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- Publisher
- Springer Journals
- Copyright
- Copyright © Springer Nature B.V. 2020
- ISSN
- 0167-8019
- eISSN
- 1572-9036
- DOI
- 10.1007/s10440-020-00368-0
- Publisher site
- See Article on Publisher Site
Abstract
The paper is concerned with the construction of goodness-of-fit tests for testing a hypothesis H0\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$H_{0}$\end{document} that the hypothetical spectral density of a stationary Gaussian process X(t)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$X(t)$\end{document} has the specified form, based on the tapered data. We show that in the case where the hypothetical spectral density of X(t)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$X(t)$\end{document} does not depend on unknown parameters (the hypothesis H0\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$H_{0}$\end{document} is simple), then the suggested test statistic has a limiting chi-square distribution. In the case where the hypothesis H0\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$H_{0}$\end{document} is composite, that is, the hypothetical spectral density of X(t)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$X(t)$\end{document} depends on an unknown parameter, we choose an appropriate estimator for unknown parameter and describe the limiting distribution of the test statistic. This distribution is similar to that of obtained by Chernov and Lehman (Ann. Math. Stat. 25(3):579–586, 1954) in the case of independent observations.
Journal
Acta Applicandae Mathematicae – Springer Journals
Published: Dec 7, 2020