Increasing Domain Asymptotics Research Articles

A Composite Likelihood-based Approach for Change-point Detection in Spatio-temporal Processes

This paper develops a unified and computationally efficient method for change-point estimation along the time dimension in a non-stationary spatio-temporal process. By modeling a non-stationary spatio-temporal process as a piecewise stationary spatio-temporal process, we consider simultaneous estimation of the number and locations of change-points, and model parameters in each segment. A composite likelihood-based criterion is developed for change-point and parameter estimation. Under the framework of increasing domain asymptotics, theoretical results including consistency and distribution of the estimators are derived under mild conditions. In contrast to classical results in fixed dimensional time series that the localization error of change-point estimator is , exact recovery of true change-points is possible in the spatio-temporal setting. More surprisingly, the consistency of change-point estimation can be achieved without any penalty term in the criterion function. In addition, we further establish consistency of the change-point estimator under the infill asymptotics framework where the time domain is increasing while the spatial sampling domain is fixed. A computationally efficient pruned dynamic programming algorithm is developed for the challenging criterion optimization problem. Extensive simulation studies and an application to the U.S. precipitation data are provided to demonstrate the effectiveness and practicality of the proposed method.

Strong law of large numbers for functionals of random fields with unboundedly increasing covariances

The paper proves the Strong Law of Large Numbers for integral functionals of random fields with unboundedly increasing covariances. The case of functional data and increasing domain asymptotics is studied. Conditions to guarantee that the Strong Law of Large Numbers holds true are provided. The considered scenarios include wide classes of non stationary random fields. The discussion about application to weak and long-range dependent random fields and numerical examples are given.

Increasing domain asymptotics for the first Minkowski functional of spherical random fields

The restriction to the sphere of a homogeneous and isotropic random field defines a spherical isotropic random field. This paper derives central and noncentral limit results for the first Minkowski functional subordinated to homogeneous and isotropic Gaussian and chi-squared random fields, restricted to the sphere in R3. Both scenarios are motivated by their interesting applications in the analysis of the Cosmic Microwave Background (CMB) radiation.

An example of inconsistent MLE of spatial covariance parameters under increasing domain asymptotics

Asymptotic properties of estimators of covariance parameters in spatial statistics are commonly considered under the frameworks of increasing domain and fixed domain asymptotics, respectively. Although inconsistency is a general conclusion under the framework of fixed domain asymptotics, it is generally believed that consistency should generally hold under the framework of increasing domain asymptotics. This article provides an example in which the maximum likelihood estimator (MLE) of covariance parameters is still inconsistent under the framework of increasing domain asymptotics. Therefore, consistency may still be a problem under the framework of increasing domain asymptotics.

Asymptotic properties of the minimum contrast estimators for projections of inhomogeneous space-time shot-noise Cox processes

We consider a flexible class of space-time point process models—inhomogeneous shot-noise Cox point processes. They are suitable for modelling clustering phenomena, e.g. in epidemiology, seismology, etc. The particular structure of the model enables the use of projections to the spatial and temporal domain. They are used to formulate a stepwise estimation method to estimate different parts of the model separately. In the first step, the Poisson likelihood approach is used to estimate the inhomogeneity parameters. In the second and third steps, the minimum contrast estimation based on K-functions of the projected processes is used to estimate the interaction parameters. We study the asymptotic properties of the resulting estimators and formulate a set of conditions sufficient for establishing consistency and asymptotic normality of the estimators under the increasing domain asymptotics.

Composite Likelihood Inference for Multivariate Gaussian Random Fields

In the recent years, there has been a growing interest in proposing covariance models for multivariate Gaussian random fields. Some of these covariance models are very flexible and can capture both the marginal and the cross-spatial dependence of the components of the associated multivariate Gaussian random field. However, effective estimation methods for these models are somehow unexplored. Maximum likelihood is certainly a useful tool, but it is impractical in all the circumstances where the number of observations is very large. In this work, we consider two possible approaches based on composite likelihood for multivariate covariance model estimation. We illustrate, through simulation experiments, that our methods offer a good balance between statistical efficiency and computational complexity. Asymptotic properties of the proposed estimators are assessed under increasing domain asymptotics. Finally, we apply the method for the analysis of a bivariate dataset on chlorophyll concentration and sea surface temperature in the Chilean coast.

Variable selection for binary spatial regression: Penalized quasi-likelihood approach.

We consider the problem of selecting covariates in a spatial regression model when the response is binary. Penalized likelihood-based approach is proved to be effective for both variable selection and estimation simultaneously. In the context of a spatially dependent binary variable, an uniquely interpretable likelihood is not available, rather a quasi-likelihood might be more suitable. We develop a penalized quasi-likelihood with spatial dependence for simultaneous variable selection and parameter estimation along with an efficient computational algorithm. The theoretical properties including asymptotic normality and consistency are studied under increasing domain asymptotics framework. An extensive simulation study is conducted to validate the methodology. Real data examples are provided for illustration and applicability. Although theoretical justification has not been made, we also investigate empirical performance of the proposed penalized quasi-likelihood approach for spatial count data to explore suitability of this method to a general exponential family of distributions.

