Discrete time parametric models with long memory and infinite variance

We study a large class of infinite variance time series that display long memory. They can be represented as linear processes (infinite order moving averages) with coefficients that decay slowly to zero and with innovations that are in the domain of attraction of a stable distribution with index 1 < α < 2 (stable fractional ARIMA is a particular example). Assume that the coefficients of the linear process depend on an unknown parameter vector β which is to be estimated from a series of length n. We show that a Whittle-type estimator β_n for β is consistent (β_n converges to the true value β₀ in probability as n → ∞), and, under some additional conditions, we characterize the limiting distribution of the rescaled differences (n/logn)^1/ga(β_n − β₀).     

E.P.R. of a mixed metal two-dimensional system: Mn1−fCof(HCOO)2·2H2O

An X-band E.P.R. study of Mn_1?fCo_f(HCOO)₂·2H₂O has been carried out for 0.0 ? f ? 0.99. Although some temperature dependent effects were noted, the dominant result was a strong inverted U-shaped dependence in the Mn(II) resonant linewidth with increasing Co(II) concentration. The data suggest the presence of both isotropic and anisotropic exchange interactions between the dissimilar metal ions. A phenomenological theoretical model is proposed which requires only one adjustable parameter to fit the linewidth vs concentration data.     

Estimation of the memory parameter by fitting fractionally differenced autoregressive models

Estimation of the memory parameter, d, by fitting a fractionally differenced autoregression of order p, where p approaches infinity simultaneously with the observed series length, n, is examined. Under some conditions on growth of p with respect to n and on the short-memory component, which admits an infinite autoregressive representation with coefficients a_j, the estimator is shown to be consistent and asymptotically normal, where p may be taken to be proportional to logn. The joint asymptotic distribution of the estimators of d and of the a_j is also derived.     

Almost sure convergence of the Bartlett estimator

Summary We study the almost sure convergence of the Bartlett estimator for the asymptotic variance of the sample mean of a stationary weekly dependent process. We also study the a.\ s.\ behavior of this estimator in the case of long-range dependent observations. In the weakly dependent case, we establish conditions under which the estimator is strongly consistent. We also show that, after appropriate normalization, the estimator converges a.s. in the long-range dependent case as well. In both cases, our conditions involve fourth order cumulants and assumptions on the rate of growth of the truncation parameter appearing in the definition of the Bartlett estimator.     

Testing for parameter constancy in GARCH() models

We propose a test for a change in the parameters of a GARCH(p,q) model. The test is based on approximate likelihood scores and does not require the observations to have finite variance. We show that the test has asymptotically correct size under weak assumptions on model errors.     

Subsampling Unit Root Tests for Heavy-Tailed Observations

It is shown that the subsampling methodology can be used to develop unit root tests when the noise sequence is heavy-tailed with infinite variance. Using least-squares residuals, we construct processes which approximately satisfy the null hypothesis and then, using subsampling, we approximate the null distribution of test statistics. We establish the asymptotic validity of this method and demonstrate its applicability in finite samples by means of a simulation study and a data example.     

Change-point in the mean of dependent observations

We prove the consistency of a family of CUSUM-type estimators of the point of change in the mean of dependent observations and derive the rates of convergence. The result is valid under weak assumptions on the dependence structure.     

Estimation of a change-point in the mean function of functional data

The paper develops a comprehensive asymptotic theory for the estimation of a change-point in the mean function of functional observations. We consider both the case of a constant change size, and the case of a change whose size approaches zero, as the sample size tends to infinity. We show how the limit distribution of a suitably defined change-point estimator depends on the size and location of the change. The theoretical insights are confirmed by a simulation study which illustrates the behavior of the estimator in finite samples.     

On the Power of R/S-Type Tests under Contiguous and Semi-Long Memory Alternatives

The paper deals with the power and robustness of the R/S type tests under contiguous alternatives. We briefly review some long memory models in levels and volatility, and describe the R/S-type tests used to test for the presence of long memory. The empirical power of the tests is investigated when replacing the fractional difference operator (1–L) ^d by the operator (1–rL) ^d , with r<1 close to 1, in the FARIMA, LARCH and ARCH time series models. We also investigate the Gegenbauer process with a pole of the spectral density at frequency close to zero.