Multivariate versions of Bartlett’s formula

This paper quantifies the form of the asymptotic covariance matrix of the sample autocovariances in a multivariate stationary time series—the classic Bartlett formula. Such quantification is useful in many statistical inferences involving autocovariances. While joint asymptotic normality of the sample autocovariances is well-known in univariate settings, explicit forms of the asymptotic covariances have not been investigated in the general multivariate non-Gaussian case. We fill this gap by providing such an analysis, bookkeeping all skewness terms. Additionally, following a recent univariate paper by Francq and Zakoian, we consider linear processes driven by non-independent errors, a feature that permits consideration of multivariate GARCH processes.     

An asymptotic theory for sample covariances of Bernoulli shifts

Covariances play a fundamental role in the theory of stationary processes and they can naturally be estimated by sample covariances. There is a well-developed asymptotic theory for sample covariances of linear processes. For nonlinear processes, however, many important problems on their asymptotic behaviors are still unanswered. The paper presents a systematic asymptotic theory for sample covariances of nonlinear time series. Our results are applied to the test of correlations.     

Gaussian fields and Gaussian sheets with generalized Cauchy covariance structure

Two types of Gaussian processes, namely the Gaussian field with generalized Cauchy covariance (GFGCC) and the Gaussian sheet with generalized Cauchy covariance (GSGCC) are considered. Some of the basic properties and the asymptotic properties of the spectral densities of these random fields are studied. The associated self-similar random fields obtained by applying the Lamperti transformation to GFGCC and GSGCC are studied.     

The rank of the covariance matrix of an evanescent field

Evanescent random fields arise as a component of the 2D Wold decomposition of homogeneous random fields. Besides their theoretical importance, evanescent random fields have a number of practical applications, such as in modeling the observed signal in the space-time adaptive processing (STAP) of airborne radar data. In this paper we derive an expression for the rank of the low-rank covariance matrix of a finite dimension sample from an evanescent random field. It is shown that the rank of this covariance matrix is completely determined by the evanescent field spectral support parameters, alone. Thus, the problem of estimating the rank lends itself to a solution that avoids the need to estimate the rank from the sample covariance matrix. We show that this result can be immediately applied to considerably simplify the estimation of the rank of the interference covariance matrix in the STAP problem.     

Estimation of high-dimensional prior and posterior covariance matrices in Kalman filter variants

This work studies the effects of sampling variability in Monte Carlo-based methods to estimate very high-dimensional systems. Recent focus in the geosciences has been on representing the atmospheric state using a probability density function, and, for extremely high-dimensional systems, various sample-based Kalman filter techniques have been developed to address the problem of real-time assimilation of system information and observations. As the employed sample sizes are typically several orders of magnitude smaller than the system dimension, such sampling techniques inevitably induce considerable variability into the state estimate, primarily through prior and posterior sample covariance matrices. In this article, we quantify this variability with mean squared error measures for two Monte Carlo-based Kalman filter variants: the ensemble Kalman filter and the ensemble square-root Kalman filter. Expressions of the error measures are derived under weak assumptions and show that sample sizes need to grow proportionally to the square of the system dimension for bounded error growth. To reduce necessary ensemble size requirements and to address rank-deficient sample covariances, covariance-shrinking (tapering) based on the Schur product of the prior sample covariance and a positive definite function is demonstrated to be a simple, computationally feasible, and very effective technique. Rules for obtaining optimal taper functions for both stationary as well as non-stationary covariances are given, and optimal taper lengths are given in terms of the ensemble size and practical range of the forecast covariance. Results are also presented for optimal covariance inflation. The theory is verified and illustrated with extensive simulations.     

Fourier transform methods for pathwise covariance estimation in the presence of jumps

We provide a new non-parametric Fourier procedure to estimate the trajectory of the instantaneous covariance process (from discrete observations of a multidimensional price process) in the presence of jumps extending the seminal work of Malliavin and Mancino (2002, 2009). Our approach relies on a modification of (classical) jump-robust estimators of integrated realized covariance to estimate the Fourier coefficients of the covariance trajectory. Using Fourier–Féjer inversion we reconstruct the path of the instantaneous covariance. We prove consistency and a central limit theorem (CLT) and in particular that the asymptotic estimator variance is smaller by a factor 2/3 in comparison to classical local estimators.     

