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162
On the Selfsimilar Nature of Ethernet Traffic (Extended Version)
, 1994
"... We demonstrate that Ethernet LAN traffic is statistically selfsimilar, that none of the commonly used traffic models is able to capture this fractallike behavior, that such behavior has serious implications for the design, control, and analysis of highspeed, cellbased networks, and that aggrega ..."
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Cited by 2213 (46 self)
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We demonstrate that Ethernet LAN traffic is statistically selfsimilar, that none of the commonly used traffic models is able to capture this fractallike behavior, that such behavior has serious implications for the design, control, and analysis of highspeed, cellbased networks, and that aggregating streams of such traffic typically intensifies the selfsimilarity (“burstiness”) instead of smoothing it. Our conclusions are supported by a rigorous statistical analysis of hundreds of millions of high quality Ethernet traffic measurements collected between 1989 and 1992, coupled with a discussion of the underlying mathematical and statistical properties of selfsimilarity and their relationship with actual network behavior. We also present traffic models based on selfsimilar stochastic processes that provide simple, accurate, and realistic descriptions of traffic scenarios expected during BISDN deployment.
On the Detection and Estimation of Long Memory in Stochastic Volatility
, 1995
"... Recent studies have suggested that stock markets' volatility has a type of longrange dependence that is not appropriately described by the usual Generalized Autoregressive Conditional Heteroskedastic (GARCH) and Exponential GARCH (EGARCH) models. In this paper, different models for describing ..."
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Cited by 214 (6 self)
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Recent studies have suggested that stock markets' volatility has a type of longrange dependence that is not appropriately described by the usual Generalized Autoregressive Conditional Heteroskedastic (GARCH) and Exponential GARCH (EGARCH) models. In this paper, different models for describing this longrange dependence are examined and the properties of a LongMemory Stochastic Volatility (LMSV) model, constructed by incorporating an Autoregressive Fractionally Integrated Moving Average (ARFIMA) process in a stochastic volatility scheme, are discussed. Strongly consistent estimators for the parameters of this LMSV model are obtained by maximizing the spectral likelihood. The distribution of the estimators is analyzed by means of a Monte Carlo study. The LMSV is applied to daily stock market returns providing an improved description of the volatility behavior. In order to assess the empirical relevance of this approach, tests for longmemory volatility are described and applied to an e...
Estimators for LongRange Dependence: An Empirical Study
, 1995
"... Various methods for estimating the selfsimilarity parameter and/or the intensity of longrange dependence in a time series are available. Some are more reliable than others. To discover the ones that work best, we apply the different methods to simulated sequences of fractional Gaussian noise and f ..."
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Cited by 157 (6 self)
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Various methods for estimating the selfsimilarity parameter and/or the intensity of longrange dependence in a time series are available. Some are more reliable than others. To discover the ones that work best, we apply the different methods to simulated sequences of fractional Gaussian noise and fractional ARIMA(0, d, 0). We also provide here a theoretical justification for the method of residuals of regression.
Central Limit Theorems for Quadratic Forms with Time Domain Conditions
, 1996
"... We establish the central limit theorem for quadratic forms P N t;s=1 b(t\Gammas)P m;n (X t ; X s ) of the bivariate Appell polynomials P m;n (X t ; X s ) under time domain conditions. These conditions relate the weights b(t) and the covariances of the sequences (P m;n (X t ; X s )) and (X t ). The ..."
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Cited by 102 (8 self)
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We establish the central limit theorem for quadratic forms P N t;s=1 b(t\Gammas)P m;n (X t ; X s ) of the bivariate Appell polynomials P m;n (X t ; X s ) under time domain conditions. These conditions relate the weights b(t) and the covariances of the sequences (P m;n (X t ; X s )) and (X t ). The time domain approach, together with the spectral domain approach developed earlier, yield a general set of conditions for central limit theorems. 1 Introduction We study the central limit theorem (CLT) for quadratic forms QN := N X t;s=1 b(t \Gamma s)P m;n (X t ; X s ) (1.1) where P m;n (X t ; X s ) =: X t ; : : : ; X t  z m ; X s ; : : : ; X s  z n : is a bivariate Appell polynomial (Wick power) of the linear variables X t and X s , m;n 0; m+ n 1. Here X t = X s2Z a(t \Gamma u)¸ u ; t 2 Z (1.2) is a linear process, that is, the random variables ¸ t ; t 2 Z are independent and identically distributed, E¸ 0 = 0 E¸ 2 0 = 1, the sequence a(t); t 2 Z of realvalu...
On Estimating the Intensity of LongRange Dependence in Finite and Infinite Variance Time Series
, 1996
"... The goal of this paper is to provide benchmarks to the practitioner for measuring the intensity of longrange dependence in time series. It provides a detailed comparison of eight estimators for longrange dependence, using simulated FARIMA(p; d; q) time series with different finite and infinite var ..."
