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1 Erratum to: Stat Papers (2014) 55:349–374 DOI 10.1007/s00362-012-0482-x
The proof of Lemma 3(ii) in the paper of Ciuperca (2014), contains a mistake, although the claim is correct. We are grateful to Fuqi Chen and Dr. Sévérien Nkurunziza, from the University of Windsor, for pointing out the mistake. We give here a corrected proof.
In the proof of Lemma 3(ii) of the paper of Ciuperca, the author proves that \( |t_n(\hat{\phi }_{n_1+n_2})| = o_{I\!\!P}(1)\), using the inequality \(|a^2 -b^2| \le (a-b)^2\). This inequality is wrong. The result \( |t_n(\hat{\phi }_{n_1+n_2})| = o_{I\!\!P}(1)\) can be proved easily otherwise, by elementary and short calculation. By the claim (i) of Lemma 3, we have, with probability 1, that \(\Vert \hat{\phi }_{n_1+n_2}-\phi ^0_1\Vert \le n^{-(u-v-\delta )/2}\). Then \(\hat{\phi }_{n_1+n_2}=\phi ^0_1 + \mathbf C n^{-(u-v-\delta )/2} \), where \(\mathbf C \) is a deterministic bounded \(p\)-vector \(\Vert \mathbf C \Vert < \infty \).
Let us consider the following decomposition for \(t_n(\hat{\phi }_{n_1+n_2})\):
For the first sum of (T1), using assumption (H2) and condition (2) of the paper of Ciuperca, together with the fact \(\Vert \mathbf C \Vert < \infty \), we have:
For the second sum of (T1), using assumptions (H2), (H3) and condition (2) of the paper of Ciuperca, together with the Cauchy-Schwarz inequality and the fact \(\Vert \mathbf C \Vert < \infty \), we have
and
Relations (T1), (T2), (T3), (T4) imply that \(|t_n(\hat{\phi }_{n_1+n_2})| =o_{I\!\!P}(1)\).
References
Ciuperca G (2014) Model selection by LASSO methods in a change-point model. Statistical Papers 55: 349–374. doi:10.1007/s00362-012-0482-x
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The online version of the original article can be found under doi:10.1007/s00362-012-0482-x.
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Ciuperca, G. Erratum to: Model selection by LASSO methods in a change-point model. Stat Papers 55, 1231–1232 (2014). https://doi.org/10.1007/s00362-014-0593-7
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DOI: https://doi.org/10.1007/s00362-014-0593-7