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Asymptotic minimax results for stochastic process families with critical points

Author

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  • Greenwood, P. E.
  • Wefelmeyer, W.

Abstract

We give two local asymptotic minimax bounds for models which admit a local quadratic approximation at every parameter point, but are not necessarily locally asymptotically normal or mixed normal. Such parameter points appear as critical points for stochastic process models exhibiting both stationary and explosive behavior. The first result shows that, for estimators normalized with the random Fisher information, the classical bound for the mixed normal case remains valid. However, the bound is not attained by asymptotically centering estimators. The second result refers to filtered models. It gives a sharp bound for estimators based on observing the path of a process until the random Fisher information exceeds a given constant.

Suggested Citation

  • Greenwood, P. E. & Wefelmeyer, W., 1993. "Asymptotic minimax results for stochastic process families with critical points," Stochastic Processes and their Applications, Elsevier, vol. 44(1), pages 107-116, January.
    • Handle: RePEc:eee:spapps:v:44:y:1993:i:1:p:107-116
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    Citations

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    Cited by:

    1. Gushchin, Alexander A. & Kuchler, Uwe, 1997. "Asymptotic inference for a linear stochastic differential equation with time delay," SFB 373 Discussion Papers 1997,43, Humboldt University of Berlin, Interdisciplinary Research Project 373: Quantification and Simulation of Economic Processes.
    2. Luschgy, Harald, 1995. "Local asymptotic quadraticity of stochastic process models based on stopping times," Stochastic Processes and their Applications, Elsevier, vol. 57(2), pages 305-317, June.
    3. George Roussas & Debasis Bhattacharya, 2009. "Hájek-Inagaki convolution representation theorem for randomly stopped locally asymptotically mixed normal experiments," Statistical Inference for Stochastic Processes, Springer, vol. 12(2), pages 185-201, June.
    4. N. Lin & S. Lototsky, 2014. "Second-order continuous-time non-stationary Gaussian autoregression," Statistical Inference for Stochastic Processes, Springer, vol. 17(1), pages 19-49, April.
    5. Galtchouk, Leonid & Konev, Victor, 2010. "On asymptotic normality of sequential LS-estimate for unstable autoregressive process AR(2)," Journal of Multivariate Analysis, Elsevier, vol. 101(10), pages 2616-2636, November.

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