• Communications for Statistical Applications and Methods
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• v.30 no.2
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• pp.149-162
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• 2023

The ℓ₁-trend filtering method is one of the most widely used methods for extracting underlying trends from noisy observations. Contrary to the Hodrick-Prescott filtering, the ℓ₁-trend filtering gives piecewise linear trends. One of the advantages of the ℓ₁-trend filtering is that it can be used for identifying change points in piecewise linear trends. However, since the ℓ₁-trend filtering employs total variation as a penalty term, estimated piecewise linear trends tend to be biased. In this study, we demonstrate the biasedness of the ℓ₁-trend filtering in trend level estimation and propose a two-stage bias-reduction procedure. The newly suggested estimator is based on the estimated change points of the ℓ₁-trend filtering. Numerical examples illustrate that the proposed method yields less biased estimates for piecewise linear trends.

• The Korean Journal of Applied Statistics
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• v.35 no.3
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• pp.371-384
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• 2022

The fused LASSO signal approximator (FLSA) can be applied to find change points from the data having piecewise constant mean structure. It is well-known that the FLSA is inconsistent in change points detection. This inconsistency is due to a total-variation denoising penalty of the FLSA. ℓ₁ trend filter, one of the popular tools for finding an underlying trend from data, can be used to identify change points of piecewise linear trends. Since the ℓ₁ trend filter applies the sum of absolute values of slope differences, it can be inconsistent for change points recovery as the FLSA. However, there are few studies on the inconsistency of the ℓ₁ trend filtering. In this paper, we demonstrate the inconsistency of the ℓ₁ trend filtering with a numerical study.