Abstract
A gap was noticed in the proof of Theorem 4.2 of Kim and Kim [1]. The purpose of this corrigendum is to rectify the argument. Specifically, it was discovered that in the proof of Lemma 4.1, one more term – in addition to those in (i)’ and (ii)’ – needed to be considered for H5(θ,τ),H6(θ,γ), and H7(θ,τ,γ), and thus the proof of Lemma 4.1 was incomplete. Theorem 2.2, which was used to establish (i)’ and (ii)’ in the proof of Lemma 4.1, is insufficient to derive the asymptotic behavior of the additional term. The limiting behavior of all terms can now be established in terms of τ as in the proof of revised Lemma 4.1 below, using similar arguments as in the proofs of Lemmas 4.3–4.5 of [1]. In this corrigendum, we present a new version of Lemma 4.1 and minor modifications are made to Lemmas 4.3–4.5, namely a change of constant in Lemma 4.3 and editorial changes in the presentation of Lemmas 4.4 and 4.5. Using Lemmas 4.1 and 4.3–4.5, we first prove Theorem 4.6, which states the Op(1/n) rate of convergence of τˆ (or γˆ), and then use Theorem 4.6 to complete the proof of Theorem 4.2. Thus, the order in which results are presented here differs from [1]. Lemma 4.1 could have been stated first but we chose to state it after Lemma 4.5, mainly because its new proof calls on a similar method used in the proofs of Lemmas 4.3–4.5 in the original paper. To prove the Op(1/n) rate of convergence of τˆ (or γˆ), we need to carefully examine the terms with τ (or γ) in ∑ig(ξ,xi,ei)/n. Here, we consider only the case where τ>τ(0), and similar arguments can be applied to get the same result in other cases. Let H∗∗(θ,γ)=H1(θ)+H3(γ)+H6(θ,γ), so that (1)∑ig(ξ,xi,ei)/n=H2(τ)+H4(τ,γ)+H5(θ,τ)+H7(θ,τ,γ)+H∗∗(θ,γ).Note that (1) replaces (6) in Kim and Kim [1]. Let Jτ(θ,τ,γ) be the sum of the terms in (1) that include τ, i.e., (2)Jτ(θ,τ,γ)=H2(τ)+H4(τ,γ)+H5(θ,τ)+H7(θ,τ,γ)=∑i∑k=01{(θ0k(0)−θ1k(0))⊤xi}21Uk+1(xi)/n−2∑i∑k=01ei(θ0k(0)−θ1k(0))⊤xi1Uk+1(xi)/n+H4(τ,γ)+H5(θ,τ)+H7(θ,τ,γ). Lemmas 4.3–4.5 in [1], with minor modifications, deal with the first three terms on the righthand side of (2), and Lemma 4.1 below deals with the last two terms. The new and revised proofs of Lemma 4.1, Theorems 4.6 and 4.2 are given below, and the proof of Lemma 4.3 can be completed with the modified constant. In Lemma 4.3, we need the discontinuity condition described in Assumption 2. For the lemmas and theorems below, let Aτ(δ)={τ:τ(0)<τ≤τ(0)+δ}, Aγ(δ)={γ:γ(0)<γ≤γ(0)+δ}, and Aθ(δ)={θ:‖θ−θ(0)‖≤δ} for δ>0. Also let Dnτ(M)={τ:0<τ−τ(0)≤Mn−1} and Anτ(M,δ)={τ:Mn−1<τ−τ(0)≤δ} for δ>0 and M<∞. Lemma 4.3 Under Assumptions 1 and 2, for anyϵ>0, there existζ>0,δ>0, andM<∞ such that for largen, Pr∃τ∈Anτ(M,δ):∑i∑k=01{(θ0k(0)−θ1k(0))⊤xi}21Uk+1(xi)≤4ζ(τ−τ(0))n<ϵ. Lemma 4.4 Under Assumption 1, for any givenϵ>0,δ>0, andζ>0, there existsM<∞ such that for largen, Pr∃τ∈Anτ(M,δ):2∑iei∑k=01(θ0k(0)−θ1k(0))⊤xi1Uk+1(xi)>ζ(τ−τ(0))n<ϵ. Lemma 4.5 Under Assumption 1, for any givenϵ>0,ζ>0, there existsδ>0 such that for largen, Pr∃(τ,γ)∈Aτ(δ)×Aγ(δ):H4(τ,γ)>ζ(τ−τ(0)+1/n)<ϵ. Lemma 4.1 Under Assumption 1, for any givenϵ>0 andζ>0, there existsδ>0 such that for largen, Pr∃(θ,τ,γ)∈Aθ(δ)×Aτ(δ)×Aγ(δ):H5(θ,τ)+H7(θ,τ,γ)>ζ(τ−τ(0)+1/n)<ϵ. Proof First, we consider H5(θ,τ) and show that for any given ϵ>0 and ζ>0, there exists δ>0 such that for large n, Pr∃(θ,τ)∈Aθ(δ)×Aτ(δ):H5(θ,τ)>12ζ(τ−τ(0)+1/n)<ϵ/2.For that, we decompose Aτ(δ) into the union of two disjoint parts as Aτ(δ)=Dnτ(1)∪Anτ(1,δ), which