Quadratic-mean differentiability at a parameter

Let $\mathsf{P}:\mathrm{\Theta }\to \mathcal{P}(\mathcal{X},\mathcal{A})$ be a parametric model, let $\mathrm{\Theta }$ be open in ${\mathbb{R}}^d$, and let a supplied sigma-finite measure $\mu$ dominate every $P_{\theta }$ (Parametric statistical models, Dominated statistical experiments). Supply as part of the setup a family of nonnegative finite-valued measurable functions $p_{\theta }:\mathcal{X}\to [0,\mathrm{\infty })$ satisfying $P_{\theta }(A)={\int }_A{p}_{\theta }d\mu$ for every $A\in \mathcal{A}$ (The measure with density $f$ relative to $\mu$). Thus each $p_{\theta }$ integrates to one. The density family is supplied, rather than selected from domination alone; no choice principle, joint measurability in $(\theta ,x)$, or common exceptional null set is asserted. A supplied extended-valued probability density can be made finite-valued by setting it to zero on its measurable infinite-value set, which has measure zero because its integral is one. The QMD condition is unchanged by replacing any supplied density by an almost-everywhere equal version: for each fixed $h$, the norm in the condition below changes only on the union of two null sets. At an interior point ${\theta }_0$, the model is quadratic-mean differentiable (QMD) with score $s_{{\theta }_0}$ if $s_{{\theta }_0}:\mathcal{X}\to {\mathbb{R}}^d$ is measurable, ${\mathbb{E}}_{{\theta }_0}\mathrm{\parallel }{s}_{{\theta }_0}(X){\mathrm{\parallel }}^2<\mathrm{\infty }$, and

$$\int {(\sqrt{p_{{\theta }_0+h}}-\sqrt{p_{{\theta }_0}}-\frac{1}{2}{h}^T{s}_{{\theta }_0}\sqrt{p_{{\theta }_0}})}^{2}d\mu =o(\mathrm{\parallel }h{\mathrm{\parallel }}^2)$$

as $h\to 0$ with ${\theta }_0+h\in \mathrm{\Theta }$. Values of the score on $\{p_{{\theta }_0}=0\}$ do not affect this condition. The Fisher information associated with this QMD score is $I_{{\theta }_0}={\mathbb{E}}_{{\theta }_0}[s_{{\theta }_0}{s}_{{\theta }_0}^T]$; it may be singular. QMD is an $L^2(\mu )$ condition and does not assert pointwise differentiability of $p_{\theta }$ or local mutual absolute continuity of $P_{\theta }$ and $P_{{\theta }_0}$.