Parameter Estimation for Inhomogeneous Space‐Time Shot‐Noise Cox Point Processes

AbstractWe consider the problem of parameter estimation for inhomogeneous space‐time shot‐noise Cox point processes. We explore the possibility of using a stepwise estimation method and dimensionality‐reducing techniques to estimate different parts of the model separately.We discuss the estimation method using projection processes and propose a refined method that avoids projection to the temporal domain. This remedies the main flaw of the method using projection processes – possible overlapping in the projection process of clusters, which are clearly separated in the original space‐time process. This issue is more prominent in the temporal projection process where the amount of information lost by projection is higher than in the spatial projection process.For the refined method, we derive consistency and asymptotic normality results under the increasing domain asymptotics and appropriate moment and mixing assumptions. We also present a simulation study that suggests that cluster overlapping is successfully overcome by the refined method.

Covariance tapering for multivariate Gaussian random fields estimation

In recent literature there has been a growing interest in the construction of covariance models for multivariate Gaussian random fields. However, effective estimation methods for these models are somehow unexplored. The maximum likelihood method has attractive features, but when we deal with large data sets this solution becomes impractical, so computationally efficient solutions have to be devised. In this paper we explore the use of the covariance tapering method for the estimation of multivariate covariance models. In particular, through a simulation study, we compare the use of simple separable tapers with more flexible multivariate tapers recently proposed in the literature and we discuss the asymptotic properties of the method under increasing domain asymptotics.

Decorrelation Property of Discrete Wavelet Transform Under Fixed-Domain Asymptotics

Theoretical aspects of the decorrelation property of the discrete wavelet transform when applied to stochastic processes have been studied exclusively from the increasing-domain perspective, in which the distance between neighboring observations stays roughly constant as the number of observations increases. To understand the underlying data-generating process and to obtain good interpolations, fixed-domain asymptotics, in which the number of observations increases in a fixed region, is often more appropriate than increasing-domain asymptotics. In the fixed-domain setting, we prove that, for a general class of inhomogeneous covariance functions, with suitable choice of wavelet filters, the wavelet transform of a nonstationary process has mostly asymptotically uncorrelated components.

Nonparametric test for spatial geometric anisotropy

Paper deals with a problem of testing isotropy against geometric anisotropy for Gaussian spatial data. The original simple test statistic based on directional empirical semivariograms is proposed. Under the assumption of independence of the classical semivariogram estimators and for increasing domain asymptotics, the distribution of test statistics is approximated by chi-squared distribution. The simulation experiments demonstrate the efficacy of the proposed test.

Bootstrapping the Empirical Distribution Function of a Spatial Process

In this article, we consider a stationary α-mixing random field in IRd. Under a large-sample scheme that is a mixture of the so-called “infill” and “increasing domain” asymptotics, we establish a functional central limit theorem for the empirical processes of this random field. Further, we apply a blockwise bootstrap to the samples. Under the condition that the side length of the block \({\lambda_l = {\cal O}(\lambda_n^{\beta})}\) for some 0 < β < 1, where λ n is the growth rate in the increasing domain asymptotics, we show that the bootstrapped empirical process converges weakly to the same limiting Gaussian process almost surely. Extension to multivariate random fields and application to differentiable statistical functionals are also given. A spatial version of the Bernstein’s inequality is developed, which may be of some independent interest.

Modelling the effects of partially observed covariates on Poisson process intensity

We propose an estimating function for parameters in a model for Poisson process intensity when time- or space-varying covariates are observed for both the events of the process and at sample times or locations selected from a probability-based sampling design. We investigate the large-sample properties of the proposed estimator under increasing domain asymptotics, demonstrating that it is consistent and asymptotically normally distributed. We illustrate our approach using data from an ecological momentary assessment of smoking.

Towards reconciling two asymptotic frameworks in spatial statistics

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 distributions 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.

INVESTIGATING ESTIMATOR CONSISTENCY OF NONLINEAR GROWTH CURVE PARAMETERS

Researchers often model biological growth data as a nonlinear function of time. This function involves unknown parameters that have intrinsic meaning to the researchers. Maximum likelihood provides a way to estimate these parameters. We present asymptotic results for the maximum likelihood estimators when adopting the generalized logistic function. Under increasing-domain asymptotics, the maximum likelihood estimators are consistent. Under infill asymptotics, evidence suggests they are not.

On the quality of likelihood-based estimators in spatial autoregressive models when the data dependence structure is misspecified

This paper seeks to help recast and extend existing results for the specification and the estimation of spatial statistical autoregressive models. In particular, a graph-theoretic formulation of the neighborhood structure allows an intuitive, but general, specification of the dependence structure among data. Moreover, issues of asymptotic efficiency and consistency of maximum likelihood (ML) estimators (MLE) are addressed in terms of correlational structure specification error. Parameters of interest are those for the mean response, the variance, and the spatial autocorrelation. Findings reported frequently differ from those derived for independent observations, and from those derived for time series correlation structures. Some findings are in keeping with those traditionally derived for generalized least squares (GLS) estimators. Moreover, theorems presented in this paper reveal that the quality of ML spatial autoregressive model parameter estimates for Gaussian geo-referenced data distributed across a lattice is such that, under specification errors, (1) the GLS mean response ML estimator is unbiased, consistent, and asymptotically efficient in selected situations, (2) the GLS error variance ML estimator is consistent in selected situations, but usually inefficient, and (3) spatial autocorrelation parameters MLEs fail to converge to their true values under increasing domain asymptotics. Finally, a model taking into account partial information about the data neighborhood structure is presented. © 1998 Elsevier Science B.V. All rights reserved.