Robust estimation of the scale and of the autocovariance function of Gaussian short- and long-range dependent processes

Lévy‐Leduc, Céline; Boistard, Hélène; Moulines, Éric; Taqqu, Murad S.; Reisen, Valdério Anselmo

doi:10.1111/j.1467-9892.2010.00688.x

Cited by 36 publications

(51 citation statements)

References 29 publications

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“…In the case of extreme long-range dependence, we prove a non-Gaussian limit result for the IQR, consistent with results found previously for the sample standard deviation and Q n . In contrast with the results of Lévy-Leduc et al [2] for a single U-quantile, namely Q n , the proof for the IQR, a difference of two quantiles, relies on the higher-order [3] Quantile based estimation of scale and dependence 175 terms in the Bahadur representation of Wu [9]. Simulation suggests that an equivalent result holds for P n ; we state the conjectured result which will require the analogous Bahadur representation for U-quantiles under long-range dependence.…”

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confidence: 56%

“…A reliable estimate of the scale of the residuals from a regression model is often of interest, whether it be parametrically estimating confidence intervals, determining a goodness-of-fit measure, performing model selection, or identifying G. Tarr [2] unusual observations. The robustness of quantile regression parameter estimates to y-outliers does not extend to the error distribution.…”

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confidence: 99%

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Quantile Based Estimation of Scale and Dependence

Tarr¹

2015

Bull. Aust. Math. Soc.

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The sample quantile has a long history in statistics. The aim of this thesis is to explore some further applications of quantiles as simple, convenient and robust alternatives to classical procedures. The first application we consider is estimating confidence intervals for quantile regression coefficients, however, the core of this thesis is the development of a new, quantile based, robust scale estimator and its extension to autocovariance estimation in the time series setting and precision matrix estimation in the multivariate setting.Chapter 1 addresses the need for reliable confidence intervals for quantile regression coefficients, particularly in small samples. The existing methods for constructing confidence intervals tend to be based on complex asymptotic arguments and little is known about their finite sample performance. We consider taking xy-pair bootstrap samples and calculating the corresponding quantile regression coefficient estimates for each sample. Instead of estimating a covariance matrix based on these bootstrap samples, our approach is to take the appropriate upper and lower quantiles of the bootstrap sample estimates as the bounds of the confidence interval. The resulting confidence interval estimate is not necessarily symmetric, only covers admissible parameter values and is shown to have good coverage properties. This work demonstrates the competitive performance of our quantile based approach in a broad range of model designs with a focus on small and moderate sample sizes. These results were published in [5].A reliable estimate of the scale of the residuals from a regression model is often of interest, whether it be parametrically estimating confidence intervals, determining a goodness-of-fit measure, performing model selection, or identifying

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confidence: 99%

Quantile Based Estimation of Scale and Dependence

Tarr¹

2015

Bull. Aust. Math. Soc.

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“…This robust estimator is based on the robust scale estimator proposed by Rousseeuw and Croux [28] which has already been widely used in different frameworks of non periodic correlated processes, see for instance [23,12,19]. The main contribution of this section is to provide central limit theorems for the robust scale and PeACV estimators when the underlying time series follows a PAR process.…”

Section: Robust Procedures To Estimate Parameters and Select A Modelmentioning

confidence: 99%

“…, X n ). As explained in [19], a robust estimator of the autocovariance function of x can be achieved by using a robust scale estimator, hereafter denoted by Q n (x). Let the functionals T 1 and T 2 be defined as follows:…”

Section: Estimation Of Peacv Based On a Robust Scale Functionmentioning

confidence: 99%

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Robust estimation of periodic autoregressive processes in the presence of additive outliers

Sarnaglia

Reisen

Lévy‐Leduc

2010

Journal of Multivariate Analysis

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a b s t r a c tThis paper suggests a robust estimation procedure for the parameters of the periodic AR (PAR) models when the data contains additive outliers. The proposed robust methodology is an extension of the robust scale and covariance functions given in, respectively, Rousseeuw and Croux (1993) [28], and Ma and Genton (2000) [23] to accommodate periodicity. These periodic robust functions are used in the Yule-Walker equations to obtain robust parameter estimates. The asymptotic central limit theorems of the estimators are established, and an extensive Monte Carlo experiment is conducted to evaluate the performance of the robust methodology for periodic time series with finite sample sizes. The quarterly Fraser River data was used as an example of application of the proposed robust methodology. All the results presented here give strong motivation to use the methodology in practical situations in which periodically correlated time series contain additive outliers.

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An Overview of Robust Spectral Estimators

Reisen

Lévy‐Leduc

Cotta

et al. 2019

Applied Condition Monitoring

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The periodogram function is widely used to estimate the spectral density of time series processes and it is well-known that this function is also very sensitive to outliers. In this context, this paper deals with robust estimation functions to estimate the spectral density of univariate and periodic time series with short and long-memory properties. The two robust periodogram functions discussed and compared here were previously explicitly and analytically derived in Fajardo et al. (2018), Reisen et al. (2017) and Fajardo et al. (2009 in the case of long-memory processes. The first two references introduce the robust periodogram based on M -regression estimator. The third reference is based on the robust autocovariance function introduced in Ma and Genton (2000) and studied theoretically and empirically in Lévy-Leduc et al. (2011). Here, the theoretical results of these estimators are discussed in the case of short and long-memory univariate time series and periodic processes. A special attention is given to the M -periodogram for short-memory processes. In this case, Theorem 1 and Corollary 1 derive the asymptotic distribution of this spectral estimator. As the application of the methodologies, robust estimators for the parameters of AR, ARFIMA and PARMA processes are discussed. Their finite sample size properties are addressed and compared in the context of absence and presence of atypical observations. Therefore, the contributions of this paper come to fill some gaps in the literature of modeling univariate and periodic time series to handle additive outliers.

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Robust estimation of the scale and of the autocovariance function of Gaussian short- and long-range dependent processes

Cited by 36 publications

References 29 publications

Quantile Based Estimation of Scale and Dependence

Quantile Based Estimation of Scale and Dependence

Robust estimation of periodic autoregressive processes in the presence of additive outliers

An Overview of Robust Spectral Estimators

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