implies that (3) Pr∃(θ,τ)∈Aθ(δ)×Aτ(δ):H5(θ,τ)>12ζ(τ−τ(0)+1/n)≤Pr∃(θ,τ)∈Aθ(δ)×Anτ(1,δ):H5(θ,τ)>12ζ(τ−τ(0))+Pr∃(θ,τ)∈Aθ(δ)×Dnτ(1):H5(θ,τ)>12ζ(1/n). Note that H5(θ,τ) is the sum of the following terms, where j,k∈{0,1}: (i) ∑i{(θjk−θjk(0))⊤xi}21Uk+1(xi)/n,(ii) ∑i2ei(θjk−θjk(0))⊤xi1Uk+1(xi)/n,(iii) ∑i2(θ0k−θ0k(0))⊤xi(θ0k(0)−θ1k(0))⊤xi1Uk+1(xi)/n.We also note that for all j,k∈{0,1}, and x with a bounded Euclidean norm, supθ∈Aθ(δ)(θjk−θjk(0))⊤xi≤C1δ,for some positive constant C1<∞. Thus, to show that the sum of the probabilities on the righthand side of (3) is less than ϵ/2, it suffices to show that for k∈{0,1}, each of the following four terms is Op(1): supτ∈Anτ(1,δ)∑i1Uk+1(xi)n(τ−τ(0)),supτ∈Anτ(1,δ)∑iei1Uk+1(xi)n(τ−τ(0)),supτ∈Dnτ(1)∑i1Uk+1(xi),supτ∈Dnτ(1)∑iei1Uk+1(xi). For the first two terms, we can get the desired results by following similar arguments as used in the proofs of Lemmas 4.3 and 4.4, and some details regarding the second term are provided below. Let fi(τ)=ei1{x:τ(0)<x1≤τ}(xi)n(τ−τ(0)),for which, by Assumption 1, we have supτ∈Anτ(1,δ)∑iEfi(τ)=O(1).Then, in order to show that the second term is Op(1), it suffices to show that for any given ϵ′>0, there exists an L∈(0,∞) such that (4) Prsupτ∈Anτ(1,δ)∑i{fi(τ)−Efi(τ)}>L<ϵ′.For some constant b>1, let Bn(b,j)={τ:bjn−1<τ−τ(0)≤bj+1n−1}, so that Anτ(1,δ)=∪j=0∞Bn(b,j). Then the probability in (4) is bounded above by (5) ∑j=0∞Prsupτ∈Bn(b,j)∑i{fi(τ)−Efi(τ)}>L.Define Fi(j) as follows, similarly as in the proofs of Lemmas 4.3 and 4.4: supτ∈Bn(b,j)fi(τ)≤ei1{x:τ(0)<x1≤τ(0)+bj+1/n}(xi)bj=Fi(j).Since ∑iEFi2(j)≤C2b1−j for some positive constant C2<∞, it follows from Lemma 2.1 that the probability in (5) is less than or equal to 1L∑j=0∞C3b1−jfor some positive constant C3<∞. By choosing L large enough, we can thus make the probability in (4) less than ϵ′. For the third term, let fi(τ)=1{x:τ(0)<x1≤τ}(xi). Then supτ∈Dnτ(1)∑iEfi(τ)≤C4for some positive constant C4<∞, and supτ∈Dnτ(1)fi(τ)≤1{x:τ(0)<x1≤τ(0)+1/n}(xi)=Fi.Since ∑iEFi2≤C5 for some positive constant C5<∞, we can deduce the desired result from Lemma 2.1. For the fourth term, we proceed similarly with fi(τ)=ei1{x:τ(0)<x1≤τ}(xi). We find Pr∃(θ,τ,γ)∈Aθ(δ)×Aτ(δ)×Aγ(δ):H7(θ,τ,γ)>12ζ(τ−τ(0)+1/n)<ϵ/2,which completes the proof of Lemma 4.1. □ Theorem 4.6 Under Assumptions 1–2, τˆ−τ(0)=Op(1/n). Proof By Lemmas 4.1, 4.3, 4.4 and 4.5, for any given ϵ>0, there exist constants ζ>0, δ>0 and M∈(2,∞) such that for large n, Pr∀(θ,τ,γ)∈Aθ(δ)×Anτ(M,δ)×Aγ(δ):nJτ(θ,τ,γ)>ζ(M−2)>0>1−ϵ.Given that nJτ(θˆ,τˆ,γˆ)≤0=nJτ(θˆ,τ(0),γˆ), we get 0<τˆ−τ(0)≤M/n with a large probability. □ Similarly, we can show that γˆ−γ(0)=Op(1/n). Theorem 4.2 Under Assumptions 1–2, n(θˆn−θ(0))⇝N(0,σ2T−1) asn→∞. Proof Note that θˆ minimizes Ln(θ,τˆ,γˆ)=H1(θ)+H5(θ,τˆ)+H6(θ,γˆ)+H7(θ,τˆ,γˆ). Also, supθ∈Aθ(δ)H5(θ,τˆ)+H6(θ,γˆ)+H7(θ,τˆ,γˆ)=op(1/n),by Lemma 4.1, Theorem 4.6, and similar results for γˆ. Invoking Lemma 3.1, we can then write Ln(θ,τˆ,γˆ)=(θ−θ(0))⊤{T+op(1)}(θ−θ(0))−2{W+op(1)}⊤(θ−θ(0))n+op(1/n), uniformly in θ in Aθ(δ). By following a similar method as used in the proof of Theorem 3.3, we can then show that ‖θˆn−θ(0)−T−1W+op(1)n‖2=op(1/n).This concludes the argument. □
Original language  English (US) 

Pages (fromto)  134137 
Number of pages  4 
Journal  Journal of Multivariate Analysis 
Volume  159 
DOIs 

State  Published  Jul 2017 
ASJC Scopus subject areas
 Statistics and Probability
 Numerical Analysis
 Statistics, Probability and Uncertainty