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The use of subseries values for estimating the variance of a general statistic from a stationary sequence. (English) Zbl 0602.62029

Let \(t_ n=t(Z_ 1,...,Z_ n)\) be a statistic from a stationary sequence, such that \(Var(t_ n)\to \sigma^ 2\) as \(n\to \infty\). Under a mixing condition on the Z-sequence, it is shown that the sample variance of the statistics \[ t_{m_ n}^{im_ n}=t(Z_{im_ n+1},...,Z_{(i+1)m_ n}),\quad 1\leq i\leq [n/m_ n], \] is consistent for \(\sigma^ 2\) whenever \(m_ n\to \infty\). The optimal length \(m_ n\) is derived in the case of first-order autoregressive schemes.
Reviewer: W.Stute

MSC:

62G05 Nonparametric estimation
60G10 Stationary stochastic processes
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