Subsampling for Non-stationary Time Series with Long Memory and Heavy Tails Using Weak Dependence Condition

• Published in: Cyclostationarity: Theory and Methods III Vol. 6; pp. 17 - 34
• Main Authors: Gajecka-Mirek, Elżbieta, Knapik, Oskar
• Format: Book Chapter
• Language: English
• Published: Switzerland Springer International Publishing AG 2017; Springer International Publishing
• Series: Applied Condition Monitoring
• Subjects: Asymptotic Distribution; Central Limit Theorem; Dynamics & vibration; Engines & power transmission; Heavy Tail; Imaging systems & technology; Time Series; Weak Dependence
• ISBN: 9783319514444; 331951444X
• ISSN: 2363-698X; 2363-6998
• DOI: 10.1007/978-3-319-51445-1_2

Statistical inference for unknown distributions of statistics or estimators may be based on asymptotic distributions. Unfortunately, in the case of dependent data the structure of such statistical procedures is often ineffective. In the last three decades we can observe an intensive development the of so-called resampling methods. Using these methods, it is possible to directly approximate the unknown distributions of statistics and estimators. A problem that needs to be solved during the study of the resampling procedures is the consistency. Their consistency for independent or stationary observations has been extensively studied. Resampling for time series with a specific non-stationarity, i.e. the periodic and almost periodic strong mixing dependence structure also been the subject of research. Recent research on resampling methods focus mainly on the time series with the weak dependence structure, defined by Paul Doukhan, Louhichi et al. and simultaneously Bickel and B $$\ddot{u}$$ hlmann. In the article a time series model with specific features i.e.: long memory, heavy tails (with at least a fourth moment, e.g.: GED, t-Student), weak dependence and periodic structure is presented. and the estimator of the mean function in the above-mentioned time series is investigated. In the article the necessary central limit theorems and consistency theorems for the mean function estimator (for one of the resampling techniques—the subsampling) are proven.