On higher spectral densities of stationary processes with mixing

Zhurbenko, I. G.; Zuev, N. M.

doi:10.1007/bf01085581

Cited by 10 publications

(6 citation statements)

References 9 publications

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“…Furthermore, since their strong mixing condition with exponential decay rate [cf. condition A.4(3) of He et al (2020)] implies our cumulant assumption 2.1 [see Andrews (1991), Zhurbenko and Zuev (1975)], our overall assumption is weaker than condition A.4 in He et al (2020). Despite the weaker assumptions, our results are stronger, as we derived the asymptotic independence of several stochastic process indexed by q ∈ 2N, which implies the asymptotic independence of U-statistics indexed by q ∈ 2N.…”

Section: Limiting Null Distributionmentioning

confidence: 74%

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Adaptive Inference for Change Points in High-Dimensional Data

Zhang

Wang

Shao

2021

Preprint

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In this article, we propose a class of test statistics for a change point in the mean of high-dimensional independent data. Our test integrates the U-statistic based approach in a recent work by and the L q -norm based high-dimensional test in He et al. (2020), and inherits several appealing features such as being tuning parameter free and asymptotic independence for test statistics corresponding to even qs. A simple combination of test statistics corresponding to several different qs leads to a test with adaptive power property, that is, it can be powerful against both sparse and dense alternatives. On the estimation front, we obtain the convergence rate of the maximizer of our test statistic standardized by sample size when there is one change-point in mean and q = 2, and propose to combine our tests with a wild binary segmentation (WBS) algorithm to estimate the change-point number and locations when there are multiple change-points. Numerical comparisons using both simulated and real data demonstrate the advantage of our adaptive test and its corresponding estimation method.

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Section: Limiting Null Distributionmentioning

confidence: 74%

“…Proposition 2 of Wu and Shao (2004)] or physical dependence measure proposed by Wu (2005) [cf. Section 4 of Shao and Wu (2007)], or α-mixing [Andrews (1991), Zhurbenko and Zuev (1975)] in the time series setting. It basically imposes weak dependence among the p components in the data.…”

Section: Limiting Null Distributionmentioning

confidence: 99%

Adaptive Inference for Change Points in High-Dimensional Data

Zhang

Wang

Shao

2021

Preprint

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“…Moreover under the above conditions we have C z ,,._, u k _ , = -m I~k (~I ,..., U k -l ) l < a for all 2~k~8 (cf. the proof of theorem 1 and theorem 3 of Zhurbenko and Zuev (1975)). The result now follows with the above theorems.…”

Section: Tends Weakly To H Y ( a ) And Z T ( A ) Tends Weakly To H Z mentioning

confidence: 92%

ON THE ASYMPTOTIC DISTRIBUTION OF BARTLETT'S U_p‐STATISTIC

Dahlhaus

1985

Journal Time Series Analysis

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In this paper the asymptotic behaviour of Bartlett's Up-statistic for a goodnessof-fit test for stationary processes, is considered. The asymptotic distribution of the test process is given under the assumption that a central limit theorem for the empirical spectral distribution function holds. It is shown that the Up-statistic tends to the supremum of a tied down Brownian motion. By a counterexample we refute the conjecture that this distribution is in general of the Kolmogorov-Smirnov type. The validity of the central limit theorem for the spectral distribution function is then discussed. Finally a goodness-offit test for ARMA-processes based on the estimated innovation sequence is given, and it is shown that this test statistic is asymptotically Kolmogorov-Smirnov distributed.

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“…Zhurbenko (1979bZhurbenko ( ), (1980c has proved such a central limit theorem for strong mixing processes with exponential decreasing mixing coefficient. If all moments of the process exist this implies the continuity of the spectra of all order (see Zhurbenko and Zuev (1975)). In the following theorem we assume that the higher-order spectra are bounded.…”

Section: The Central Limit Theoremmentioning

confidence: 99%

On a spectral density estimate obtained by averaging periodograms

Dahlhaus

1985

Journal of Applied Probability

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A spectral density statistic obtained by averaging periodograms over overlapping time intervals is considered where the periodograms are calculated using a data window. The asymptotic mean square error of this estimate for scale parameter windows is determined and, as an example, it is shown that the use of the Tukey–Hanning window leads partially to a smaller mean square error than a window suggested by Kolmogorov and Zhurbenko. Furthermore the Tukey–Hanning window is independent of the unknown spectral density, which is not the case for the Kolmogorov–Zhurbenko window. The mean square error of this estimate is also less than the mean square error of commonly used window estimates. Finally, a central limit theorem for the estimate is established.

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On higher spectral densities of stationary processes with mixing

Cited by 10 publications

References 9 publications

Adaptive Inference for Change Points in High-Dimensional Data

Adaptive Inference for Change Points in High-Dimensional Data

ON THE ASYMPTOTIC DISTRIBUTION OF BARTLETT'S U_p‐STATISTIC

On a spectral density estimate obtained by averaging periodograms

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On higher spectral densities of stationary processes with mixing

Cited by 10 publications

References 9 publications

Adaptive Inference for Change Points in High-Dimensional Data

Adaptive Inference for Change Points in High-Dimensional Data

ON THE ASYMPTOTIC DISTRIBUTION OF BARTLETT'S Up‐STATISTIC

On a spectral density estimate obtained by averaging periodograms

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Product

Resources

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ON THE ASYMPTOTIC DISTRIBUTION OF BARTLETT'S U_p‐STATISTIC