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Asymptotic Properties of Estimators in Standard and Nonstandard Situations

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

Abstract

Below we describe the properties of the MLEs \({{\hat{\theta }}_{\varepsilon }}\) and the BEs \({{\hat{\theta }}_{\varepsilon }}\) of parameterθ,constructed by observations of the diffusion-type process
$$d{{X}_{t}} = {{S}_{t}}\left( {0,X} \right)dt + \varepsilon d{{W}_{t}},{{X}_{0}} = {{x}_{0}},0 \leqslant t \leqslant T$$
in a usual (standard) situation of a regular statistical experiment (the trend S(•) is a smooth function of its arguments, etc.) when it is possibly a consistent and asymptotically normal estimation of the unknown finite-dimensional parameter θ ∈ Θand, in nonstandard but “close to regular” situations when trend is not differentiable with respect toθ, the initial value X0 is a random variable, the observations correspond to another equation, trend is the same for the different values of an unknown parameter, and the true value lies on the boundary of Θ.The problems of parameter estimation for the linear schemes are considered for a slightly different model which is not included in the general scheme of observations and corresponds to the “large signals” observed in white Gaussian noise.

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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, Universitédu MaineLe MansFrance

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