计量经济分析 = Econometric analysis

Convergence of Probability Measures. By P. Billingsley. Chichester, Sussex, Wiley, 1968. xii, 253 p. 9 1/4“. 117s.

Convergence of Probability Measures

Stable non-Gaussian random processes , by G. Samorodnitsky and M. S. Taqqu. Pp. 632. £49.50. 1994. ISBN 0-412-05171-0 (Chapman and Hall).

This article surveys bootstrap methods for producing good approximate confidence intervals. The goal is to improve by an order of magnitude upon the accuracy of the standard intervals $\hat{\theta} \pm z^{(\alpha)} \hat{\sigma}$, in a way that allows routine application even to very complicated problems. Both theory and examples are used to show how this is done. The first seven sections provide a heuristic overview of four bootstrap confidence interval procedures: $BC_a$, bootstrap-t , ABC and calibration. Sections 8 and 9 describe the theory behind these methods, and their close connection with the likelihood-based confidence interval theory developed by Barndorff-Nielsen, Cox and Reid and others.

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Bootstrap Confidence Intervals

Data snooping occurs when a given set of data is used more than once for purposes of inference or model selection When such data reuse occurs, there is always the possibility that any satisfactory results obtained may simply be due to chance rather than to any merit inherent in the method yielding the results This problem is practically unavoidable in the analysis of time-series data, as typically only a single history measuring a given phenomenon of interest is available for analysis It is widely acknowledged by empirical researchers that data snooping is a dangerous practice to be avoided, but in fact it is endemic The main problem has been a lack of sufficiently simple practical methods capable of assessing the potential dangers of data snooping in a given situation Our purpose here is to provide such methods by specifying a straightforward procedure for testing the null hypothesis that the best model encountered in a specification search has no predictive superiority over a given benchmark model This permits data snooping to be undertaken with some degree of confidence that one will not mistake results that could have been generated by chance for genuinely good results

A Reality Check for Data Snooping

SUMMARY We address the issue of optimal block choice in applications of the block bootstrap to dependent data It is shown that optimal block size depends significantly on context, being equal to n"/3, n"/4 and nll5 in the cases of variance or bias estimation, estimation of a onesided distribution function, and estimation of a two-sided distribution function, respectively A clear intuitive explanation of this phenomenon is given, together with outlines of theoretical arguments in specific cases It is shown that these orders of magnitude of block sizes can be used to produce a simple, practical rule for selecting block size empirically That technique is explored numerically

On blocking rules for the bootstrap with dependent data

Empirical likelihood has been found very useful in many different occasions. However, when applied directly to some more complicated statistics such as U-statistics, it runs into serious computational difficulties. In this paper, we introduce a so-called jackknife empirical likelihood (JEL) method. The new method is extremely simple to use in practice. In particular, the JEL is shown to be very effective in handling one and two-sample U-statistics. The JEL can be potentially useful for other nonlinear statistics.

Jackknife Empirical Likelihood

Let $X_1, X_2, \ldots $ be independent random variables with zero means and finite variances. It is well known that a finite exponential moment assumption is necessary for a Cramer-type large deviation result for the standardized partial sums. In this paper, we show that a Cramer-type large deviation theorem holds for self-normalized sums only under a finite $(2+\delta)$th moment, $0< \delta \leq 1$. In particular, we show $P(S_n /V_n \geq x)=\break (1-\Phi(x)) (1+O(1) (1+x)^{2+\delta} /d_{n,\delta}^{2+\delta})$ for $0 \leq x \leq d_{n,\delta}$,\vspace{1pt} where $d_{n,\delta} = (\sum_{i=1}^n EX_i^2)^{1/2}/(\sum_{i=1}^n E|X_i|^{2+\delta})^{1/(2+\delta)}$ and $V_n= (\sum_{i=1}^n X_i^2)^{1/2}$. Applications to the Studentized bootstrap and to the self-normalized law of the iterated logarithm are discussed.

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Self-normalized Cramér-type large deviations for independent random variables

Asymptotic properties for estimates of nonparametric regression models based on negatively associated sequences

We suggest a sample reuse method for dependent data, based on a cross between the block bootstrap and Richardson extrapolation. Instead of simulating a same size resample by resampling blocks and placing them end to end, it analyses the blocks directly and employs a variant of Richardson extrapolation to adjust for block size. A simple empirical rule, also based on Richardson extrapolation, is suggested for empirically selecting the block size. Performance in the contexts of distribution and bias estimation is discussed via theoretical analysis and numerical simulation.

Bing-Yi Jing

Papers

On blocking rules for the bootstrap with dependent data

Jackknife Empirical Likelihood

Self-normalized Cramér-type large deviations for independent random variables

Asymptotic properties for estimates of nonparametric regression models based on negatively associated sequences

On Sample Reuse Methods for Dependent Data