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 $f_{\theta}(x)$ is continuously differentiable in $\theta$ for μ-almost all x.
2. The score function $z_{\theta} = {\nabla f_{\theta}}{f_{\theta}} 1_{\{f_{\theta} > 0\}}$ 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.