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Expansions

  • Yu. Kutoyants
Part of the Mathematics and Its Applications book series (MAIA, volume 300)

Abstract

The consistency and asymptotic normality of the MLE \({{\hat{\theta }}_{\varepsilon }}\) (in the regular case)allow us to write a formal representation

$${{\hat{\theta }}_{\varepsilon }} = 0 + \varepsilon \xi \left( {1 + o\left( 1 \right)} \right),$$

where ξis Gaussian, and to understand this equality as an expansion of MLE by the powers of ε. So it is natural to consider the other terms also in the asymptotic expansion

$${{\hat{\theta }}_{\varepsilon }} = 0 + \sum\limits_{{j = 1}}^{k} {{{\Psi }_{j}}{{\varepsilon }^{j}} + o\left( {{{\varepsilon }^{k}}} \right),}$$

where the variables \({{\psi }_{j}},j = 1,...,k\) do not depend on ε and the rest are small in a certain sense.

Keywords

Asymptotic Expansion Wiener Process Asymptotic Normality Mathematical Expectation Inverse Fourier Transform 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer Science+Business Media Dordrecht 1994

Authors and Affiliations

  • Yu. Kutoyants
    • 1
  1. 1.Département de MathématiquesFaculté des Sciences, Université du MaineLe MansFrance

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