Subsampling the mean of heavy-tailed dependent observations

We determine the exact large-buffer asymptotics for a mixture of light-tailed and heavy-tailed input flows. Earlier studies have found a ‘reduced-load equivalence’ in situations where the peak rate of the heavy-tailed flows plus the mean rate of the light-tailed flows is larger than the service rate. In that case, the workload is asymptotically equivalent to that in a reduced system, which consists of a certain ‘dominant’ subset of the heavy-tailed flows, with the service rate reduced by the mean rate of all other flows. In the present paper, we focus on the opposite case where the peak rate of the heavy-tailed flows plus the mean rate of the light-tailed flows is smaller than the service rate. Under mild assumptions, we prove that the workload distribution is asymptotically equivalent to that in a somewhat ‘dual’ reduced system, multiplied by a certain prefactor. The reduced system now consists of only the light-tailed flows, with the service rate reduced by the peak rate of the heavy-tailed flows. The prefactor represents the probability that the heavy-tailed flows have sent at their peak rate for more than a certain amount of time, which may be interpreted as the ‘time to overflow’ for the light-tailed flows in the reduced system. The results provide crucial insight into the typical overflow scenario.

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A Bias-Reduced Estimator for the Mean of a Heavy-Tailed Distribution with an Infinite Second Moment

SSRN Electronic Journal ◽

10.2139/ssrn.2013083 ◽

2012 ◽

Author(s):

Brahim Brahimi ◽

Djamel Meraghni ◽

Necir Abdelhakim ◽

Yahia Djabrane

Keyword(s):

Second Moment ◽

Heavy Tailed Distribution ◽

The Mean ◽

Heavy Tailed

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Comparison of power of modified Jarque-Bera normality tests and selected tests of normality

Acta Universitatis Agriculturae et Silviculturae Mendelianae Brunensis ◽

10.11118/actaun200856060137 ◽

2008 ◽

Vol 56 (6) ◽

pp. 137-148 ◽

Cited By ~ 1

Author(s):

Luboš Střelec

Keyword(s):

Stock Exchange ◽

Financial Time Series ◽

Simulation Studies ◽

Test Paper ◽

Monthly Average ◽

Financial Time ◽

Uniform Distributions ◽

Source Data ◽

The Mean ◽

Heavy Tailed

The aim of this paper is to modify the classical Jarque-Bera test and the robust Jarque-Bera test of normality. We use the median as an estimator instead of the mean in the classical Jarque-Bera test and in the robust Jarque-Bera test. This leads to the modified Jarque-Bera test and the modified robust Jarque-Bera test. Paper also demonstrates results of simulation studies of power of such tests with the various alternatives – light tailed alternatives as exponential, lognormal and gamma distribution, heavy tailed alternatives as Cauchy, Laplace, t3, t5 and logistic distributions and short tailed alternatives as beta and uniform distributions. These tests of normality are also used for normality testing of selected datasets of financial time series. Source data include logarithmic returns of monthly average prices of Prague stock exchange index PX and monthly average prices of CZK/EUR exchange rate in the period from 2000 to 2007.

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A robust method for shift detection in time series

Biometrika ◽

10.1093/biomet/asaa004 ◽

2020 ◽

Vol 107 (3) ◽

pp. 647-660

Author(s):

H Dehling ◽

R Fried ◽

M Wendler

Keyword(s):

Time Series ◽

Nonparametric Test ◽

Test Statistic ◽

Moment Conditions ◽

Limit Theory ◽

Dependent Observations ◽

U Statistics ◽

Shift Detection ◽

Heavy Tailed ◽

The Stability

Summary We present a robust and nonparametric test for the presence of a changepoint in a time series, based on the two-sample Hodges–Lehmann estimator. We develop new limit theory for a class of statistics based on two-sample U-quantile processes in the case of short-range dependent observations. Using this theory, we derive the asymptotic distribution of our test statistic under the null hypothesis of a constant level. The proposed test shows better overall performance under normal, heavy-tailed and skewed distributions than several other modifications of the popular cumulative sums test based on U-statistics, one-sample U-quantiles or M-estimation. The new theory does not involve moment conditions, so any transform of the observed process can be used to test the stability of higher-order characteristics such as variability, skewness and kurtosis.

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Variance asymptotics and central limit theorems for volumes of unions of random closed sets

Advances in Applied Probability ◽

10.1017/s0001867800011745 ◽

2002 ◽

Vol 34 (03) ◽

pp. 520-539 ◽

Cited By ~ 6

Author(s):

Tomasz Schreiber

Keyword(s):

Central Limit ◽

Borel Measure ◽

Strong Law ◽

Closed Sets ◽

Random Closed Sets ◽

Large Numbers ◽

Multidimensional Ball ◽

Mean Width ◽

The Mean ◽

Heavy Tailed

Let X, X 1, X 2, … be a sequence of i.i.d. random closed subsets of a certain locally compact, Hausdorff and separable base space E. For a fixed normalised Borel measure μ on E, we investigate the behaviour of random variables μ(E \ (X 1 ∪ ∙ ∙ ∙ ∪ X n )) for large n. The results obtained include a description of variance asymptotics, strong law of large numbers and a central limit theorem. As an example we give an application of the developed methods for asymptotic analysis of the mean width of convex hulls generated by uniform samples from a multidimensional ball. Another example deals with unions of random balls in ℝ d with centres distributed according to a spherically-symmetric heavy-tailed law.

