General Asymptotic Theory Research Articles

A general leading-order asymptotic theory of thin microstructural plates and uncertainty quantification of the elastic performance of composite laminates

We propose an asymptotic framework that may be used as a platform to model general thin microstructural plates, and our initiative is twofold. First, it provides an explanation to why so many tailored plate theories have been introduced for modelling plates bearing microstructures. In hypotheses-based plate theories, the microstructural details within each ply are usually smeared out. But it is shown that with such in-ply details properly taken into account, one manages to address, even only with leading-order terms, several key issues, such as transverse shear strain effects, shear locking, etc., whose rationalisation is believed to necessitate higher-order terms. To a certain degree, the proposed formulation can be used as a target for tailoring kinematic shape functions in both Reissner-Mindlin types of plate theories and zig-zag theories. Second, being slender in nature, a microstructural plate should accommodate a stress state that is close to the state of plane stress. This agrees with our findings that the (leading-order) mechanical units of a microstructural plate should be two-dimensional slices parameterised with the thickness coordinate, distinguishing the proposed framework from other plate theories based on asymptotic analysis. To demonstrate what the present framework can predict beyond existing theories, the effects on the behaviour of fibrous laminates due to manufacturing uncertainties are investigated.

Randomized empirical processes by algebraic groups, and tests for weak null hypotheses

Randomization tests are based on a re-randomization of existing data to gain data-dependent critical values that lead to exact hypothesis tests under special circumstances. However, it is not always possible to re-randomize data in accordance with the physical randomization from which the data have been obtained. As a consequence, statistical tests typically cannot control the type I error probability if the available sample is not overly large. Still, similarly as the bootstrap, data re-randomization can be used to improve the type I error control. However, in contrast to bootstrap and permutation test theory, a general asymptotic theory under weak null hypotheses has not yet been developed for randomization tests. It is thus the aim of this paper to provide a conveniently applicable theory on the asymptotic validity of randomization tests with test statistics that are asymptotically normal under weak null hypotheses. This can also be used to justify the asymptotic reliability of randomization-based confidence intervals. To achieve this, the present article creates a link between two well-established fields in mathematical statistics: empirical processes and inference based on randomization via algebraic groups. To this end, a broadly applicable conditional weak convergence theorem is developed for empirical processes that are based on randomized observations. Random elements of an algebraic group are applied to the data vectors from which the randomized version of a statistic is derived. Combining a variant of the functional delta-method with a suitable studentization of the statistic, asymptotically exact hypothesis tests can be deduced, while the finite sample exactness property under group-invariant sub-hypotheses is preserved. The methodology is exemplified with three examples: the Pearson correlation coefficient, a Mann-Whitney effect based on right-censored paired data, and a competing risks analysis. The practical usefulness of the approaches is assessed through simulation studies and an application to data from patients suffering from diabetic retinopathy.

Bayes factor asymptotics for variable selection in the Gaussian process framework

We investigate Bayesian variable selection in models driven by Gaussian processes, which allows us to treat linear, nonlinear and nonparametric models, in conjunction with even dependent setups, in the same vein. We consider the Bayes factor route to variable selection, and develop a general asymptotic theory for the Gaussian process framework in the “large p, large n” settings even with \(p\gg n\), establishing almost sure exponential convergence of the Bayes factor under appropriately mild conditions. The fixed p setup is included as a special case. To illustrate, we apply our result to variable selection in linear regression, Gaussian process model with squared exponential covariance function accommodating the covariates, and a first-order autoregressive process with time-varying covariates. We also follow up our theoretical investigations with ample simulation experiments in the above regression contexts and variable selection in a real, riboflavin data consisting of 71 observations and 4088 covariates. For implementation of variable selection using Bayes factors, we develop a novel and effective general-purpose transdimensional, transformation-based Markov chain Monte Carlo algorithm, which has played a crucial role in simulated and real data applications.

