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PreprintCC BY 4.0HAL

Beyond the delta method

  • Antoine Lejay
  • Sara Mazzonetto
Open at sourcePDFCite

Abstract

We give an asymptotic development of the maximum likelihood estimator (MLE), or any other estimator defined implicitly, in a way which involves the limiting behavior of the score and its higher-order derivatives. This development, which is explicitly computable, gives some insights about the non-asymptotic behavior of the renormalized MLE and its departure from its limit. We highlight that the results hold whenever the score and its derivative converge, including to non Gaussian limits. Our approach is based on an asymptotic implicit function theorem, inspired from perturbative approaches.

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Cite

BibTeXRIS
BibTeX
@misc{lejay2026beyond,
  title = {{Beyond the delta method}},
  author = {Antoine Lejay and Sara Mazzonetto},
  howpublished = {HAL},
  year = {2026},
  month = sep,
  eprint = {2207.13954},
  archivePrefix = {arXiv},
  url = {https://inria.hal.science/hal-03738371v5},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Beyond the delta method
AU  - Antoine Lejay
AU  - Sara Mazzonetto
PY  - 2026
DA  - 2026/09/22
PB  - HAL
UR  - https://inria.hal.science/hal-03738371v5
AB  - We give an asymptotic development of the maximum likelihood estimator (MLE), or any other estimator defined implicitly, in a way which involves the limiting behavior of the score and its higher-order derivatives. This development, which is explicitly computable, gives some insights about the non-asymptotic behavior of the renormalized MLE and its departure from its limit. We highlight that the results hold whenever the score and its derivative converge, including to non Gaussian limits. Our approach is based on an asymptotic implicit function theorem, inspired from perturbative approaches.
ER  -