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A bias-reduced estimator for the mean of a heavy-tailed distribution with an infinite second moment

Journal of Statistical Planning and Inference ◽

10.1016/j.jspi.2012.12.010 ◽

2013 ◽

Vol 143 (6) ◽

pp. 1064-1081 ◽

Cited By ~ 3

Author(s):

Brahim Brahimi ◽

Djamel Meraghni ◽

Abdelhakim Necir ◽

Djabrane Yahia

Keyword(s):

Second Moment ◽

Heavy Tailed Distribution ◽

The Mean ◽

Heavy Tailed

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Confidence Intervals for the Mean Based on Exponential Type Inequalities and Empirical Likelihood

ISRN Biomathematics ◽

10.1155/2013/765752 ◽

2013 ◽

Vol 2013 ◽

pp. 1-8

Author(s):

Sandra Vucane ◽

Janis Valeinis ◽

George Luta

Keyword(s):

Confidence Intervals ◽

Empirical Likelihood ◽

Exponential Type ◽

Likelihood Method ◽

Interval Length ◽

Simple Extension ◽

Dependent Observations ◽

Special Cases ◽

Coverage Accuracy ◽

The Mean

For independent observations, recently, it has been proposed to construct the confidence intervals for the mean using exponential type inequalities. Although this method requires much weaker assumptions than those required by the classical methods, the resulting intervals are usually too large. Still in special cases, one can find some advantage of using bounded and unbounded Bernstein inequalities. In this paper, we discuss the applicability of this approach for dependent data. Moreover, we propose to use the empirical likelihood method both in the case of independent and dependent observations for inference regarding the mean. The advantage of empirical likelihood is its Bartlett correctability and a rather simple extension to the dependent case. Finally, we provide some simulation results comparing these methods with respect to their empirical coverage accuracy and average interval length. At the end, we apply the above described methods for the serial analysis of a gene expression (SAGE) data example.

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Type I Error Rates and Power Estimates of Selected Parametric and Nonparametric Tests of Scale

Journal of Educational Statistics ◽

10.3102/10769986012001045 ◽

1987 ◽

Vol 12 (1) ◽

pp. 45-61 ◽

Cited By ~ 22

Author(s):

Stephen F. Olejnik ◽

James Algina

Keyword(s):

Type I Error ◽

Nonparametric Tests ◽

Error Rates ◽

Type I ◽

Normal Distributions ◽

Type I Error Rates ◽

Heavy Tailed Distributions ◽

The Mean ◽

Heavy Tailed ◽

Distribution Type

Estimated Type I error rates and power are reported for the Brown-Forsythe, O’Brien, Klotz, and Siegel-Tukey procedures. The effect of aligning the data, by using deviations from group means or group medians, is investigated for the latter two tests. Normal and non-normal distributions, equal and unequal sample-size combinations, and equal and unequal means are investigated for a two-group design. No test is robust and most powerful for all distributions, however, using O’Brien’s procedure will avoid the possibility of a liberal test and provide power almost as large as what would be provided by choosing the most powerful test for each distribution type. Using the Brown-Forsythe procedure with heavy-tailed distributions and O’Brien’s procedure for other distributions will increase power modestly and maintain robustness. Using the mean-aligned Klotz test or the unaligned Klotz test with appropriate distributions can increase power, but only at the risk of increased Type I error rates if the tests are not accurately matched to the distribution type.

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A fluid queue with a finite buffer and subexponential input

Advances in Applied Probability ◽

10.1239/aap/1013540031 ◽

2000 ◽

Vol 32 (1) ◽

pp. 221-243 ◽

Cited By ~ 18

Author(s):

A. P. Zwart

Keyword(s):

Stationary Distribution ◽

Fluid Model ◽

Buffer Size ◽

Finite Buffer ◽

Asymptotic Results ◽

Fluid Queue ◽

Buffer Content ◽

The Mean ◽

Heavy Tailed ◽

Finite Dam

We consider a fluid model similar to that of Kella and Whitt [32], but with a buffer having finite capacity K. The connections between the infinite buffer fluid model and the G/G/1 queue established by Kella and Whitt are extended to the finite buffer case: it is shown that the stationary distribution of the buffer content is related to the stationary distribution of the finite dam. We also derive a number of new results for the latter model. In particular, an asymptotic expansion for the loss fraction is given for the case of subexponential service times. The stationary buffer content distribution of the fluid model is also related to that of the corresponding model with infinite buffer size, by showing that the two corresponding probability measures are proportional on [0,K) if the silence periods are exponentially distributed. These results are applied to obtain large buffer asymptotics for the loss fraction and the mean buffer content when the fluid queue is fed by N On-Off sources with subexponential on-periods. The asymptotic results show a significant influence of heavy-tailed input characteristics on the performance of the fluid queue.

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