Hierarchy of geometrical frustration in elastic ribbons: Shape-transitions and energy scaling obtained from a general asymptotic theory

Geometrically frustrated elastic ribbons exhibit, in many cases, significant changes in configuration depending on the relation between their width and thickness. We show that the existence of such a transition, and the scaling at which it occurs, strongly depend on the system considered. Using an asymptotic approach, treating the width as a small parameter, we find the leading energy terms resulting from the frustration and predict the existence and scaling of the shape transition. We study in detail 5 different types of frustrated ribbons with a different morphological dependence on ribbon’s width: a sharp shape-transition at a critical width, a moderate transition with an intermediate regime, and no transition at all. We show that the predictions of our approach match experimental results from two different experimental systems: prestressed rubber bilayers and 4D printed thermoplastics, in a wide variety of geometric settings.

Modified spectral function based DOA estimation in colored noise

With the wide application of large-scale antenna arrays in the field of radar, sonar and wireless communication, direction of arrival (DOA) estimation based on the general asymptotic theory (GAT) has received more and more attentions recently, which can effectively conquer the inconsistent estimation problem of sampling covariance matrix (SCM) in the scenarios that the number of sensor M is large and of the same order of magnitude as the number of snapshots N. From the GAT perspective, a new DOA estimation method (termed as SEMDT-M-MUSIC) in colored noise is investigated in this paper, which mainly comprises three steps, the first step is to obtain an improved SCM through covariance matrix shrinkage estimation, the second step is to eliminate the influence of the colored noise via the matrix difference technology, and the last step is to enhance the DOA estimation via the modified spectral function, where the phase transformation result of the spike covariance matrix is employed. Numerical simulations validate the effectiveness of the proposed method.

Bootstrapping Hill estimator and tail array sums for regularly varying time series

In the extreme value analysis of stationary regularly varying time series, tail array sums form a broad class of statistics suitable to analyze their extremal behavior. This class includes for example, the Hill estimator or estimators of the extremogram and the tail dependence coefficient. General asymptotic theory for tail array sums has been developed by Rootzén et al. (Ann. Appl. Probab. 8 (1998) 868–885) under mixing conditions and in Kulik et al. (Stochastic Process. Appl. 129 (2019) 4209–4238) for functions of geometrically ergodic Markov chains. A more general framework of cluster functionals is presented in Drees and Rootzén (Ann. Statist. 38 (2010) 2145–2186). However, the resulting limiting distributions turn out to be very complex and cumbersome to estimate as they usually depend on the whole extremal dependence structure of the time series. Hence, a suitable bootstrap procedure is desired, but available bootstrap consistency results for tail array sums are scarce. In this paper, following Drees (Drees (2015)), we consider a multiplier block bootstrap to estimate the limiting distribution of tail array sums. We prove that, conditionally on the data, an appropriately constructed multiplier block bootstrap statistic converges to the correct limiting distribution. Interestingly, in contrast, it turns out that an apparently natural, but naïve application of the multiplier block bootstrap scheme does not yield the correct limit. In simulations, we provide numerical evidence of our theoretical findings and illustrate the superiority of the proposed multiplier block bootstrap over some obvious competitors. The proposed bootstrap scheme proves to be computationally efficient in comparison to other approaches.

Preliminary test estimation in uniformly locally asymptotically normal models

AbstractPreliminary test estimation is a methodology that combines goodness‐of‐fit testing and estimation. It is a classical procedure when it is suspected that the parameter to be estimated satisfies some prespecified constraints. In the present paper, we establish general results on the asymptotic behavior of preliminary test estimators. More precisely, we show that, in uniformly locally asymptotically normal (ULAN) models, a general asymptotic theory can be derived for preliminary test estimators based on estimators admitting generic Bahadur‐type representations. This allows for a detailed comparison between classical estimators and preliminary test estimators in ULAN models. Our results, that, in standard linear regression models, are shown to reduce to some classical results, are also illustrated in more modern and involved setups, such as the multisample one where m covariance matrices are to be estimated when it is suspected that these matrices might be equal, might be proportional, or might share a common “scale”. Simulation results confirm our theoretical findings and an illustration on a real data example is provided.

