FARIMA Processes Research Articles

Estimation of hurst parameter and minimum variance spectrum

Consider FARIMA time series with innovations that have infinite variances. We are interested in the estimation of self-similarities $H_n$ of FARIMA$(0, d, 0)$ by using modified $R/S$ statistic. We can confirm that the $H_n$ converges to Hurst parameter $H=d+\frac12$. Finally, we figure out ARMA and minimum variance power spectrum density of FARIMA processes.

Memory properties of transformations of linear processes

In this paper, we study the memory properties of transformations of linear processes. Dittmann and Granger (2002) studied the polynomial transformations of Gaussian FARIMA(0,d,0) processes by applying the orthonormality of the Hermite polynomials under the measure for the standard normal distribution. Nevertheless, the orthogonality does not hold for transformations of non-Gaussian linear processes. Instead, we use the decomposition developed by Ho and Hsing (1996, 1997) to study the memory properties of nonlinear transformations of linear processes, which include the FARIMA(p,d,q) processes, and obtain consistent results as in the Gaussian case. In particular, for stationary processes, the transformations of short-memory time series still have short-memory and the transformation of long-memory time series may have different weaker memory parameters which depend on the power rank of the transformation. On the other hand, the memory properties of transformations of non-stationary time series may not depend on the power ranks of the transformations. This study has application in econometrics and financial data analysis when the time series observations have non-Gaussian heavy tails. As an example, the memory properties of call option processes at different strike prices are discussed in details.

PARAMETER ESTIMATION AND SPECTRUM OF FRACTIONAL ARIMA PROCESS

We consider fractional Brownian motion and FARIMA process with Gaussian innovations and show that the suitably scaled distributions of the FARIMA processes converge to fractional Brownian motion in the sense of finite dimensional distributions. We figure out ACF function and estimate the self-similarity parameter H of FARIMA(0, d, 0) by using R/S method. Finally, we display power spectrum density of FARIMA process.

The point process approach for fractionally differentiated random walks under heavy traffic

We prove some heavy-traffic limit theorems for some nonstationary linear processes which encompass the fractionally differentiated random walk as well as some FARIMA processes, when the innovations are in the domain of attraction of a non-Gaussian stable distribution. The results are based on an extension of the point process methodology to linear processes with nonsummable coefficients and make use of a new maximal type inequality.

Asymptotic properties of sieve bootstrap prediction intervals for [formula omitted] processes

The sieve bootstrap is a resampling technique that uses autoregressive approximations of order p to model invertible linear time series, where p is allowed to go to infinity with sample size n. The asymptotic properties of sieve bootstrap prediction intervals for stationary invertible linear processes with short memory have been established in the past. In this paper, we extend these results to long memory (FARIMA) processes. We show that under certain regularity conditions the sieve bootstrap provides consistent estimators of the conditional distribution of future values of FARIMA processes, given the observed data.

Heavy-traffic approximations for fractionally integrated random walks in the domain of attraction of a non-Gaussian stable distribution

We prove some heavy-traffic limit theorems for processes which encompass the fractionally integrated random walk as well as some FARIMA processes, when the innovations are in the domain of attraction of a non-Gaussian stable distribution.

An explicit representation of Verblunsky coefficients

We prove a representation of the partial autocorrelation function (PACF) of a stationary process, or of the Verblunsky coefficients of its normalized spectral measure, in terms of the Fourier coefficients of the phase function. It is not of fractional form, whence simpler than the existing one obtained by the second author. We apply it to show a general estimate on the Verblunsky coefficients for short-memory processes as well as the precise asymptotic behavior, with remainder term, of those for FARIMA processes.

Invariance principles for some FARIMA and nonstationary linear processes in the domain of a stable distribution

We prove some invariance principles for processes which generalize FARIMA processes, when the innovations are in the domain of attraction of a nonGaussian stable distribution. The limiting processes are extensions of the fractional Lévy processes. The technique used is interesting in itself; it extends an older idea of splitting a sample into a central part and an extreme one, analyzing each part with different techniques, and then combining the results. This technique seems to have the potential to be useful in other problems in the domain of nonGaussian stable distributions.

Stability and lack of memory of the returns of the Hang Seng index

In this paper we show that the logarithmic returns of the Hang Seng index from January 2, 1987 to November 14, 2005 statistically resemble a sequence of independent identically distributed Lévy stable random variables. This is in stark contrast to Xiu and Jin (2007) [39], where long-memory FARIMA processes with Gaussian noise were suggested as well fitted to the data. The lack of memory is checked by using Lo’s modified R/S statistic and a new method of estimation of the memory parameter d which applies the notion of empirical mean-squared displacement. In order to test stability of the data we employ several statistical tests based on the empirical distribution function. Finally, we also show that the returns possess no conditional heteroscedasticity property thus excluding the ARCH/GARCH family of processes as possible underlying models.

