Second-order Characteristics for Stochastic Structures Connected with Gibbs Point Processes

The points of stationary and isotropic GIBBS point process are marked by exp(corresponding local energy). For the mean mark and two mark product density functions very simple formulae are true which contain the intensity and the pair potential function of the process. Furthermore, there is a close connection between the pair correlation function of the process and the covariance function of the random field given by the conditional intensity.     

Convex Optimization and Feasible Circulant Matrix Embeddings in Synthesis of Stationary Gaussian Fields

Circulant matrix embedding is one of the most popular and efficient methods for the exact generation of Gaussian stationary univariate series. Although the idea of circulant matrix embedding has also been used for the generation of Gaussian stationary random fields, there are many practical covariance structures of random fields where classical embedding methods break down. In this work, we propose a novel methodology that adaptively constructs feasible circulant embeddings based on convex optimization with an objective function measuring the distance of the covariance embedding to the targeted covariance structure over the domain of interest. The optimal value of the objective function will be zero if and only if there exists a feasible embedding for the a priori chosen embedding size.     

The Poisson limit theorem for high extrema of a time series with seasonal component and monotone trend

Poisson’s limit theorem is considered in the paper for high extrema of a stationary time series with a monotone trend and an almost periodic component. It is assumed that the distribution function of the time series is maximum stable and the time series satisfies the weak dependence condition. The limit behavior of the random process of high extrema for this model is considered for the first time.     

Time Varying Isotropic Vector Random Fields on Spheres

For a vector random field that is isotropic and mean square continuous on a sphere and stationary on a temporal domain, this paper derives a general form of its covariance matrix function and provides a series representation for the random field, which involve the ultraspherical polynomials. The series representation is somehow an imitator of the covariance matrix function, but differs from the spectral representation in terms of the ordinary spherical harmonics, and is useful for modeling and simulation. Some semiparametric models are also illustrated.     

A Case Study of the Residual-Based Cointegration Procedure

The study of long-run equilibrium processes is a significant component of economic and finance theory. The Johansen technique for identifying the existence of such long-run stationary equilibrium conditions among financial time series allows the identification of all potential linearly independent cointegrating vectors within a given system of eligible financial time series. The practical application of the technique may be restricted, however, by the pre-condition that the underlying data generating process fits a finite-order vector autoregression (VAR) model with white noise. This paper studies an alternative method for determining cointegrating relationships without such a precondition. The method is simple to implement through commonly available statistical packages. This 'residual-based cointegration' (RBC) technique uses the relationship between cointegration and univariate Box-Jenkins ARIMA models to identify cointegrating vectors through the rank of the covariance matrix of the residual processes which result from the fitting of univariate ARIMA models. The RBC approach for identifying multivariate cointegrating vectors is explained and then demonstrated through simulated examples. The RBC and Johansen techniques are then both implemented using several real-life financial time series.     

ESTIMATION OF THE MIXED AR AND HIDDEN PERIODIC MODEL

ThisresearchissupportedbytheNationalNaturalScienceFoundationofChina.1.IntroductionGeneralizedhiddenperiodicmodelhasthefollowingformwhereacisthesetofallpositiveintegers,('~{((t);tEac}isastationarysequencewithzeromeanandcontinuousspectraldensity,i=n,qisanonnegativeinteger,'f=0,X=(Al,Az,',A,)isarealvectorwith--T相似文献   

Comparison theorems and asymptotic behavior of correlation estimators in spaces of continuous functions. II

This paper is the second part of [12]. Using the comparison theorems which were proved in the first part, the asymptotic normality of the estimator — in a model of a series of several samples — of the correlation function of a stationary Gaussian random process in spaces of continuous functions with weights is established. A method for constructing functional confidence intervals for an unknown correlation function in these spaces is described.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, No. 5, pp. 579–583, May, 1991.     

