Results 1  10
of
20
NarrowBand Analysis Of Nonstationary Processes
, 1999
"... The behaviour of averaged periodograms and crossperiodograms of a broad class of nonstationary processes is studied. The processes include nonstationary ones that are fractional of any order, as well as asymptotically stationary fractional ones, and the crossperiodogram can involve two nonstationa ..."
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Cited by 48 (13 self)
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The behaviour of averaged periodograms and crossperiodograms of a broad class of nonstationary processes is studied. The processes include nonstationary ones that are fractional of any order, as well as asymptotically stationary fractional ones, and the crossperiodogram can involve two nonstationary processes of possibly di#erent orders, or a nonstationary and an asymptotically stationary one. The averaging takes place either over the whole frequency band, or on one that degenerates slowly to zero frequency as sample size increases. In some cases it is found to make no asymptotic di#erence, and in particular we indicate how the behaviour of the mean and variance changes across the twodimensional space of integration orders. The results employ only localtozero assumptions on the spectra of the underlying weakly stationary sequences. It is shown how the results can be readily applied in case of fractional cointegration with unknown integration orders. 1 1. INTRODUCTION In the analy...
The Supremum of a Negative Drift Random Walk with Dependent HeavyTailed Steps
, 1998
"... . Many important probabilistic models in queuing theory, insurance and finance deal with partial sums of a negative mean stationary process (a negative drift random walk), and the law of the supremum of such a process is used to calculate, depending on the context, the ruin probability, the steady s ..."
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Cited by 40 (25 self)
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. Many important probabilistic models in queuing theory, insurance and finance deal with partial sums of a negative mean stationary process (a negative drift random walk), and the law of the supremum of such a process is used to calculate, depending on the context, the ruin probability, the steady state distribution of the number of customers in the system or the value at risk. When the stationary process is heavytailed, the corresponding ruin probabilities are high and the stationary distributions are heavytailed as well. If the steps of the random walk are independent, then the exact asymptotic behavior of such probability tails was described by Embrechts and Veraverbeke (1982). We show that this asymptotic behavior may be different if the steps of the random walk are not independent, and the dependence affects the joint probability tails of the stationary process. Such type of dependence can be modeled, for example, by a linear process. 1. Introduction In various applied fields...
SemiParametric Graphical Estimation Techniques for LongMemory Data.
, 1996
"... This paper reviews several periodogrambased methods for estimating the longmemory parameter H in time series and suggests a way to robustify them. The high frequencies tend to bias the estimates. Using only low frequencies eliminates the bias but increases the variance. We hence suggest plotting t ..."
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Cited by 16 (4 self)
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This paper reviews several periodogrambased methods for estimating the longmemory parameter H in time series and suggests a way to robustify them. The high frequencies tend to bias the estimates. Using only low frequencies eliminates the bias but increases the variance. We hence suggest plotting the estimates of H as a function of a parameter which balances bias versus variance and, if the plot flattens in a central region, to use the flat part for estimating H. We apply this technique to the periodogram regression method, the Whittle approximation to maximum likelihood and to the local Whittle method. We investigate its effectiveness on several simulated fractional ARIMA series and also apply it to estimate the longmemory parameter H in computer network traffic. 1 Introduction Time series with long memory have been considered in many fields including hydrology, biology and computer networks. Unfortunately, estimating the long memory (longrange dependence) parameter H in a given d...
Robustness of Whittletype Estimators for Time Series with LongRange Dependence.
, 1997
"... We study the robustness of the "standard Whittle", "local Whittle " and "aggregated Whittle" estimators by using a large number of simulated Gaussian time series with longrange dependence. We also consider what happens when the Gaussian innovations are replaced by infi ..."
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Cited by 15 (6 self)
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We study the robustness of the "standard Whittle", "local Whittle " and "aggregated Whittle" estimators by using a large number of simulated Gaussian time series with longrange dependence. We also consider what happens when the Gaussian innovations are replaced by infinite variance symmetric stable ones. The standard Whittle estimator is a parametric estimator, the local Whittle estimator is a semiparametric one recently developed by Robinson (1995) and the aggregated Whittle estimator smoothes out the high frequencies. The goal is to estimate H; the intensity of longrange dependence. We investigate the standard deviation and bias of these estimators in order to determine when they are reliable. These estimators are then applied to reallife Ethernet data. 1 Introduction The recent discovery that longrange dependence is present in highspeed networks has given a new urgency to the development of reliable methods for estimating the intensity of longrange dependence in extremely lon...
Periodic moving averages of random variables with regularly varying tails, Ann
 Stat
, 1997
"... In this paper we establish the basic asymptotic theory for periodic moving averages of i.i.d. random variables with regularly varying tails. The moving average coefficients are allowed to vary according to the season. A simple reformulation yields the corresponding results for moving averages of ran ..."
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Cited by 10 (5 self)
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In this paper we establish the basic asymptotic theory for periodic moving averages of i.i.d. random variables with regularly varying tails. The moving average coefficients are allowed to vary according to the season. A simple reformulation yields the corresponding results for moving averages of random vectors. Our main result is that when the underlying random variables have finite variance but infinite fourth moment, the sample autocorrelations are asymptotically stable. It is well known in this case that sample autocorrelations in the classical stationary moving average model are asymptotically normal. Introduction. Regular variation is used to characterize those i.i.d. sequences of random variables for which a version of the central limit theorem holds. When these random variables have infinite variance, the sum is asymptotically stable instead of asymptotically normal. Stable random variables have found many practical applications beginning with the work of Holts
The Averaged Periodogram For Nonstationary Vector Stochastic Processes
, 1999
"... Frequency domain statistics are studied in the presence of fractional deterministic and stochastic trends. It is shown how the behaviour of the sample variancecovariance matrix of nonstationary processes can be dominated by components corresponding to a possibly degenerating band around zero freque ..."
