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Bounds for the asymptotic normality of the maximum likelihood estimator using the Delta method

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Abstract
The asymptotic normality of the Maximum Likelihood Estimator (MLE) is a cornerstone of statistical theory. In the present paper, we provide sharp explicit upper bounds on Zolotarev-type distances between the exact, unknown distribution of the MLE and its limiting normal distribution. Our approach to this fundamental issue is based on a sound combination of the Delta method, Stein's method, Taylor expansions and conditional expectations, for the classical situations where the MLE can be expressed as a function of a sum of independent and identically distributed terms. This result is tailored for the broad class of one-parameter exponential family distributions.
Keywords
Delta method, Maximum likelihood estimator, Normal approximation, Stein's method

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MLA
Anastasiou, Andreas, and Christophe Ley. “Bounds for the Asymptotic Normality of the Maximum Likelihood Estimator Using the Delta Method.” ALEA-LATIN AMERICAN JOURNAL OF PROBABILITY AND MATHEMATICAL STATISTICS 14.1 (2017): 153–171. Print.
APA
Anastasiou, A., & Ley, C. (2017). Bounds for the asymptotic normality of the maximum likelihood estimator using the Delta method. ALEA-LATIN AMERICAN JOURNAL OF PROBABILITY AND MATHEMATICAL STATISTICS, 14(1), 153–171.
Chicago author-date
Anastasiou, Andreas, and Christophe Ley. 2017. “Bounds for the Asymptotic Normality of the Maximum Likelihood Estimator Using the Delta Method.” Alea-latin American Journal of Probability and Mathematical Statistics 14 (1): 153–171.
Chicago author-date (all authors)
Anastasiou, Andreas, and Christophe Ley. 2017. “Bounds for the Asymptotic Normality of the Maximum Likelihood Estimator Using the Delta Method.” Alea-latin American Journal of Probability and Mathematical Statistics 14 (1): 153–171.
Vancouver
1.
Anastasiou A, Ley C. Bounds for the asymptotic normality of the maximum likelihood estimator using the Delta method. ALEA-LATIN AMERICAN JOURNAL OF PROBABILITY AND MATHEMATICAL STATISTICS. 2017;14(1):153–71.
IEEE
[1]
A. Anastasiou and C. Ley, “Bounds for the asymptotic normality of the maximum likelihood estimator using the Delta method,” ALEA-LATIN AMERICAN JOURNAL OF PROBABILITY AND MATHEMATICAL STATISTICS, vol. 14, no. 1, pp. 153–171, 2017.
@article{8548269,
  abstract     = {The asymptotic normality of the Maximum Likelihood Estimator (MLE) is a cornerstone of statistical theory. In the present paper, we provide sharp explicit upper bounds on Zolotarev-type distances between the exact, unknown distribution of the MLE and its limiting normal distribution. Our approach to this fundamental issue is based on a sound combination of the Delta method, Stein's method, Taylor expansions and conditional expectations, for the classical situations where the MLE can be expressed as a function of a sum of independent and identically distributed terms. This result is tailored for the broad class of one-parameter exponential family distributions.},
  author       = {Anastasiou, Andreas and Ley, Christophe},
  issn         = {1980-0436},
  journal      = {ALEA-LATIN AMERICAN JOURNAL OF PROBABILITY AND MATHEMATICAL STATISTICS},
  keywords     = {Delta method,Maximum likelihood estimator,Normal approximation,Stein's method},
  language     = {eng},
  number       = {1},
  pages        = {153--171},
  title        = {Bounds for the asymptotic normality of the maximum likelihood estimator using the Delta method},
  volume       = {14},
  year         = {2017},
}

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