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The Mean Value Theorem and Taylor's Expansion in Statistics

Author

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  • Changyong Feng
  • Hongyue Wang
  • Yu Han
  • Yinglin Xia
  • Xin M. Tu

Abstract

The mean value theorem and Taylor's expansion are powerful tools in statistics that are used to derive estimators from nonlinear estimating equations and to study the asymptotic properties of the resulting estimators. However, the mean value theorem for a vector-valued differentiable function does not exist. Our survey shows that this nonexistent theorem has been used for a long time in statistical literature to derive the asymptotic properties of estimators and is still being used. We review several frequently cited papers and monographs that have misused this "theorem" and discuss the flaws in these applications. We also offer methods to fix such errors.

Suggested Citation

  • Changyong Feng & Hongyue Wang & Yu Han & Yinglin Xia & Xin M. Tu, 2013. "The Mean Value Theorem and Taylor's Expansion in Statistics," The American Statistician, Taylor & Francis Journals, vol. 67(4), pages 245-248, November.
    • Handle: RePEc:taf:amstat:v:67:y:2013:i:4:p:245-248
    DOI: 10.1080/00031305.2013.844203
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    Cited by:

    1. Amir T. Payandeh Najafabadi & Maryam Omidi Najafabadi, 2016. "On the Bayesian estimation for Cronbach's alpha," Journal of Applied Statistics, Taylor & Francis Journals, vol. 43(13), pages 2416-2441, October.
    2. Timo Dimitriadis & Yannick Hoga, 2022. "Dynamic CoVaR Modeling," Papers 2206.14275, arXiv.org, revised Feb 2024.
    3. Wang, Fa, 2022. "Maximum likelihood estimation and inference for high dimensional generalized factor models with application to factor-augmented regressions," Journal of Econometrics, Elsevier, vol. 229(1), pages 180-200.

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