A criterion for a continuous spectral density

For a given weakly stationary random field indexed by the integer lattice of an arbitrary finite dimension, a necessary and sufficient condition is given for the existence of a continuous spectral density. The condition involves the covariances of pairs of sums of the random variables, with the two index sets being “separated” from each other (but possibly “interlaced”) by a certain distance along a coordinate direction.     

Multi-operator scaling random fields

In this paper, we define and study a new class of random fields called harmonizable multi-operator scaling stable random fields. These fields satisfy a local asymptotic operator scaling property which generalizes both the local asymptotic self-similarity property and the operator scaling property. Actually, they locally look like operator scaling random fields, whose order is allowed to vary along the sample paths. We also give an upper bound of their modulus of continuity. Their pointwise Hölder exponents may also vary with the position x and their anisotropic behavior is driven by a matrix which may also depend on x.     

Asymptotic theory for maximum deviations of sample covariance matrix estimates

We consider asymptotic distributions of maximum deviations of sample covariance matrices, a fundamental problem in high-dimensional inference of covariances. Under mild dependence conditions on the entries of the data matrices, we establish the Gumbel convergence of the maximum deviations. Our result substantially generalizes earlier ones where the entries are assumed to be independent and identically distributed, and it provides a theoretical foundation for high-dimensional simultaneous inference of covariances.     

Small ball probabilities for smooth Gaussian fields and tensor products of compact operators

We find the logarithmic L₂‐small ball asymptotics for a class of zero mean Gaussian fields with covariances having the structure of “tensor product”. The main condition imposed on marginal covariances is slow growth at the origin of counting functions of their eigenvalues. That is valid for Gaussian functions with smooth covariances. Another type of marginal functions considered as well are classical Wiener process, Brownian bridge, Ornstein–Uhlenbeck process, etc., in the case of special self‐similar measure of integration. Our results are based on a new theorem on spectral asymptotics for the tensor products of compact self‐adjoint operators in Hilbert space which is of independent interest. Thus, we continue to develop the approach proposed in the paper 6 , where the regular behavior at infinity of marginal eigenvalues was assumed.     

Innovations algorithm asymptotics for periodically stationary time series with heavy tails

The innovations algorithm can be used to obtain parameter estimates for periodically stationary time series models. In this paper we compute the asymptotic distribution for these estimates in the case where the underlying noise sequence has infinite fourth moment but finite second moment. In this case, the sample covariances on which the innovations algorithm are based are known to be asymptotically stable. The asymptotic results developed here are useful to determine which model parameters are significant. In the process, we also compute the asymptotic distributions of least squares estimates of parameters in an autoregressive model.     

Limit distribution of the sum and maximum from multivariate Gaussian sequences

In this paper we study the asymptotic joint behavior of the maximum and the partial sum of a multivariate Gaussian sequence. The multivariate maximum is defined to be the coordinatewise maximum. Results extend univariate results of McCormick and Qi. We show that, under regularity conditions, if the maximum has a limiting distribution it is asymptotically independent of the partial sum. We also prove that the maximum of a stationary sequence, when normalized in a special sense which includes subtracting the sample mean, is asymptotically independent of the partial sum (again, under regularity conditions). The limiting distributions are also obtained.     

Fractional Generalized Random Fields on Bounded Domains

Using the theory of generalized random fields on fractional Sobolev spaces on bounded domains, and the concept of dual generalized random field, this paper introduces a class of random fields with fractional-order pure point spectra. The covariance factorization of an α-generalized random field having a dual is established, leading to a white-noise linear-filter representation, which reduces to the usual Markov representation in the ordinary case when α∈N and the covariance operator of the dual random field is local. Fractional-order differential models commonly arising from anomalous diffusion in disordered media can be studied within this framework.     

On the covariance of the asymptotic empirical copula process

Conditions are given under which the empirical copula process associated with a random sample from a bivariate continuous distribution has a smaller asymptotic covariance function than the standard empirical process based on observations from the copula. Illustrations are provided and consequences for inference are outlined.     

