesian nonparametric covariance regression model by relating these location specific trajectories to a lower dimensional subspace through a latent factor model with predictor dependent factor loadings these loadings are in terms of a collection of basis functions that vary nonparametrically over the predictor space such low rank approximations are in contrast to sparse precision assumptions and are appropriate in a wide range of applications our formulation aims to address three challenges scaling to large p domains coping with missing values and allowing an irregular grid of observations the model is shown to be highly exible while leading to a computationally feasible implementation via gibbs sampling the ability to scale to large p domains and cope with missing values is fundamental in analyzing the google flu trends data

fixed domain asymptotic properties of tapered maximum likelihood estimators when the spatial sample size is extremely large which occurs in many environmental and ecological studies operations on the large covariance matrix are a numerical challenge covariance tapering is a technique to alleviate the numerical challenges under the assumption that data are collected along a line in a bounded region we investigate how the tapering affects the asymptotic efficiency of the maximum likelihood estimator mle for the microergodic parameter in the matern covariance function by establishing the fixed domain asymptotic distribution of the exact mle and that of the tapered mle our results imply that under some conditions on the taper the tapered mle is asymptotically as efficient as the true mle for the microergodic parameter in the matern model

global bifurcation of periodic solutions to some autonomous differential delay equations we are concerned with slowly oscillating periodic solutions of the parametrized differential delay equation x xdot t email protected x t 1 l 0 where is an odd continuous and nonlinear function satisfying email protected x 0 for all x 0 due to the oddness of we can identify odd harmonic periodic solutions having the period t with other solutions of the periods 2t t 2 from this observation and a result of nussbaum we obtain the existence of periodic solutions with a period close to 2 moreover we can reach conclusions about a secondary bifurcation of periodic solutions we present a combination of a galerkin scheme with a continuation method which yields a feasable numerical procedure for the effective computation and continuation of periodic solutions we report the results of an extensive case study for the nonlinearities x x 1 x p p 1 for p 8 we show many different continua of periodic solutions which are either disjoint or bifurcating from continua of periodic solutions that contain a higher symmetry for p 1 and l p 2 it is known that periodic solutions are unique we discuss two homotopies linking the cases p 8 and p 1 which exhibit very different transitions from multiplicity to uniqueness of solutions

on the consistent separation of scale and variance for gaussian random fields we present fixed domain asymptotic results that establish consistent estimates of the variance and scale parameters for a gaussian random field with a geometric anisotropic matern autocovariance in dimension d 4 when d 4 we show that one can estimate the coefficient on the principle irregular term accurately enough to get a consistent estimate of the coefficient on the second irregular term these two coefficients can then be used to separate the scale and variance we extend our results to the general problem of estimating a variance and geometric anisotropy for more general autocovariance functions our results illustrate the interaction between the accuracy of estimation the smoothness of the random field the dimension of the observation space and the number of increments used for estimation as a corollary our results establish the orthogonality of matern gaussian random fields with different parameters when d 4 the case d 4 is still open

towards reconciling two asymptotic frameworks in spatial statistics summary two asymptotic frameworks increasing domain asymptotics and infill asymptotics have been advanced for obtaining limiting distributions of maximum likelihood estimators of covariance parameters in gaussian spatial models with or without a nugget effect these limiting distributions are known to be different in some cases it is therefore of interest to know for a given finite sample which framework is more appropriate we consider the possibility of making this choice on the basis of how well the limiting distri butions obtained under each framework approximate their finite sample counterparts we investigate the quality of these approximations both theoretically and empirically showing that for certain consistently estimable parameters of exponential covariograms approximations corresponding to the two frameworks perform about equally well for those parameters that cannot be estimated consistently however the infill asymptotic approximation is preferable

optimal stopping of partially observable markov processes a filtering based duality approach in this note we develop a numerical approach to the problem of optimal stopping of discrete time continuous state partially observable markov processes pomps our motivation is to find approximate solutions that provide lower and upper bounds on the value function such that the gap between the bounds can provide a practical measure of the quality of the solutions to this end we develop a filtering based duality approach which relies on the martingale duality formulation of the optimal stopping problem and the particle filtering technique we show that this approach complements an asymptotic lower bound derived from a suboptimal stopping time with an asymptotic upper bound on the value function we carry out error analysis and illustrate the effectiveness of our method on an example of pricing american options under partial observation of stochastic volatility

smooth bifurcation of symmetric periodic solutions of functional differential equations with odd f c r in contrast to the one dimensional ordinary differential equation this equation can have periodic solutions as the example i t 7r 2 x t 1 with solutions x t z sin rc 2 t z 0 shows numerical bifurcation diagrams e g 6 143 indicate a complicated structure of the period solutions depending on the delay parameter u one finds various intersecting smooth curves one task of mathematics is to prove the existence of periodic solutions in this paper we are mainly interested in symmetric periodic solutions i e solutions x of c which fultil the symmetry property

