4,090 research outputs found

### Consistency of the jackknife-after-bootstrap variance estimator for the bootstrap quantiles of a studentized statistic

Efron [J. Roy. Statist. Soc. Ser. B 54 (1992) 83--111] proposed a
computationally efficient method, called the jackknife-after-bootstrap, for
estimating the variance of a bootstrap estimator for independent data. For
dependent data, a version of the jackknife-after-bootstrap method has been
recently proposed by Lahiri [Econometric Theory 18 (2002) 79--98]. In this
paper it is shown that the jackknife-after-bootstrap estimators of the variance
of a bootstrap quantile are consistent for both dependent and independent data.
Results from a simulation study are also presented.Comment: Published at http://dx.doi.org/10.1214/009053605000000507 in the
Annals of Statistics (http://www.imstat.org/aos/) by the Institute of
Mathematical Statistics (http://www.imstat.org

### Asymptotic expansions for sums of block-variables under weak dependence

Let $\{X_i\}_{i=-\infty}^{\infty}$ be a sequence of random vectors and
$Y_{in}=f_{in}(\mathcal{X}_{i,\ell})$ be zero mean block-variables where
$\mathcal{X}_{i,\ell}=(X_i,...,X_{i+\ell-1}),i\geq 1$, are overlapping blocks
of length $\ell$ and where $f_{in}$ are Borel measurable functions. This paper
establishes valid joint asymptotic expansions of general orders for the joint
distribution of the sums $\sum_{i=1}^nX_i$ and $\sum_{i=1}^nY_{in}$ under weak
dependence conditions on the sequence $\{X_i\}_{i=-\infty}^{\infty}$ when the
block length $\ell$ grows to infinity. In contrast to the classical Edgeworth
expansion results where the terms in the expansions are given by powers of
$n^{-1/2}$, the expansions derived here are mixtures of two series, one in
powers of $n^{-1/2}$ and the other in powers of $[\frac{n}{\ell}]^{-1/2}$.
Applications of the main results to (i) expansions for Studentized statistics
of time series data and (ii) second order correctness of the blocks of blocks
bootstrap method are given.Comment: Published at http://dx.doi.org/10.1214/009053607000000190 in the
Annals of Statistics (http://www.imstat.org/aos/) by the Institute of
Mathematical Statistics (http://www.imstat.org

### Edgeworth expansions for studentized statistics under weak dependence

In this paper, we derive valid Edgeworth expansions for studentized versions
of a large class of statistics when the data are generated by a strongly mixing
process. Under dependence, the asymptotic variance of such a statistic is given
by an infinite series of lag-covariances, and therefore, studentizing factors
(i.e., estimators of the asymptotic standard error) typically involve an
increasing number, say, $\ell$ of lag-covariance estimators, which are
themselves quadratic functions of the observations. The unboundedness of the
dimension $\ell$ of these quadratic functions makes the derivation and the form
of the expansions nonstandard. It is shown that in contrast to the case of the
studentized means under independence, the derived Edgeworth expansion is a
superposition of three distinct series, respectively, given by one in powers of
$n^{-1/2}$, one in powers of $[n/\ell]^{-1/2}$ (resulting from the standard
error of the studentizing factor) and one in powers of the bias of the
studentizing factor, where $n$ denotes the sample size.Comment: Published in at http://dx.doi.org/10.1214/09-AOS722 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org

### Resampling methods for spatial regression models under a class of stochastic designs

In this paper we consider the problem of bootstrapping a class of spatial
regression models when the sampling sites are generated by a (possibly
nonuniform) stochastic design and are irregularly spaced. It is shown that the
natural extension of the existing block bootstrap methods for grid spatial data
does not work for irregularly spaced spatial data under nonuniform stochastic
designs. A variant of the blocking mechanism is proposed. It is shown that the
proposed block bootstrap method provides a valid approximation to the
distribution of a class of M-estimators of the spatial regression parameters.
Finite sample properties of the method are investigated through a moderately
large simulation study and a real data example is given to illustrate the
methodology.Comment: Published at http://dx.doi.org/10.1214/009053606000000551 in the
Annals of Statistics (http://www.imstat.org/aos/) by the Institute of
Mathematical Statistics (http://www.imstat.org

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