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 \(t_n(\hat{\phi }_{n_1+n_2})\):

$$\begin{aligned} t_n(\hat{\phi }_{n_1+n_2}) = \sum ^{n_1+n_2}_{i=n_1+1} [(\varepsilon _i-\mathbf {X}_i'(\phi ^0_1-\phi ^0_2+ \mathbf C n^{-(u-v-\delta )/2}))^2-(\varepsilon _i-\mathbf {X}_i'(\phi ^0_1-\phi ^0_2))^2 ] \end{aligned}$$
$$\begin{aligned} =n^{-(u-v-\delta )} \sum ^{n_1+n_2}_{i=n_1+1} (\mathbf {X}_i' \mathbf C )^2 - 2 n^{-(u-v-\delta )/2} \sum ^{n_1+n_2}_{i=n_1+1} \mathbf {X}_i' \mathbf C (\varepsilon _i-\mathbf {X}_i'(\phi ^0_1-\phi ^0_2)). \end{aligned}$$
(T1)

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:

$$\begin{aligned} n^{-(u-v-\delta )} \sum ^{n_1+n_2}_{i=n_1+1} (\mathbf {X}_i' \mathbf C )^2= n^{-(u-v-\delta )} O(n_2)=O(n^{-(u-2v-\delta )})=o(1) . \end{aligned}$$
(T2)

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

$$\begin{aligned} n^{-(u-v-\delta )/2} \sum ^{n_1+n_2}_{i=n_1+1} \mathbf {X}_i' \mathbf C \varepsilon _i = n^{-(u-v-\delta )/2} O_{I\!\!P}(n^{v/2}) =O_{I\!\!P}(n^{-(u-2v-\delta )/2})=o_{I\!\!P}(1) \end{aligned}$$
(T3)

and

$$\begin{aligned}&n^{-(u-v-\delta )/2} \sum ^{n_1+n_2}_{i=n_1+1} \mathbf {X}_i' \mathbf C \mathbf {X}_i' (\phi ^0_1-\phi ^0_2)=n^{-(u-v-\delta )/2} O(n_2)\\&\quad =O(n^{-(u-3v-\delta )/2})=o(1) . \end{aligned}$$
(T4)

Relations (T1), (T2), (T3), (T4) imply that \(|t_n(\hat{\phi }_{n_1+n_2})| =o_{I\!\!P}(1)\).