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Differentiability in quadratic mean (sometimes Hellinger differentiability) is a property of certain statistical models introduced by Lucien Le Cam and detailed in a paper in 1970. Differentiability in quadratic mean of a model guarantees certain asymptotic results, such as the asymptotic normality of the associated maximum likelihood estimator, or local asymptotic normality.

Definition
Let $$P_\theta$$ be a statistical model depending on aparameter $$\theta \in \mathbb{R}^k$$ with dimension $$k \in \mathbb{N}^*$$, generating a random variable $$X$$ in a space $$\mathcal{X}$$. Write $$p(\theta_0;x)$$ the likelihood of an observation $$x$$ under this model with a parameter value $$\theta = \theta_0$$.

The model $$P_\theta$$ is differentiable in quadratic mean in $$\theta_0$$ if there exist a measurable function $$s : \mathbb{R}^k \times \mathcal{X} \mapsto \mathbb{R}$$ such that, for all $$h$$ belonging to $$\mathbb{R}^k$$ in a neighborood of $$0$$,
 * $$\int_{\mathcal{X}} \left[\sqrt{p(\theta_0 + h; x)} - \sqrt{p(\theta_0;x)} - \frac{1}{2} h ^T s(\theta_0, x) \sqrt{p(\theta_0; x)}\right]^2\,dx = o(|| h|| ^{2})\,.$$