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Register a new userVolterra bootstrap: Resampling higher-order statistics for strictly stationary univariate time series
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Added by
Thorsten Dickhaus
Publication date
2020
fields
Mathematical Statistics
and research's language is
English
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We are concerned with nonparametric hypothesis testing of time series functionals. It is known that the popular autoregressive sieve bootstrap is, in general, not valid for statistics whose (asymptotic) distribution depends on moments of order higher than two, irrespective of whether the data come from a linear time series or a nonlinear one. Inspired by nonlinear system theory we circumvent this non-validity by introducing a higher-order bootstrap scheme based on the Volterra series representation of the process. In order to estimate coefficients of such a representation efficiently, we rely on the alternative formulation of Volterra operators in reproducing kernel Hilbert space. We perform polynomial kernel regression which scales linearly with the input dimensionality and is independent of the degree of nonlinearity. We illustrate the applicability of the suggested Volterra-representation-based bootstrap procedure in a simulation study where we consider strictly stationary linear and nonlinear processes.
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Mediation analyses are a statistical tool for testing the hypothesis about how the relationship between two variables may be direct or indirect via a third variable. Assessing statistical significance has been an area of active research; however, assessment of statistical power has been hampered by the lack of closed form calculations and the need for substantial amounts of computational simulations. The current work provides a detailed explanation of implementing large scale simulation procedures within a shared computing cluster environment. In addition, all results and code for implementing these procedures is publicly available. The resulting power analyses compare the effects of sample size and strength and direction of the relationships between the three variables. Comparisons of three confidence interval calculation methods demonstrated that the bias-corrected method is optimal and requires approximately ten less participants than the percentile method to achieve equivalent power. Differing strengths of distal and proximal effects were compared and did not differentially affect the power to detect mediation effects. Suppression effects were explored and demonstrate that in the presence of no observed relationship between two variables, entrance of the mediating variable into the model can reveal a suppressed relationship. The power to detect suppression effects is similar to unsuppressed mediation. These results and their methods provide important information about the power of mediation models for study planning. Of greater importance is that the methods lay the groundwork for assessment of statistical power of more complicated models involving multiple mediators and moderators.
We study causality between bivariate curve time series using the Granger causality generalized measures of correlation. With this measure, we can investigate which curve time series Granger-causes the other; in turn, it helps determine the predictability of any two curve time series. Illustrated by a climatology example, we find that the sea surface temperature Granger-causes the sea-level atmospheric pressure. Motivated by a portfolio management application in finance, we single out those stocks that lead or lag behind Dow-Jones industrial averages. Given a close relationship between S&P 500 index and crude oil price, we determine the leading and lagging variables.
A new time series bootstrap scheme, the time frequency toggle (TFT)-bootstrap, is proposed. Its basic idea is to bootstrap the Fourier coefficients of the observed time series, and then to back-transform them to obtain a bootstrap sample in the time domain. Related previous proposals, such as the surrogate data approach, resampled only the phase of the Fourier coefficients and thus had only limited validity. By contrast, we show that the appropriate resampling of phase and magnitude, in addition to some smoothing of Fourier coefficients, yields a bootstrap scheme that mimics the correct second-order moment structure for a large class of time series processes. As a main result we obtain a functional limit theorem for the TFT-bootstrap under a variety of popular ways of frequency domain bootstrapping. Possible applications of the TFT-bootstrap naturally arise in change-point analysis and unit-root testing where statistics are frequently based on functionals of partial sums. Finally, a small simulation study explores the potential of the TFT-bootstrap for small samples showing that for the discussed tests in change-point analysis as well as unit-root testing, it yields better results than the corresponding asymptotic tests if measured by size and power.
We introduce the notion of intensity reweighted moment pseudostationary point processes on linear networks. Based on arbitrary general regular linear network distances, we propose geometrically correct
We study the critical probability for the metastable phase transition of the two-dimensional anisotropic bootstrap percolation model with $(1,2)$-neighbourhood and threshold $r = 3$. The first order asymptotics for the critical probability were recently determined by the first and second authors. Here we determine the following sharp second and third order asymptotics: [ p_cbig( [L]^2,mathcal{N}_{(1,2)},3 big) ; = ; frac{(log log L)^2}{12log L} , - , frac{log log L , log log log L}{ 3log L} + frac{left(log frac{9}{2} + 1 pm o(1) right)log log L}{6log L}. ] We note that the second and third order terms are so large that the first order asymptotics fail to approximate $p_c$ even for lattices of size well beyond $10^{10^{1000}}$.
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