Corrigendum: A Self-Normalized Approach to Confidence Interval Construction in Time Series
Xiaofeng Shao · Journal of the Royal Statistical Society Series B (Statistical Methodology) · 2010
J. R. Statist. Soc. B, 72 (2010), 343–366 Assumptions 1 and 2 in the paper are not sufficient to ensure the validity of theorem 1. Assumption 2 on page 346 should be replaced by the following. Assumption 2. Assume that RN = op(N−1/2) and N−2Σt=1NtRt2=op(1). Then theorem 1 holds under assumption 1 and this assumption 2. The problem lies in the derivation of the limiting distribution of WN. Let Note that It is easy to see that I1N→D ΔVqΔ′ follows from assumption 1 and the continuous mapping theorem. Hence it suffices to show that I3N = op(1), as it implies that I2N = op(1) by the Cauchy–Schwarz inequality. It can be shown that the relationship I3N = op(1) is directly implied by assumption 2, but not by the original assumption 2 in the paper, because the following statement is in general false. Let ZkNk=1N be a sequence of random variables with ZkN = op(1) uniformly in k = 1,…, N (i.e. supk=1,…,NPZkN>ε→0 as N → ∞ for any ε > 0). Then N−1Σk=1NZkN=op(1). Consequently, to verify assumption 2 in remark 1 on page 346 for the smooth function model, we need to impose the assumption that Yt∈L4 and the absolute summability condition on the jth-order cumulants of Yt, j = 2,3,4. Furthermore, the assumptions on Rtnt=1n and TBnρBnt−θ^tt=1n in theorem 2 on page 351 should be changed to Rnn = op(n−1/2) and n−2Σt=1ntRtn2=op(1) and TBnρBnn−θ^n=opn−1/2 and n−2∑t=1n|tTBnρBnt−θ^t|2=op(1) respectively. A revised version of the manuscript can be found at http://arxiv.org/PS_cache/arxiv/pdf/1005/1005.2137v1.pdf.