An extension of a logarithmic form of Cramér’s ruin theorem to some FARIMA and related processes

Cramér’s theorem provides an estimate for the tail probability of the maximum of a random walk with negative drift and increments having a moment generating function finite in a neighborhood of the origin. The class of ( g , F ) -processes generalizes in a natural way random walks and fractional ARIMA models used in time series analysis. For those ( g , F ) -processes with negative drift, we obtain a logarithmic estimate of the tail probability of their maximum, under conditions comparable to Cramér’s. Furthermore, we exhibit the most likely paths as well as the most likely behavior of the innovations leading to a large maximum.

Detection of Multiple Changes in Fractional Integrated ARMA Processes

This paper addresses the problem of changepoint detection in FARIMA processes. The received signal is modeled as a FARIMA process, with abrupt changes in the Hurst and ARMA parameters. The proposed changepoint detection method first estimates the model parameters over small segments. The changes are then detected in the parameter vector sequence by minimizing an appropriate least-squares criterion. The cases of known, as well as unknown, number of changes are investigated. Dynamic programming is used to solve this optimization problem. A theoretical analysis of the statistical properties of the change point estimates is provided. Simulation results on synthetic data and real network traffic data are presented.

From solar flare time series to fractional dynamics

We demonstrate that continuous-time FARIMA processes with α -stable noise provide a new stochastic tool for studying the solar flare phenomenon in the framework of fractional Langevin equation. Simple computer tests to check the origins of α -stability and self-similarity are implemented for empirical time series describing the energy of solar flares. Based on observed physical time series we solve the challenging problem of how to detect long-range dependence from real data and how to model it via fractional dynamics (Langevin or Fokker–Planck). We employ here codifference as a proper measure for long-range dependence. It is applicable to empirical data from the distribution lacking the second moment.

Nonparametric estimation under long memory dependence

Modelling dependence structure by means of long memory processes appears of great interest for many applications. This paper aims to show how nonparametric kernel estimators behave asymptotically under this dependence structure. We will consider a wide class of functional estimation problems, including distribution function, hazard function and density function, and we will give rates of weak consistency and of mean squared consistency. A main feature of this paper is to deal with a general form of long dependence processes, without assuming any Gaussian shape for the data or any parametric form for the covariances sequence. Finally, we will show how these results apply naturally to many classical parametric processes, and special attention will be paid to the Fractional Brownian Motion and to FARIMA processes.

Adaptive Estimation of the Fractional Differencing Coefficient

Semi-parametric estimation of the fractional differencing coefficient d of a long-range dependent stationary time series has received substantial attention in recent years. Some of the so-called local estimators introduced early on were proved rate-optimal over relevant classes of spectral densities. The rates of convergence of these estimators are limited to n2/5, where n is the sample size. This paper focuses on the fractional exponential (FEXP) or broadband estimator of d. Minimax rates of convergence over classes of spectral densities which are smooth outside the zero frequency are obtained, and the FEXP estimator is proved rate-optimal over these classes. On a certain functional class which contains the spectral densities of FARIMA processes, the rate of convergence of the FEXP estimator is (n/log(n))1/2, thus making it a reasonable alternative to parametric estimators. As usual in semiparametric estimation problems, these rate-optimal estimators are infeasible, since they depend on an unknown smoothness parameter defining the functional class. A feasible adaptive version of the broadband estimator is constructed. It is shown that this estimator is minimax rate-optimal up to a factor proportional to the logarithm of the sample size.

Prediction intervals for farima processes by bootstrap methods

In this paper we introduce a procedure to compute prediction intervals for FARIMA (p d q) processes, taking into account the variability due to model identification and parameter estimation. To this aim, a particular bootstrap technique is developed. The performance of the prediction intervals is then assessed and compared to that of stand­ard bootstrap percentile intervals. The methods are applied to the time series of Nile River annual minima.

On the use of self-similar processes in network simulation

Several recent traffic measurement studies have convincingly shown the presence of self-similarity in modern high-speed networks, involving a very important revolution in the stochastic modeling of traffic. Thus the use of self-similar processes has opened new problems and research fields in network performance analysis, mainly in simulation studies, where the efficient synthetic generation of sample paths (traces) corresponding to self-similar traffic is one of the main topics. In this article, we justify the selection of interarrival time instead of counting processes for modeling arrivals. Also, we discuss the advantages and drawbacks of the most important self-similar processes when applied to traffic modeling in simulation studies, proposing the use of models based in F-ARIMA, mainly due to their flexibility to capture both long- and short-range correlations. However, F-ARIMA processes have been little used in simulation studies, mainly because the synthetic generation methods available in the literature are very inefficient compared with those for FGN. In order to solve this problem, we propose a new method that can generate high-quality traces corresponding to a F-ARIMA(p, d, q) process. A comparison with existing methods shows that the new method is significantly more efficient, and even slightly better than the best method for FGN.