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Cited by 63 (4 self)
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The goal of this paper is to provide benchmarks to the practitioner for measuring the intensity of longrange dependence in time series. It provides a detailed comparison of eight estimators for longrange dependence, using simulated FARIMA(p; d; q) time series with different finite and infinite variance innovations. FARIMA time series model both longrange dependence (through the parameter d) and shortrange dependence (through the parameters p and q). We evaluate the biases and standard deviations of several estimators of d and compare them for each type of series used. We consider Gaussian, exponential, lognormal, Pareto, symmetric and skewed stable innovations. Detailed tables and graphs have been included. We find that the estimators tend to perform less well when p and q are not zero, that is, when there is additional shortrange dependence structure. For most of the estimators, however, the use of infinite variance instead of finite variance innovations does not cause a great dec...
Microeconomic Models for LongMemory in the Volatility of Financial Time Series
"... We show that a class of microeconomic behavioral models with interacting agents, derived from Kirman (1991, 1993), can replicate the empirical longmemory properties of the two first conditional moments of financial time series. The essence of these models is that the forecasts and thus the desired ..."
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Cited by 61 (2 self)
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We show that a class of microeconomic behavioral models with interacting agents, derived from Kirman (1991, 1993), can replicate the empirical longmemory properties of the two first conditional moments of financial time series. The essence of these models is that the forecasts and thus the desired trades of the individuals in the markets are influenced, directly, or indirectly by those of the other participants. These "field effects" generate "herding" behaviour which affects the structure of the asset price dynamics. The series of returns generated by these models display the same empirical properties as financial returns: returns are I(0), the series of absolute and squared returns display strong dependence, while the series of absolute returns do not display a trend. Furthermore, this class of models is able to replicate the common longmemory properties in the volatility and covolatility of financial time series, revealed by Teyssière (1997, 1998a). These properties are investigated by using various model independent tests and estimators, i.e., semiparametric and nonparametric, introduced by Lo (1991), Kwiatkowski, Phillips, Schmidt and Shin (1992), Robinson (1995), Lobato and Robinson (1998), Giraitis, Kokoszka Leipus and Teyssière (2000, 2001). The relative performance of these tests and estimators for longmemory in a nonstandard data generating process is then assessed.
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 56 (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...
Whittle pseudomaximum likelihood estimation for nonstationary time series
 J. Am. Statist. Assoc
, 2000
"... Exact and approximate maximum likelihood estimation for the parameters of stationary time series has been justi�ed under various sets of conditions, including spectral densities with a peak at the origin due to a persistent behaviour �e.g. Fox and Taqqu �1986��. Thus it is often assumed that the spe ..."
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Cited by 52 (12 self)
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Exact and approximate maximum likelihood estimation for the parameters of stationary time series has been justi�ed under various sets of conditions, including spectral densities with a peak at the origin due to a persistent behaviour �e.g. Fox and Taqqu �1986��. Thus it is often assumed that the spectral density f�� � of an observed covariance stationary sequence satis�es, for 0 �G�1, �1 � f�� � � Gj�j,2d as � ! 0; where d 2 �, 1 1; � is the parameter that governs the degree of memory of the series. This is 2 2 the interval of values of d for which the process is stationary and invertible. If d 2 �0; 1 � then 2 we say that the series exhibits long memory or long range dependence. When the observations are nonstationary, they are usually di�erenced an integer number of times to achieve stationarity. If an observed processes fXtg has covariance stationary increments �X t, with spectral density satisfying f �X�� � � G j�j,2�d,1 � as � ! 0, d � 1
Using Wavelets to Obtain a Consistent Ordinary Least Squares Estimator of the Longmemory Parameter
 Journal of Forecasting
, 1999
"... We develop an ordinary least squares estimator of the long memory parameter from a fractionally integrated process that is an alternative to the Geweke PorterHudak estimator. Using the wavelet transform from a fractionally integrated process, we establish a loglinear relationship between the wavel ..."
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Cited by 49 (7 self)
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We develop an ordinary least squares estimator of the long memory parameter from a fractionally integrated process that is an alternative to the Geweke PorterHudak estimator. Using the wavelet transform from a fractionally integrated process, we establish a loglinear relationship between the wavelet coe cients ' variance and the scaling parameter equal to the long memory parameter. This loglinear relationship yields a consistent ordinary least squares estimator of the long memory parameter when the wavelet coe cients ' population variance is replaced by their sample variance. We derive the small sample bias and variance of the ordinary least squares estimator and test it against the Geweke PorterHudak estimator and the McCoy Walden maximum likelihood wavelet estimator by conducting a numberofMonte Carlo experiments. Based upon the criterion of choosing the estimator which minimizes the mean squared error, the wavelet OLS approach was superior to the Geweke PorterHudak estimator, but inferior to the McCoy Walden wavelet estimator for the processes simulated. However, given the simplicity of programming and running the wavelet OLS estimator and its statistical inference of the long memory parameter we feel the general practitioner will be attracted to wavelet OLS estimator. Keywords