bounds on the efficiency of linear predictions using an incorrect covariance function suppose z is a random process defined on a bounded set r r d with finite second moments consider the behavior of linear predictions based on z t 1 z t n where t 1 t 2 is a dense sequence of points in r a general method is described for obtaining rates of convergence when the covariance function is misspecified but compatible with the correct covariance function when z is gaussian these bounds are related to the entropy distance between the measures for the random field under the actual and presumed covariance functions explicit bounds are given when r 0 1 and z is stationary with spectral density of the form f a 2 2 p where is a known positive integer and a is the parameter that is misspecified

fixed domain asymptotics for a subclass of matern type gaussian random fields stein statist sci 4 1989 432 433 proposed the matern type gaussian random fields as a very flexible class of models for computer experiments this article considers a subclass of these models that are exactly once mean square differentiable in particular the likelihood function is determined in closed form and under mild conditions the sieve maximum likelihood estimators for the parameters of the covariance function are shown to be weakly consistent with respect to fixed domain asymptotics

asymptotically efficient prediction of a random field with a misspecified covariance function on peut obtenir les meilleurs predicteurs lineaires sans biais d un champ aleatoire si la fonction de covariance du champ aleatoire est specifiee correctement

a frequency domain empirical likelihood method for irregularly spaced spatial data this paper develops empirical likelihood methodology for irregularly spaced spatial data in the frequency domain unlike the frequency domain empirical likelihood fdel methodology for time series on a regular grid the formulation of the spatial fdel needs special care due to lack of the usual orthogonality properties of the discrete fourier transform for irregularly spaced data and due to presence of nontrivial bias in the periodogram under different spatial asymptotic structures a spatial fdel is formulated in the paper taking into account the effects of these factors the main results of the paper show that wilks phenomenon holds for a scaled version of the logarithm of the proposed empirical likelihood ratio statistic in the sense that it is asymptotically distribution free and has a chi squared limit as a result the proposed spatial fdel method can be used to build nonparametric asymptotically correct confidence regions and tests for covariance parameters that are defined through spectral estimating equations for irregularly spaced spatial data in comparison to the more common studentization approach a major advantage of our method is that it does not require explicit estimation of the standard error of an estimator which is itself a very difficult problem as the asymptotic variances of many common estimators depend on intricate interactions among several population quantities including the spectral density of the spatial process the spatial sampling density and the spatial asymptotic structure results from a numerical study are also reported to illustrate the methodology and its finite sample properties

circulant matrices and differential delay equations on etudie les solutions periodiques non triviales de familles parametrisees d equations differentielles a retard

a resampling based stochastic approximation method for analysis of large geostatistical data the gaussian geostatistical model has been widely used in modeling of spatial data however it is challenging to computationally implement this method because it requires the inversion of a large covariance matrix particularly when there is a large number of observations this article proposes a resampling based stochastic approximation method to address this challenge at each iteration of the proposed method a small subsample is drawn from the full dataset and then the current estimate of the parameters is updated accordingly under the framework of stochastic approximation since the proposed method makes use of only a small proportion of the data at each iteration it avoids inverting large covariance matrices and thus is scalable to large datasets the proposed method also leads to a general parameter estimation approach maximum mean log likelihood estimation which includes the popular maximum log likelihood estimation mle approach as a special case and is expected to play an important role

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

propagation of population pulses and fronts in a cell replication problem non locality and dependence on the initial function abstract here we study the solution types in a simplified equation arising from a cell population dynamics model formulated as a non linear first order partial differential equation for the cell density u t x in the presence of non local maturational x interactions this can be considered as a reaction convection equation in which the cell density is convected with maturation velocity r the non local action affects only the reaction birth rate an initial function representing a pulse of small amplitude on an almost extinguished population may be dissipated may produce multiple pulses or may remain intact as it travels driven by reaction and convection depending on the magnitude of the non local action alternately for localized initial perturbations the equation also has positive traveling front solutions positive fronts correspond to the invasion of the zero amplitude solution by a finite amplitude solution for weak convection r 1 which is the only case treated here reaction driven fronts arise if the localized initial perturbation acts at a non zero maturation on the zero amplitude state a classification arises according to whether the magnitude of the maturation delay is larger or smaller than a critical value and scaling arguments are used to determine the critical delay for delays larger than critical the reaction driven fronts are oscillatory and correspond to the case for which the correlation length for non local action is of the same order as the characteristic length of maturation population gradients that yield oscillatory birth rates simulation results are validated and interpreted using scaling arguments and solutions for the local equations