The Annals of Statistics

The BLUE in continuous-time regression models with correlated errors

Holger Dette, Andrey Pepelyshev, and Anatoly Zhigljavsky

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In this paper, the problem of best linear unbiased estimation is investigated for continuous-time regression models. We prove several general statements concerning the explicit form of the best linear unbiased estimator (BLUE), in particular when the error process is a smooth process with one or several derivatives of the response process available for construction of the estimators. We derive the explicit form of the BLUE for many specific models including the cases of continuous autoregressive errors of order two and integrated error processes (such as integrated Brownian motion). The results are illustrated on many examples.

Article information

Ann. Statist., Volume 47, Number 4 (2019), 1928-1959.

Received: October 2017
Revised: May 2018
First available in Project Euclid: 21 May 2019

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62M10: Time series, auto-correlation, regression, etc. [See also 91B84]
Secondary: 62M09: Non-Markovian processes: estimation

Linear regression correlated observations signed measures optimal design BLUE AR processes continuous autoregressive model


Dette, Holger; Pepelyshev, Andrey; Zhigljavsky, Anatoly. The BLUE in continuous-time regression models with correlated errors. Ann. Statist. 47 (2019), no. 4, 1928--1959. doi:10.1214/18-AOS1734.

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Supplemental materials

  • Supplement to “The BLUE in continuous-time regression models with correlated errors”. We exemplarily demonstrate that the covariance matrix of the BLUE for the model (1.1) with observations on the interval can be obtained as a limit of the covariance matrices of the BLUE in the discrete regression model (1.2) with observations at equidistant points and a discrete $\operatorname{AR}(2)$ error process.