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Register a new userEstimation of cluster functionals for regularly varying time series: runs estimators
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Cluster indices describe extremal behaviour of stationary time series. We consider runs estimators of cluster indices. Using a modern theory of multivariate, regularly varying time series, we obtain central limit theorems under conditions that can be easily verified for a large class of models. In particular, we show that blocks and runs estimators have the same limiting variance.
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Suppose that particles are randomly distributed in $bR^d$, and they are subject to identical stochastic motion independently of each other. The Smoluchowski process describes fluctuations of the number of particles in an observation region over time. This paper studies properties of the Smoluchowski processes and considers related statistical problems. In the first part of the paper we revisit probabilistic properties of the Smoluchowski process in a unified and principled way: explicit formulas for generating functionals and moments are derived, conditions for stationarity and Gaussian approximation are discussed, and relations to other stochastic models are highlighted. The second part deals with statistics of the Smoluchowki processes. We consider two different models of the particle displacement process: the undeviated uniform motion (when a particle moves with random constant velocity along a straight line) and the Brownian motion displacement. In the setting of the undeviated uniform motion we study the problems of estimating the mean speed and the speed distribution, while for the Brownian displacement model the problem of estimating the diffusion coefficient is considered. In all these settings we develop estimators with provable accuracy guarantees.
We provide some asymptotic theory for the largest eigenvalues of a sample covariance matrix of a p-dimensional time series where the dimension p = p_n converges to infinity when the sample size n increases. We give a short overview of the literature on the topic both in the light- and heavy-tailed cases when the data have finite (infinite) fourth moment, respectively. Our main focus is on the heavytailed case. In this case, one has a theory for the point process of the normalized eigenvalues of the sample covariance matrix in the iid case but also when rows and columns of the data are linearly dependent. We provide limit results for the weak convergence of these point processes to Poisson or cluster Poisson processes. Based on this convergence we can also derive the limit laws of various function als of the ordered eigenvalues such as the joint convergence of a finite number of the largest order statistics, the joint limit law of the largest eigenvalue and the trace, limit laws for successive ratios of ordered eigenvalues, etc. We also develop some limit theory for the singular values of the sample autocovariance matrices and their sums of squares. The theory is illustrated for simulated data and for the components of the S&P 500 stock index.
This paper deals with the dimension reduction for high-dimensional time series based on common factors. In particular we allow the dimension of time series $p$ to be as large as, or even larger than, the sample size $n$. The estimation for the factor loading matrix and the factor process itself is carried out via an eigenanalysis for a $ptimes p$ non-negative definite matrix. We show that when all the factors are strong in the sense that the norm of each column in the factor loading matrix is of the order $p^{1/2}$, the estimator for the factor loading matrix, as well as the resulting estimator for the precision matrix of the original $p$-variant time series, are weakly consistent in $L_2$-norm with the convergence rates independent of $p$. This result exhibits clearly that the `curse is canceled out by the `blessings in dimensionality. We also establish the asymptotic properties of the estimation when not all factors are strong. For the latter case, a two-step estimation procedure is preferred accordingly to the asymptotic theory. The proposed methods together with their asymptotic properties are further illustrated in a simulation study. An application to a real data set is also reported.
A general class of time-varying regression models is considered in this paper. We estimate the regression coefficients by using local linear M-estimation. For these estimators, weak Bahadur representations are obtained and are used to construct simultaneous confidence bands. For practical implementation, we propose a bootstrap based method to circumvent the slow logarithmic convergence of the theoretical simultaneous bands. Our results substantially generalize and unify the treatments for several time-varying regression and auto-regression models. The performance for ARCH and GARCH models is studied in simulations and a few real-life applications of our study are presented through analysis of some popular financial datasets.
This paper is devoted to two different two-time-scale stochastic approximation algorithms for superquantile estimation. We shall investigate the asymptotic behavior of a Robbins-Monro estimator and its convexified version. Our main contribution is to establish the almost sure convergence, the quadratic strong law and the law of iterated logarithm for our estimates via a martingale approach. A joint asymptotic normality is also provided. Our theoretical analysis is illustrated by numerical experiments on real datasets.
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