15,788 research outputs found

    Estimation of fractal dimension for a class of Non-Gaussian stationary processes and fields

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    We present the asymptotic distribution theory for a class of increment-based estimators of the fractal dimension of a random field of the form g{X(t)}, where g:R\to R is an unknown smooth function and X(t) is a real-valued stationary Gaussian field on R^d, d=1 or 2, whose covariance function obeys a power law at the origin. The relevant theoretical framework here is ``fixed domain'' (or ``infill'') asymptotics. Surprisingly, the limit theory in this non-Gaussian case is somewhat richer than in the Gaussian case (the latter is recovered when g is affine), in part because estimators of the type considered may have an asymptotic variance which is random in the limit. Broadly, when g is smooth and nonaffine, three types of limit distributions can arise, types (i), (ii) and (iii), say. Each type can be represented as a random integral. More specifically, type (i) can be represented as the integral of a certain random function with respect to Lebesgue measure; type (ii) can be represented as the integral of a second random functio

    Testing for Changes in Kendall's Tau

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    For a bivariate time series ((Xi,Yi))i=1,...,n((X_i,Y_i))_{i=1,...,n} we want to detect whether the correlation between XiX_i and YiY_i stays constant for all i=1,...,ni = 1,...,n. We propose a nonparametric change-point test statistic based on Kendall's tau and derive its asymptotic distribution under the null hypothesis of no change by means a new U-statistic invariance principle for dependent processes. The asymptotic distribution depends on the long run variance of Kendall's tau, for which we propose an estimator and show its consistency. Furthermore, assuming a single change-point, we show that the location of the change-point is consistently estimated. Kendall's tau possesses a high efficiency at the normal distribution, as compared to the normal maximum likelihood estimator, Pearson's moment correlation coefficient. Contrary to Pearson's correlation coefficient, it has excellent robustness properties and shows no loss in efficiency at heavy-tailed distributions. We assume the data ((Xi,Yi))i=1,...,n((X_i,Y_i))_{i=1,...,n} to be stationary and P-near epoch dependent on an absolutely regular process. The P-near epoch dependence condition constitutes a generalization of the usually considered LpL_p-near epoch dependence, p≥1p \ge 1, that does not require the existence of any moments. It is therefore very well suited for our objective to efficiently detect changes in correlation for arbitrarily heavy-tailed data

    Adaptive wavelet based estimator of the memory parameter for stationary Gaussian processes

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    This work is intended as a contribution to a wavelet-based adaptive estimator of the memory parameter in the classical semi-parametric framework for Gaussian stationary processes. In particular we introduce and develop the choice of a data-driven optimal bandwidth. Moreover, we establish a central limit theorem for the estimator of the memory parameter with the minimax rate of convergence (up to a logarithm factor). The quality of the estimators are attested by simulations

    An efficient semiparametric maxima estimator of the extremal index

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    The extremal index θ\theta, a measure of the degree of local dependence in the extremes of a stationary process, plays an important role in extreme value analyses. We estimate θ\theta semiparametrically, using the relationship between the distribution of block maxima and the marginal distribution of a process to define a semiparametric model. We show that these semiparametric estimators are simpler and substantially more efficient than their parametric counterparts. We seek to improve efficiency further using maxima over sliding blocks. A simulation study shows that the semiparametric estimators are competitive with the leading estimators. An application to sea-surge heights combines inferences about θ\theta with a standard extreme value analysis of block maxima to estimate marginal quantiles.Comment: 17 pages, 7 figures. Minor edits made to version 1 prior to journal publication. The final publication is available at Springer via http://dx.doi.org/10.1007/s10687-015-0221-
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