Additive regression for predictors of various natures and possibly incomplete Hilbertian responses

In this paper we consider a fully nonparametric additive regression model for responses and predictors of various natures. This includes the case of Hilbertian and incomplete (like censored or missing) responses, and continuous, nominal discrete and ordinal discrete predictors. We propose a backfitting technique that estimates this additive model, and establish the existence of the estimator and the convergence of the associated backfitting algorithm under minimal conditions. We also develop a general asymptotic theory for the estimator such as the rates of convergence and asymptotic distribution. We verify the practical performance of the proposed estimator in a simulation study. We also apply the method to various real data sets, including those for a density-valued response regressed on a mixture of continuous and nominal discrete predictors, for a compositional response regressed on a mixture of continuous and ordinal discrete predictors, and for a censored scalar response regressed on a mixture of continuous and nominal discrete predictors.

Asymptotic Theory of Eigenvectors for Random Matrices With Diverging Spikes

Characterizing the asymptotic distributions of eigenvectors for large random matrices poses important challenges yet can provide useful insights into a range of statistical applications. To this end, in this article we introduce a general framework of asymptotic theory of eigenvectors for large spiked random matrices with diverging spikes and heterogeneous variances, and establish the asymptotic properties of the spiked eigenvectors and eigenvalues for the scenario of the generalized Wigner matrix noise. Under some mild regularity conditions, we provide the asymptotic expansions for the spiked eigenvalues and show that they are asymptotically normal after some normalization. For the spiked eigenvectors, we establish asymptotic expansions for the general linear combination and further show that it is asymptotically normal after some normalization, where the weight vector can be arbitrary. We also provide a more general asymptotic theory for the spiked eigenvectors using the bilinear form. Simulation studies verify the validity of our new theoretical results. Our family of models encompasses many popularly used ones such as the stochastic block models with or without overlapping communities for network analysis and the topic models for text analysis, and our general theory can be exploited for statistical inference in these large-scale applications. Supplementary materials for this article are available online.

Asymptotic inference for AR(1) panel data

A general asymptotic theory is given for the panel data AR(1) model with time series independent in different cross sections. The theory covers the cases of stationary process, local to unity process, unit root process, mildly integrated, mildly explosive and explosive processes. It is assumed that the cross-sectional dimension and time-series dimension are respectively N and T. The results in this paper illustrate that whichever the process is, with an appropriate regularization, the least squares estimator of the autoregressive coefficient converges in distribution to a normal distribution with rate at least O(N−1/3). Since the variance is the key to characterize the normal distribution, it is important to discuss the variance of the least squares estimator. We will show that when the autoregressive coefficient ρ satisfies |ρ| 1. ρ =1 is the critical point where the convergence rate changes radically. The transition process is studied by assuming ρ depending on T and going to 1. An interesting phenomenon discovered in this paper is that, in the explosive case, the least squares estimator of the autoregressive coefficient has a standard normal limiting distribution in the panel data case while it may not has a limiting distribution in the univariate time series case.

Identification strength with a large number of moments

This paper studies how identification is affected in GMM estimation as the number of moment conditions increases. We develop a general asymptotic theory extending the set up of Chao and Swanson and Antoine and Renault to the case where moment conditions have heterogeneous identification strengths and the number of them may diverge to infinity with the sample size. We also allow the models to be locally misspecified and examine how the asymptotic theory is affected by the degree of misspecification. The theory encompasses many cases including GMM models with many moments (Han and Phillips), partially linear models, and local GMM via kernel smoothing with a large number of conditional moment restrictions. We provide an understanding of the benefits of a large number of moments that compensate the weakness of individual moments by explicitly showing how an increasing number of moments improves the rate of convergence in GMM.