Modeling covariance matrices via partial autocorrelations

We study the role of partial autocorrelations in the reparameterization and parsimonious modeling of a covariance matrix. The work is motivated by and tries to mimic the phenomenal success of the partial autocorrelations function (PACF) in model formulation, removing the positive-definiteness constraint on the autocorrelation function of a stationary time series and in reparameterizing the stationarity-invertibility domain of ARMA models. It turns out that once an order is fixed among the variables of a general random vector, then the above properties continue to hold and follow from establishing a one-to-one correspondence between a correlation matrix and its associated matrix of partial autocorrelations. Connections between the latter and the parameters of the modified Cholesky decomposition of a covariance matrix are discussed. Graphical tools similar to partial correlograms for model formulation and various priors based on the partial autocorrelations are proposed. We develop frequentist/Bayesian procedures for modelling correlation matrices, illustrate them using a real dataset, and explore their properties via simulations.     

Generalized discrete autoregressive moving‐average models

This article proposes the generalized discrete autoregressive moving‐average (GDARMA) model as a parsimonious and universally applicable approach for stationary univariate or multivariate time series. The GDARMA model can be applied to any type of quantitative time series. It allows to compute moment properties in a unique way, and it exhibits the autocorrelation structure of the traditional ARMA model. This great flexibility is obtained by using data‐specific variation operators, which is illustrated for the most common types of time series data, such as counts, integers, reals, and compositional data. The practical potential of the GDARMA approach is demonstrated by considering a time series of integers regarding votes for a change of the interest rate, and a time series of compositional data regarding television market shares.     

Multifractal detrended fluctuation analysis of analog random multiplicative processes

We investigate non-Gaussian statistical properties of stationary stochastic signals generated by an analog circuit that simulates a random multiplicative process with weak additive noise. The random noises are originated by thermal shot noise and avalanche processes, while the multiplicative process is generated by a fully analog circuit. The resulting signal describes stochastic time series of current interest in several areas such as turbulence, finance, biology and environment, which exhibit power-law distributions. Specifically, we study the correlation properties of the signal by employing a detrended fluctuation analysis and explore its multifractal nature. The singularity spectrum is obtained and analyzed as a function of the control circuit parameter that tunes the asymptotic power-law form of the probability distribution function.     

How does innovation''''s tail risk determine marginal tail risk of a stationary financial time series?

We discuss the relationship between the marginal tail risk probability and theinnovation's tail risk probability for some stationary financial time series models. We firstgive the main results on the tail behavior of a class of infinite weighted sums of randomvariables with heavy-tailed probabilities. And then, the main results are applied to threeimportant types of time series models; infinite order moving averages, the simple bilineartime series and the solutions of stochastic difference equations. The explicit formulasare given to describe how the marginal tail probabilities come from the innovation's tailprobabilities for these time series. Our results can be applied to the tail estimation of timeseries and are useful for risk analysis in finance.     

Tail dependence of random variables from ARCH and heavy-tailed bilinear models

Discussed in this paper is the dependent structure in the tails of distributions of random variables from some heavy-tailed stationary nonlinear time series. One class of models discussed is the first-order autoregressive conditional heteroscedastic (ARCH) process introduced by Engle (1982). The other class is the simple first-order bilinear models driven by heavy-tailed innovations. We give some explicit formulas for the asymptotic values of conditional probabilities used for measuring the tail dependence between two random variables from these models. Our results have significant meanings in finance.     

Estimation of population spectrum for linear processes with random coefficients

A set of time series generated by stationary linear processes with an absolutely continuous spectral distribution is analysed. The time series can then be considered realizations of a linear process of random coefficients. Likewise, each spectral density function is a realization of a stochastic process whose function of means is called a population spectrum. We propose a kernel estimator for the population spectrum and give conditions for its consistency. We then illustrate the properties of this estimator in a simulation study and compare its performance with an alternative parametric estimator that can be found in the literature.      

Local likelihood estimation for nonstationary random fields

We develop a weighted local likelihood estimate for the parameters that govern the local spatial dependency of a locally stationary random field. The advantage of this local likelihood estimate is that it smoothly downweights the influence of faraway observations, works for irregular sampling locations, and when designed appropriately, can trade bias and variance for reducing estimation error. This paper starts with an exposition of our technique on the problem of estimating an unknown positive function when multiplied by a stationary random field. This example gives concrete evidence of the benefits of our local likelihood as compared to unweighted local likelihoods. We then discuss the difficult problem of estimating a bandwidth parameter that controls the amount of influence from distant observations. Finally we present a simulation experiment for estimating the local smoothness of a local Matérn random field when observing the field at random sampling locations in [0,1]². The local Matérn is a fully nonstationary random field, has a closed form covariance, can attain any degree of differentiability or Hölder smoothness and behaves locally like a stationary Matérn. We include an appendix that proves the positive definiteness of this covariance function.     

