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Let Zᵢ: -<i <+} be a strictly stationary -mixing sequence. Without specifying the dependence model giving rise to Zᵢ and without specifying the marginal distribution of Zᵢ, we address the question of variance estimation for a general statistic tₙ=tₙ (Z₁,. . . , Zₙ). For estimating Vartₙ from just the available data (Z₁,. . . , Zₙ) we propose computing subseries values: tₘ (Z₈+₁, Z₈+₂, Z₈+₌), 0 i<i+m n. These subseries values are used as replicates to model the sampling variability of tₙ. In particular, we use adjacent nonoverlapping subseries of length m = mₙ, with m_\ infty and mₙ/n 0. Our variance estimator is just the usual sample variance computed amongst these subseries values (after appropriate standardization). This estimator is shown to be consistent under mild integrability conditions. We present optimal (i. e. , minimum m. s. e. ) choices of mₙ for the special case where tₙ=-Zₙ and Zᵢ is a normal AR (1) sequence. A simulation study is conducted, showing that those same choices of mₙ are effective when tₙ is a robust estimator of location and Zᵢ is subject to contamination.
Edward Carlstein (Mon,) studied this question.