Quasi-local method of wave decomposition in a slowly varying medium

The general asymptotic theory for wave propagation in a slowly varying medium, classically known as the Wentzel–Kramers–Brillouin–Jeffreys (WKBJ) approximation, is revisited here with the aim of constructing a new data diagnostic technique useful in atmospheric and oceanic sciences. Using the Wigner transform, a kind of mapping that associates a linear operator with a function, we analytically decompose a flow field into mutually independent wave signals. This method takes account of the variations in the polarisation relations, an eigenvector that represents the kinematic characteristics of each wave component, so as to project the variables onto their eigenspace quasi-locally. The temporal evolution of a specific mode signal obeys a single wave equation characterised by the dispersion relation that also incorporates the effect from the local gradient in the medium. Combining this method with transport theory and applying them to numerical simulation data, we can detect the transfer of energy or other conserved quantities associated with the propagation of each wave signal in a wide variety of situations.

Efficient Standard Errors in Item Response Theory Models for Short Tests.

In dichotomous item response theory (IRT) framework, the asymptotic standard error (ASE) is the most common statistic to evaluate the precision of various ability estimators. Easy-to-use ASE formulas are readily available; however, the accuracy of some of these formulas was recently questioned and new ASE formulas were derived from a general asymptotic theory framework. Furthermore, exact standard errors were suggested to better evaluate the precision of ability estimators, especially with short tests for which the asymptotic framework is invalid. Unfortunately, the accuracy of exact standard errors was assessed so far only in a very limiting setting. The purpose of this article is to perform a global comparison of exact versus (classical and new formulations of) asymptotic standard errors, for a wide range of usual IRT ability estimators, IRT models, and with short tests. Results indicate that exact standard errors globally outperform the ASE versions in terms of reduced bias and root mean square error, while the new ASE formulas are also globally less biased than their classical counterparts. Further discussion about the usefulness and practical computation of exact standard errors are outlined.

Asymptotic Optimality of Mixture Rules for Detecting Changes in General Stochastic Models

The paper addresses a sequential changepoint detection problem for a general stochastic model, assuming that the observed data may be non-i.i.d. (i.e., dependent and non-identically distributed) and prior distribution of the change point is arbitrary. Tartakovsky and Veeravalli (2005), Baron and Tartakovsky (2006), and, more recently, Tartakovsky (2017) developed a general asymptotic theory of changepoint detection for non-i.i.d. stochastic models, assuming certain stability of the log-likelihood ratio process, in the case of simple hypotheses when both pre-change and post-change models are completely specified. However, in most applications, the post-change distribution is not completely known. In the present paper, we generalize previous results to the case of parametric uncertainty, assuming that the parameter of the post-change distribution is unknown. We introduce two detection rules based on mixtures–the mixture Shiryaev rule and the mixture Shiryaev–Roberts rule–and study their asymptotic properties in the Bayesian context. In particular, we provide sufficient conditions under which these rules are first-order asymptotically optimal, minimizing moments of the delay to detection as the probability of false alarm approaches zero.

Divergence Measures Estimation and Its Asymptotic Normality Theory Using Wavelets Empirical Processes II

In Ba et al. (2017), a general normal asymptotic theory for divergence measures estimators has been provided. These estimators are constructed from the wavelets empirical process and concerned the general O -divergence measures. In this paper, we first extend the aforementioned results to symmetrized forms of divergence measures. Second, the Tsallis and Renyi divergence measures as well as the Kullback-Leibler measures are investigated in details. The question of the applicability of the results, based on the boundedness assumption is also dealt, leading to future packages. Keywords: Divergence measures estimation; Asymptotic normality; Wavelet theory; wavelets empirical processes; Besov spaces AMS 2010 Mathematics Subject Classification : 62G05; 62G20; 62G07

Divergence Measures Estimation and Its Asymptotic Normality Theory Using Wavelets Empirical Processes I

We deal with the normality asymptotic theory of empirical divergences measures based on wavelets in a series of three papers. In this first paper, we provide the asymptotic theory of the general of ϕ-divergences measures, which includes the most common divergence measures : Renyi and Tsallis families and the Kullback-Leibler measures. Instead of using the Parzen nonparametric estimators of the probability density functions whose discrepancy is estimated, we use the wavelets approach and the geometry of Besov spaces. One-sided and two-sided statistical tests are derived. This paper is devoted to the foundations the general asymptotic theory and the exposition of the mains theoretical tools concerning the ϕ-forms, while proofs and next detailed and applied results will be given in the two subsequent papers which deal important key divergence measures and symmetrized estimators.

