A Bootstrap Approach to Testing for Time-Variability of AR Process Coefficients in Regression Time Series with t-Distrib
In this paper, we intend to test whether the random deviations of an observed regression time series with unknown regression coefficients can be described by a covariance-stationary autoregressive (AR) process, or whether an AR process with time-variable
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Abstract
In this paper, we intend to test whether the random deviations of an observed regression time series with unknown regression coefficients can be described by a covariance-stationary autoregressive (AR) process, or whether an AR process with time-variable (say, linearly changing) coefficients should be set up. To account for possibly present multiple outliers, the white noise components of the AR process are assumed to follow a scaled (Student) t-distribution with unknown scale factor and degree of freedom. As a consequence of this distributional assumption and the nonlinearity of the estimator, the distribution of the test statistic is analytically intractable. To solve this challenging testing problem, we propose a Monte Carlo (MC) bootstrap approach, in which all unknown model parameters and their joint covariance matrix are estimated by an expectation maximization algorithm. We determine and analyze the power function of this bootstrap test via a closed-loop MC simulation. We also demonstrate the application of this test to a real accelerometer dataset within a vibration experiment, where the initial measurement phase is characterized by transient oscillations and modeled by a time-variable AR process. Keywords
Bootstrap test EM algorithm Monte Carlo simulation Regression time series Scaled t-distribution Time-variable autoregressive process
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Introduction
Reliable and precise estimation of geodetic time series models remains a challenging task as they frequently involve huge numbers of auto-correlated and outlier-afflicted measurements. On the one hand, a parsimonious model that allows both for the description and the estimation of autocorrelations is given by autoregressive (AR) processes (cf. Schuh 2003). On the other hand, a flexible approach to modeling multiple outliers (or more generally a heavy-tailed error H. Alkhatib () · M. Omidalizarandi Geodetic Institute, Leibniz University Hannover, Hannover, Germany e-mail: [email protected] B. Kargoll Institut für Geoinformation und Vermessung Dessau, Anhalt University of Applied Sciences, Dessau-Roßlau, Germany
law) is enabled by the assumption that the random deviations follow a scaled t-distribution (cf. Koch and Kargoll 2013). Since adjustment techniques based on least squares are sensitive to misspecifications of the functional and stochastic observation model (cf. Kutterer 1999), as well as sensitive to outliers (cf. Baarda 1968), frequently encountered data features such as functional non-linearity, colored measurement noise and heavy-tailed error distribution should be adequately taken into account. Modern geodetic sensors often involve a data sampling at a high rate, thus producing significantly auto-correlated noise (cf. Kuhlmann 2001), in potentially huge numbers of observations. In such cases, the use of a covariance matrix easily exceeds the memory of the computer. Instead, an AR process can often be used for modeling (auto-)correlations more parsimoniously (cf. Schuh 2003). Moreover, the error law of geodetic me
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