On the stability of robust filter-cleaners

We study the dependence on initial conditions of two recursive filters for cleaning a contaminated time series from additive outliers. We show that the function in the recursive equation is in general not contractive, but nevertheless there exists a stationary solution and two iterates with arbitrary initial conditions coincide after some random time T₀. However T₀ may be quite large.     

A Test of Correlation in the Random Coefficients of an Autoregressive Process

A random coefficient autoregressive process in which the coefficients are correlated is investigated. First we look at the existence of a strictly stationary causal solution, we give the second-order stationarity conditions and the autocorrelation function of the process. Then we study some asymptotic properties of the empirical mean and the usual estimators of the process, such as convergence, asymptotic normality and rates of convergence, supplied with appropriate assumptions on the driving perturbations. Our objective is to get an overview of the influence of correlated coefficients in the estimation step through a simple model. In particular, the lack of consistency is shown for the estimation of the autoregressive parameter when the independence hypothesis in the random coefficients is violated. Finally, a consistent estimation is given together with a testing procedure for the existence of correlation in the coefficients. While convergence properties rely on ergodicity, we use a martingale approach to reach most of the results.     

Cover time and mixing time of random walks on dynamic graphs

The application of simple random walks on graphs is a powerful tool that is useful in many algorithmic settings such as network exploration, sampling, information spreading, and distributed computing. This is due to the reliance of a simple random walk on only local data, its negligible memory requirements, and its distributed nature. It is well known that for static graphs the cover time, that is, the expected time to visit every node of the graph, and the mixing time, that is, the time to sample a node according to the stationary distribution, are at most polynomial relative to the size of the graph. Motivated by real world networks, such as peer‐to‐peer and wireless networks, the conference version of this paper was the first to study random walks on arbitrary dynamic networks. We study the most general model in which an oblivious adversary is permitted to change the graph after every step of the random walk. In contrast to static graphs, and somewhat counter‐intuitively, we show that there are adversary strategies that force the expected cover time and the mixing time of the simple random walk on dynamic graphs to be exponentially long, even when at each time step the network is well connected and rapidly mixing. To resolve this, we propose a simple strategy, the lazy random walk, which guarantees, under minor conditions, polynomial cover time and polynomial mixing time regardless of the changes made by the adversary.     

The geometry of the Wilks’s Λ random field

The statistical problem addressed in this paper is to approximate the P value of the maximum of a smooth random field of Wilks’s Λ statistics. So far results are only available for the usual univariate statistics (Z, t, χ², F) and a few multivariate statistics (Hotelling’s T ², maximum canonical correlation, Roy’s maximum root). We derive results for any differentiable scalar function of two independent Wishart random fields, such as Wilks’s Λ random field. We apply our results to a problem in brain shape analysis.     

Empirical distributions in marked point processes

We study the asymptotic behaviour of the empirical distribution function derived from a stationary marked point process when a convex sampling window is expanding without bounds in all directions. We consider a random field model which assumes that the marks and the points are independent and admits dependencies between the marks. The main result is the weak convergence of the empirical process under strong mixing conditions on both independent components of the model. Applying an approximation principle weak convergence can be also shown for appropriately weighted empirical process defined from a stationary d-dimensional germ-grain process with dependent grains.     

Separation of random noise models from seasonal time series data

The autocorrelation function of seasonal time series data is shown to have peaks which occur at the correlation lags equal to the integer multiples of the fundamental period that is present in the series. This property is shown to be valid even if some of the harmonics including the fundamental are removed from the time series data. Using this property, an analytical procedure is presented for estimating the variance of the white noise generating the low frequency random walk model present in the data. The procedure is similarly extended to estimate the variance of white noise generating the autoregressive (AR) and moving average (MA) noise models. The method is validated on several seasonal time series data whose components are known a priori.