On the Validity of the Formal Edgeworth Expansion

Bhattacharya, Rabi; Ghosh, Jayanta K.

doi:10.1214/aos/1176344134

Cited by 535 publications

(371 citation statements)

References 17 publications

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“…From (2.10), (2.11) and the expansion for R 3 given in Appendix A.3, the signed square root n 1/2 R can be expressed as a smooth function of U , namely there exists a smooth function h such that n 1/2 R = h(Ū ). We can then use the results given in Bhattacharya and Ghosh (1978) is an even function, the error term in (A.27) is in fact O(n −2 ). This completes the proof.…”

Section: A3 Expansion For Rmentioning

confidence: 99%

On the second-order properties of empirical likelihood with moment restrictions

Chen

Cui

2007

Journal of Econometrics

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This paper considers the second order properties of empirical likelihood for a parameter defined by moment restrictions, which is the framework operated upon by the Generalized Method of Moments. It is shown that the empirical likelihood defined for this general framework still admits the delicate second order property of Bartlett correction, which represents a substantial extension of all the established cases of Bartlett correction for the empirical likelihood. An empirical Bartlett correction is proposed, which is shown to work effectively in improving the coverage accuracy of confidence regions for the parameter. KeywordsBartlett correction, coverage accuracy, empirical likelihood, generalized method of moments Disciplines Statistics and Probability CommentsThis preprint was published as Song Xi Chen and Hengjian Cui, "On the second-order properties of empirical likelihood with moment restrictions", Journal of Econometrics (2007) Abstract. This paper considers the second order properties of empirical likelihood for a parameter defined by moment restrictions, which is the framework operated upon by the Generalized Method of Moments. It is shown that the empirical likelihood defined for this general framework still admits the delicate second order property of Bartlett correction, which represents a substantial extension of all the established cases of Bartlett correction for the empirical likelihood. An empirical Bartlett correction is proposed, which is shown to work effectively in improving the coverage accuracy of confidence regions for the parameter.

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Section: A3 Expansion For Rmentioning

confidence: 99%

On the second-order properties of empirical likelihood with moment restrictions

Chen

Cui

2007

Journal of Econometrics

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“…First, we establish an Edgeworth expansion for the ML estimator and the t statistic based on the ML estimator that holds uniformly over a compact set in the parameter space. The method of doing so is similar to that of Bhattacharya and Ghosh (1978). This method is also used by Hall and Horowitz (1996) and Andrews (2001) among others.…”

Section: Introductionmentioning

confidence: 99%

“…This method is also used by Hall and Horowitz (1996) and Andrews (2001) among others. We utilize an Edgeworth expansion for the normalized sum of strong mixing random variables due to Lahiri (1993), which is an extension of a result of Götze and Hipp (1983), whereas Bhattacharya and Ghosh (1978) consider iid random variables and use a standard Edgeworth expansion for iid random variables. Second, we convert these Edgeworth expansions into Edgeworth expansions for the bootstrap ML estimator and bootstrap t statistic using the fact that the ML estimator lies in a neighborhood of the true value with probability that goes to one at a sufficiently fast rate.…”

Section: Introductionmentioning

confidence: 99%

Higher-order improvements of the parametric bootstrap for long-memory Gaussian processes

Andrews

Lieberman

Marmer³

2006

Journal of Econometrics

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This paper provides bounds on the errors in coverage probabilities of maximum likelihood-based, percentile-t, parametric bootstrap conÞdence intervals for Markov time series processes. These bounds show that the parametric bootstrap for Markov time series provides higher-order improvements (over conÞdence intervals based on Þrst order asymptotics) that are comparable to those obtained by the parametric and nonparametric bootstrap for iid data and are better than those obtained by the block bootstrap for time series. Similar results are given for Wald-based conÞdence regions.The paper also shows that k-step parametric bootstrap conÞdence intervals achieve the same higher-order improvements as the standard parametric bootstrap for Markov processes. The k-step bootstrap conÞdence intervals are computationally attractive. They circumvent the need to compute a nonlinear optimization for each simulated bootstrap sample. The latter is necessary to implement the standard parametric bootstrap when the maximum likelihood estimator solves a nonlinear optimization problem.

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“…This follows from general results on the expansions of cumulants in Wallace (1958), Bhattacharya and Ghosh (1978) and Hall (1992). It now follows from Theorem 1 of Mykland (1999) that k p 0 for p > 3 when T n R n .…”

Section: Introductionmentioning

confidence: 63%

Likelihood Computations without Bartlett Identities

Mykland

2001

Bernoulli

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The signed square root statistic R is given by sgn(, where l is the log-likelihood and è is the maximum likelihood estimator. The pth cumulant of R is typically of the form n À pa2 k p O(n À( p2)a2 ), where n is the number of observations. This paper shows how to symbolically compute k p without invoking the Bartlett identities. As an application, we show how the family of alternatives in¯uences the coverage accuracy of R.

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On the Validity of the Formal Edgeworth Expansion

Cited by 535 publications

References 17 publications

On the second-order properties of empirical likelihood with moment restrictions

On the second-order properties of empirical likelihood with moment restrictions

Higher-order improvements of the parametric bootstrap for long-memory Gaussian processes

Likelihood Computations without Bartlett Identities

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