Likelihood Inference on Semiparametric Models: Average Derivative and Treatment Effect

Over the past few decades, much progress has been made in semiparametric modelling and estimation methods for econometric analysis. This paper is concerned with inference (i.e. confidence intervals and hypothesis testing) in semiparametric models. In contrast to the conventional approach based on t-ratios, we advocate likelihood-based inference. In particular, we study two widely applied semiparametric problems, weighted average derivatives and treatment effects, and propose semiparametric empirical likelihood and jackknife empirical likelihood methods. We derive the limiting behaviour of these empirical likelihood statistics and investigate their finite sample performance through Monte Carlo simulation. Furthermore, we extend the (delete-1) jackknife empirical likelihood toward the delete-d version with growing d and establish general asymptotic theory. This extension is crucial to deal with non-smooth objects, such as quantiles and quantile average derivatives or treatment effects, due to the well-known inconsistency phenomena of the jackknife under non-smoothness.

Asymptotic theory of generalized estimating equations based on jack-knife pseudo-observations

A general asymptotic theory of estimates from estimating functions based on jack-knife pseudo-observations is established by requiring that the underlying estimator can be expressed as a smooth functional of the empirical distribution. Using results in $p$-variation norms, the theory is applied to important estimators from time-to-event analysis, namely the Kaplan–Meier estimator and the Aalen–Johansen estimator in a competing risks model, and the corresponding estimators of restricted mean survival and cause-specific lifetime lost. Under an assumption of completely independent censorings, this allows for estimating parameters in regression models of survival, cumulative incidences, restricted mean survival, and cause-specific lifetime lost. Considering estimators as functionals and applying results in $p$-variation norms is apparently an excellent way of studying the asymptotics of such estimators.

Towards nonlinear imperfect interface models including micro-cracks and smooth roughness

The present paper deals with a general asymptotic theory aimed at deriving some imperfect interface models starting from thin interphases. The novelty of this work consists in taking into account some non-standard constitutive behaviors for the interphase material. In particular, micro-cracks, surface roughness and geometrical nonlinearity are included into the general framework of the matched-asymptotic-expansion theory. The elastic equilibrium problem of a three-composite body comprising two elastic adherents and an adhesive interphase is investigated. Higher order interface models are derived within the cases of soft and hard interphase materials. Simple FEM-based numerical applications are also presented.

On Asymptotic Optimality in Sequential Changepoint Detection: Non-iid Case

We consider a sequential Bayesian changepoint detection problem for a general stochastic model, assuming that the observed data may be dependent and non-identically distributed and the prior distribution of the change point is arbitrary, not necessarily geometric. Tartakovsky and Veeravalli (2005) developed a general asymptotic theory of changepoint detection in the case of non-identically distributed and dependent observations and discrete time, and Baron and Tartakovsky (2006) in continuous time assuming the certain stability of the log-likelihood ratio process. This stability property was formulated in terms of the $r$ -quick convergence of the normalized log-likelihood ratio process to a positive and finite number, which can be interpreted as the limiting Kullback–Leibler information between the “change” and “no change” hypotheses. In these papers, it was conjectured that the $r$ -quick convergence can be relaxed in the $r$ -complete convergence, which is typically much easier to verify in particular examples. In the present paper, we justify this conjecture by showing that the Shiryaev change detection procedure is nearly optimal, minimizing asymptotically to first order (as the probability of false alarm vanishes) moments of the delay to detection up to order $r$ whenever $r$ -complete convergence holds. We also study asymptotic properties of the Shiryaev–Roberts detection procedure in the Bayesian context.