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Cited by 9 (4 self)
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Frequency domain statistics are studied in the presence of fractional deterministic and stochastic trends. It is shown how the behaviour of the sample variancecovariance matrix of nonstationary processes can be dominated by components corresponding to a possibly degenerating band around zero frequency. This property is used to establish the limiting distribution of the averaged periodogram matrix, of memory estimates for nonstationary series, and for frequency domain regression estimates under nonstandard conditions. Keywords: Averaged periodogram, nonstationary processes, fractional Brownian motion. 1 1. INTRODUCTION For a sequence of column vectors u t , t = 1, 2, ..., n, with realvalued elements, define the discrete Fourier transform w u (#) = 1 # 2#n n # t=1 (u t  u)e it# , where u = n 1 # n t=1 u t denotes the sample mean. Given also a vector sequence v t , t = 1, ..., n, define the crossperiodogram matrix I uv (#) = w u (#)w # v (#), (1.1) the asterisk denot...
2004): “Subsampling the mean of heavytailed dependent observations
 Journal of Time Series Analysis
"... We establish the validity of subsampling confidence intervals for the mean of a dependent series with heavytailed marginal distributions. Using point process theory, we study both linear and nonlinear GARCHlike time series models. We propose a datadependent method for the optimal block size selec ..."
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Cited by 2 (0 self)
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We establish the validity of subsampling confidence intervals for the mean of a dependent series with heavytailed marginal distributions. Using point process theory, we study both linear and nonlinear GARCHlike time series models. We propose a datadependent method for the optimal block size selection and investigate its performance by means of a simulation study. JEL CLASSIFICATION NOS: C10, C14, C32. KEYWORDS: GARCH, Heavy tails, Linear time series, Subsampling.
Nonparametric Regression Under Dependent Errors With Infinite Variance
"... We consider local least absolute deviation (LLAD) estimation for trend functions of time series with heavy tails which are characterised via a symmetric stable law distribution. The setting includes both causal stable ARMA model and fractional stable ARIMA model as special cases. The asymptotic li ..."
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Cited by 2 (0 self)
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We consider local least absolute deviation (LLAD) estimation for trend functions of time series with heavy tails which are characterised via a symmetric stable law distribution. The setting includes both causal stable ARMA model and fractional stable ARIMA model as special cases. The asymptotic limit of the estimator is established under the assumption that the process has either short or long memory autocorrelation. For a short memory process, the estimator admits the same convergence rate as if the process has the finite variance. The optimal rate of convergence n−2/5 is obtainable by using appropriate bandwidths. This is distinctly different from local least squares estimation, of which the convergence is slowed down due to the existence of heavy tails. On the other hand, the rate of convergence of the LLAD estimator for a long memory process is always slower than n −2/5 and the limit is no longer normal. 1
Multiresolution Gaussian processes
 in Advances in Neural Information Processing Systems 25
, 2012
"... We propose a multiresolution Gaussian process to capture longrange, nonMarkovian dependencies while allowing for abrupt changes and nonstationarity. The multiresolution GP hierarchically couples a collection of smooth GPs, each defined over an element of a random nested partition. Longrange depe ..."
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Cited by 1 (0 self)
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We propose a multiresolution Gaussian process to capture longrange, nonMarkovian dependencies while allowing for abrupt changes and nonstationarity. The multiresolution GP hierarchically couples a collection of smooth GPs, each defined over an element of a random nested partition. Longrange dependencies are captured by the toplevel GP while the partition points define the abrupt changes. Due to the inherent conjugacy of the GPs, one can analytically marginalize the GPs and compute the marginal likelihood of the observations given the partition tree. This property allows for efficient inference of the partition itself, for which we employ graphtheoretic techniques. We apply the multiresolution GP to the analysis of magnetoencephalography (MEG) recordings of brain activity. 1
2000), “Weighted least absolute deviations estimation for ARMA models with infinite variance,” Econometric Theory 23
"... For autoregressive and movingaverage (ARMA) models with infinite variance innovations, quasilikelihood based estimators (such as Whittle’s estimators) suffer from complex asymptotic distributions depending on unknown tail indices. This makes the statistical inference for such models difficult. In ..."
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Cited by 1 (0 self)
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For autoregressive and movingaverage (ARMA) models with infinite variance innovations, quasilikelihood based estimators (such as Whittle’s estimators) suffer from complex asymptotic distributions depending on unknown tail indices. This makes the statistical inference for such models difficult. In contrast, the least absolute deviations estimators (LADE) are more appealing in dealing with heavy tailed processes. In this paper, we propose a weighted least absolute deviations estimator (WLADE) for ARMA models. We show that the proposed WLADE is asymptotically normal, unbiased and with the standard rootn convergence rate even when the variance of innovations is infinity. This paves the way for the statistical inference based on asymptotic normality for heavytailed ARMA processes. For relatively small samples numerical results illustrate that the WLADE with appropriate weight is more accurate than the Whittle estimator, the quasimaximum likelihood estimator (QMLE) and the GaussNewton estimator when the innovation variance is infinite, and that the efficiencyloss due to the use of weights in estimation is not substantial. The authors thank the two referees for their valuable suggestions. The work was partially supported by an EPSRC