For i.i.d. experiments, differentiability in quadratic mean is the intrinsic regularity condition. It does not require pointwise differentiability of densities and remains invariant under changes of dominating measure. This chapter develops its Hellinger and tangent-space consequences; the full LAN proof appears in §15.
Square-root differentiability
Definition. A dominated model \(\{p_\theta:\theta\in\Theta\subset\mathbb R^p\}\) is differentiable in quadratic mean at \(\theta_0\) if there is \(\dot\ell_{\theta_0}\in L_2^p(P_{\theta_0})\) such that \[ \left\| \sqrt{p_{\theta_0+h}}-\sqrt{p_{\theta_0}} -\frac12h^\top\dot\ell_{\theta_0}\sqrt{p_{\theta_0}} \right\|_{L_2(\mu)} =o(\|h\|). \]
The derivative of the square-root density is one half the score times \(\sqrt p\). The factor one half makes the Fisher information four times the squared \(L_2\)-speed of the square-root path.
Proposition (local Hellinger metric). Under DQM, \[ H^2(P_{\theta_0+h},P_{\theta_0}) = \frac14h^\top I_{\theta_0}h+o(\|h\|^2), \] where \[ I_{\theta_0} = P_{\theta_0} (\dot\ell_{\theta_0}\dot\ell_{\theta_0}^\top). \]
Proof. Take the squared \(L_2\)-norm of the DQM expansion: \[ \sqrt{p_{\theta_0+h}}-\sqrt{p_{\theta_0}} = \frac12h^\top\dot\ell_{\theta_0}\sqrt{p_{\theta_0}} +o_{L_2}(\|h\|). \] The squared norm of the leading term is \(\frac14h^\top I_{\theta_0}h\); the cross and remainder terms are \(o(\|h\|^2)\). \(\square\)
A practical sufficient condition
Theorem. Suppose \(p_\theta(x)\) is differentiable in \(\theta\) for \(\mu\)-almost every \(x\), and \[ \partial_\theta\sqrt{p_\theta(x)} \] is continuous in \(L_2(\mu)\) at \(\theta_0\). Then the model is DQM with \[ \dot\ell_{\theta_0} = 2\frac{\partial_\theta\sqrt{p_{\theta_0}}} {\sqrt{p_{\theta_0}}} \] on \(\{p_{\theta_0}>0\}\).
This \(L_2\) continuity is stronger than necessary but often easy to verify by dominated convergence.
Regular examples
Example (normal location). For \(N(\theta,\sigma^2)\), \[ \dot\ell_\theta(x)=\frac{x-\theta}{\sigma^2}, \qquad I_\theta=\sigma^{-2}. \] DQM follows from smoothness of the normal square-root density in \(L_2\).
Example (Bernoulli interior). For \(X\sim\operatorname{Bernoulli}(\theta)\), \(0<\theta<1\), \[ \dot\ell_\theta(X) = \frac{X-\theta}{\theta(1-\theta)}, \qquad I_\theta=\frac1{\theta(1-\theta)}. \] At the boundary \(\theta=0\), the support changes and the root-\(n\) DQM geometry fails; the local count experiment at rate \(n^{-1}\) is Poisson.
Location models with rough densities
Proposition. For the location model \(p_\theta(x)=p(x-\theta)\), DQM holds if \(\sqrt p\) is weakly differentiable with derivative in \(L_2\). Then \[ \dot\ell_\theta(x) = -2\frac{(\sqrt p)'(x-\theta)} {\sqrt p(x-\theta)} \] on the positive-density set.
Pointwise differentiability of \(p\) is not required. Conversely, support boundaries can destroy DQM even when the density formula is smooth inside its support, as in the uniform location model.
One-dimensional paths and tangent scores
Definition. A differentiable path \(t\mapsto P_t\) through \(P=P_0\) has score \(g\in L_2^0(P)\) if \[ \sqrt{\frac{\mathrm{d}P_t}{\mathrm{d}\mu}} = \sqrt{\frac{\mathrm{d}P}{\mathrm{d}\mu}} +\frac t2g\sqrt{\frac{\mathrm{d}P}{\mathrm{d}\mu}} +o_{L_2}(t). \] The tangent space is the closed linear span in \(L_2^0(P)\) of all path scores.
In a \(p\)-dimensional parametric model, tangent scores are \(h^\top\dot\ell_{\theta_0}\), and \[ \|h^\top\dot\ell_{\theta_0}\|_{P,2}^2 =h^\top I_{\theta_0}h. \] In semiparametric models, the tangent space may be infinite dimensional; efficient scores arise by orthogonal projection away from nuisance tangents.
Stability under statistics
Proposition. If \(T(X)\) is observed instead of \(X\), the score in the reduced experiment is \[ E_{\theta_0}(\dot\ell_{\theta_0}(X)\mid T). \] Its information is no larger: \[ \operatorname{Var}\{E(\dot\ell\mid T)\} \preceq\operatorname{Var}(\dot\ell). \]
Proof. Differentiate expectations of bounded functions of \(T\): \[ \left.\frac{\mathrm{d}}{\mathrm{d}t} E_{\theta_0+th}\varphi(T)\right|_{t=0} = E_{\theta_0}\{\varphi(T)h^\top\dot\ell(X)\} = E_{\theta_0}\{\varphi(T)h^\top E(\dot\ell\mid T)\}. \] This identifies the reduced score. Conditional-variance decomposition gives the information inequality. \(\square\)
A statistic is sufficient precisely when it loses no score information for every local direction, under suitable regularity. This is the infinitesimal counterpart of exact Blackwell equivalence.