Abstract
Linear relations, containing measurement errors in input and output data, are considered. Parameters of these so-called errors-in-variables models can change at some unknown moment. The aim is to test whether such an unknown change has occurred or not. For instance, detecting a change in trend for a randomly spaced time series is a special case of the investigated framework. The designed changepoint tests are shown to be consistent and involve neither nuisance parameters nor tuning constants, which makes the testing procedures effortlessly applicable. A changepoint estimator is also introduced and its consistency is proved. A boundary issue is avoided, meaning that the changepoint can be detected when being close to the extremities of the observation regime. As a theoretical basis for the developed methods, a weak invariance principle for the smallest singular value of the data matrix is provided, assuming weakly dependent and non-stationary errors. The results are presented in a simulation study, which demonstrates computational efficiency of the techniques. The completely data-driven tests are illustrated through problems coming from calibration and insurance; however, the methodology can be applied to other areas such as clinical measurements, dietary assessment, computational psychometrics, or environmental toxicology as manifested in the paper.
Highlights
Introduction and Main AimsIf measured input and output data are supposed to be in some linear relations, it is of particular interest to detect whether impact of the input characteristics has changed over time on the output observables
There is a vast literature aimed at linear relations modeled through so-called measurement error models or errors-in-variables models, but very little has been explored in the changepoint analysis for these models yet
We have proposed two tests for changepoints with desirable theoretical properties: The asymptotic size of the tests is guaranteed by a limit theorem even under non-stationarity and weak dependency; the tests and the related changepoint estimator are consistent
Summary
If measured input and output data are supposed to be in some linear relations, it is of particular interest to detect whether impact of the input characteristics has changed over time on the output observables. It becomes even more challenging to handle measurement errors in input and output data simultaneously, when the linear relations are subject to change at some unknown time point—changepoint. A change in the variance parameter of the normally distributed errors within the measurement error models was investigated in [14]. All of these mentioned contributions dealt with the changepoint estimation solely. Our main goal is to test for a possible change in the parameters relating the input and output data, both encumbered by some errors. To that, our changepoint tests are supposed to be nuisance-parameter-free, distributional-free, and to allow for very general error structures
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