A general resampling scheme for triangular arrays of \(\alpha\)-mixing random variables with application to the problem of spectral density estimation.

*(English)*Zbl 0776.62070The authors study bootstrap and jackknife resampling for estimation of a parameter \(\mu\) of the joint distribution of a dependent process. They introduce the ‘block of blocks’ technique: Given the observations \(X_ 1,\dots,X_ N\), new random variables \(T_ 1,\dots,T_ Q\) are introduced which are functions \(\phi_ M(X_ j,X_{j+1},\dots,X_{j+M-1})\) of sliding blocks of \(M\) consecutive observations. Assume that \(\overline T_ N=(T_ 1+\cdots+T_ Q)\) is an asymptotically normal and consistent estimator of \(\mu\). Then for the bootstrap again blocks of consecutive \(T_ i\)’s are formed and one resamples from these blocks. In this way the bootstrap sample \(T^*_ 1,\dots,T^*_ l\) is obtained, and the distribution of its arithmetic means is taken as bootstrap estimate of the distribution of \(\overline T_ N\).

The authors prove the validity of this method under certain technical assumptions. Also a similar jackknife procedure is introduced. The novel feature of the paper is that the length \(M\) of the initial blocks may grow with the sample size. This allows an application to the estimation of parameters that are functions of the joint distribution of the entire process, like the spectral density.

The authors prove the validity of this method under certain technical assumptions. Also a similar jackknife procedure is introduced. The novel feature of the paper is that the length \(M\) of the initial blocks may grow with the sample size. This allows an application to the estimation of parameters that are functions of the joint distribution of the entire process, like the spectral density.

Reviewer: H.Dehling (Groningen)

##### MSC:

62M15 | Inference from stochastic processes and spectral analysis |

62G09 | Nonparametric statistical resampling methods |

62M10 | Time series, auto-correlation, regression, etc. in statistics (GARCH) |