Estimation of the variance of partial sums of dependent processes

Abstract

Abstract. We study subsampling estimators for the limit varianceof partial sums of a stationary stochastic process (X k ) k≥1 . We establish L2-consistency of a non-overlapping block resampling method. Our results apply to processes that can be represented as functionals of strongly mixing processes. Motivated by recent applications to rank tests, we also study estimators for the series Var(F (X1))+2, where F is the distribution function of X1. Simulations illustrate the usefulness of the proposed estimators and of a mean squared error optimal rule for the choice of the block length.

A robust method for shift detection in time series

“…is a Brownian bridge. Finally, by Lemma A5, Lemma A6 and Theorem 1.2 of Dehling et al (2013),û k,n /σ n → u(0)/σ in probability, which completes the proof also of Corollary 1.…”

“…, X n . Dehling et al (2013) have established consistency of the non-overlapping version of this estimator under the same assumptions as made here if the block length l = l n → ∞ fulfills l n = o(n −1/2 ).…”

“…For estimation of the respective long-run variances needed for standardization of the test statistics, we follow Carlstein (1986) and Dehling et al (2013), who suggest overlapping subsampling with Carlstein's adaptive block length l n = max[⌈n 1/3 {2φ/(1 − φ 2 )} 2/3 ⌉, 1] depending on the lag-one sample autocorrelationφ. In case of the Wilcoxon and the Hodges-Lehmann change-point tests we use Spearman's rank autocorrelation for this, while it is the ordinary sample autocorrelation of the possibly transformed data in case of the ordinary and the Huberization cumulative sums test.…”

Tests for Scale Changes Based on Pairwise Differences

“…We have proposed estimators based on HAC kernel estimation, which is common in the change-point context. An alternative estimation technique is block subsampling, see, e.g., Dehling et al (2013) and the references therein. An entirely different approach, which avoids any unknown scaling constants in the limit distribution of the test statistic is the self-normalization approach as proposed by Shao and Zhang (2010).…”

A clearing system with impatient passengers: asymptotics and estimation in a bus stop model

“…For practical applications, for example, the calculation of confidence intervals or the testing of hypotheses concerning B(x), we need to estimate the variance σ 2 (x). A standard way to do this is by subsampling estimators (see for example [2]). Consistency of such estimators for α-mixing sequences X n , n ≥ 1 (here X n = 1 {min[R n−1 ,D n−1 ]≤x} ) was already shown in [1].…”