Statistical inference for stochastic processes: concepts and developments in asymptotic theory

1 Frame of the first-order asymptotic decision theory Consider a sequence of statistical experiments ET = (X T ,AT , {P T θ }θ∈Θ) (T ∈ R+). Let θ̂T : X T → Θ be a sequence of estimators of the unknown parameter θ. A basic property θ̂T should have is the consistency : θ̂T →PT θ θ (T → ∞) for every θ ∈ Θ. The analyst should not use any estimator without checking this property. For example, if one uses an estimator which is not consistent ∗Graduate School of Mathematical Sciences, University of Tokyo, 3-8-1 Komaba, Meguro-ku, Tokyo 153-8914 Japan e-mail: [email protected] http://www.ms.utokyo.ac.jp/ nakahiro/hp-naka-e