Local Asymptotic Normality

Definition:

Regular parametric model

Definition: Regular parametric model is a parametric model \( \mathcal{P} = \{ P_\theta: \theta \in \Theta \} \subseteq \mathcal{M}_\mu \), such that

  1. \(\Theta\) is an open subset of \(R^k\).
  2. (Score function?) \( s(θ) = \sqrt{ \frac{\text{d } P_{\theta}}{\text{d } \mu} } \), a mapping from \(\Theta\) to \(L_2(\mu)\) is continuously Fréchet differentiable.
  3. Fisher information matrix \( I(\theta) = 4 \int \dot{s}(\theta) \dot{s}(\theta)' \text{d } \mu \) is non-singular.

Sufficient Conditions

The parametric model is regular if the following conditions hold:

  1. The density function \( ƒ_{\theta}(x) \) is continuously differentiable in \(\theta\) for μ-almost all x.
  2. The score function \( z_{\theta} = {\nabla ƒ_{\theta}}{ƒ_{\theta}} \mathbf{1}(ƒ_{\theta} > 0) \) belongs to the space \( L^2(P_{\theta})\) of square-integrable functions with respect to the measure \(P_{\theta}\).
  3. The Fisher information matrix \( I(θ) = \int z_{\theta} z_{\theta}' \text{d } P_{\theta} \) is nonsingular and continuous in θ.

Properties

Local asymptotic normality.

If the regular parametric model is identifiable, then there exists a uniformly \(\sqrt{n}\)-consistent and efficient estimator of its parameter.

Reference

  1. Bickel, Peter J., Chris A.J. Klaassen, Ya’acov Ritov and Jon A. Wellner (1998). Efficient and adaptive estimation for semiparametric models. Springer: New York.
  2. A. W. van der Vaart. Asymptotic Statistics. Cambridge University Press, 1998.

🏷 Category=Statistics