Quantile estimations via modified Cholesky decomposition for longitudinal single-index models

Quantile regression is a powerful complement to the usual mean regression and becomes increasingly popular due to its desirable properties. In longitudinal studies, it is necessary to consider the intra-subject correlation among repeated measures over time to improve the estimation efficiency. In this paper, we focus on longitudinal single-index models. Firstly, we apply the modified Cholesky decomposition to parameterize the intra-subject covariance matrix and develop a regression approach to estimate the parameters of the covariance matrix. Secondly, we propose efficient quantile estimating equations for the index coefficients and the link function based on the estimated covariance matrix. Since the proposed estimating equations include a discrete indicator function, we propose smoothed estimating equations for fast and accurate computation of the index coefficients, as well as their asymptotic covariances. Thirdly, we establish the asymptotic properties of the proposed estimators. Finally, simulation studies and a real data analysis have illustrated the efficiency of the proposed approach.

     

Asymptotic expansions in mean and covariance structure analysis

Asymptotic expansions of the distributions of parameter estimators in mean and covariance structures are derived. The parameters may be common to, or specific in means and covariances of observable variables. The means are possibly structured by the common/specific parameters. First, the distributions of the parameter estimators standardized by the population asymptotic standard errors are expanded using the single- and the two-term Edgeworth expansions. In practice, the pivotal statistic or the Studentized estimator with the asymptotically distribution-free standard error is of interest. An asymptotic distribution of the pivotal statistic is also derived by the Cornish-Fisher expansion. Simulations are performed for a factor analysis model with nonzero factor means to see the accuracy of the asymptotic expansions in finite samples.     

Relative efficiencies of estimates using patterned covariances or correlations in the multivariate normal estimation problem

Summary The relative efficiency of maximum likelihood estimates is studied when taking advantage of underlying linear patterns in the covariances or correlations when estimating covariance matrices. We compare the variances of estimates of the covariance matrix obtained under two nested patterns with the assumption that the more restricted pattern is the true state. Formulas for the asymptotic variances are given which are exact for linear covariance patterns when explicit maximum likelihood estimates exist. Several specific examples are given using complete symmetry, circular symmetry and general covariance patterns as well as an example involving a covariance matrix with a linear pattern in the correlations.     

On fractional tempered stable motion

Fractional tempered stable motion (fTSm) is defined and studied. FTSm has the same covariance structure as fractional Brownian motion, while having tails heavier than Gaussian ones but lighter than (non-Gaussian) stable ones. Moreover, in short time it is close to fractional stable Lévy motion, while it is approximately fractional Brownian motion in long time. A series representation of fTSm is derived and used for simulation and to study some of its sample paths properties.     

A class of spectral density estimators

A class of spectral windows depending on one parameter is presented and shown to include many of the common windows. The mean square rate of convergence of the associated spectral density estimators are calculated in terms of this parameter for spectral densities which are locally Lipschitz continuous The class is shown to include certain data tapers and data windows corresponding to missing observations. This is true also for the kernels of (C−α) summability which provide means for estimating the spectral density when the covariance function is periodic.     

Asymptotics for high-dimensional covariance matrices and quadratic forms with applications to the trace functional and shrinkage

We establish large sample approximations for an arbitrary number of bilinear forms of the sample variance–covariance matrix of a high-dimensional vector time series using ${?}_1$-bounded and small ${?}_2$-bounded weighting vectors. Estimation of the asymptotic covariance structure is also discussed. The results hold true without any constraint on the dimension, the number of forms and the sample size or their ratios. Concrete and potential applications are widespread and cover high-dimensional data science problems such as tests for large numbers of covariances, sparse portfolio optimization and projections onto sparse principal components or more general spanning sets as frequently considered, e.g. in classification and dictionary learning. As two specific applications of our results, we study in greater detail the asymptotics of the trace functional and shrinkage estimation of covariance matrices. In shrinkage estimation, it turns out that the asymptotics differ for weighting vectors bounded away from orthogonality and nearly orthogonal ones in the sense that their inner product converges to 0.