Beyond smooth finite-dimensional targets in unified semiparametric data fusion

Let $X\in\mathcal X$ denote an unobserved full-data random element with unknown law $P$ in a statistical model $\mathcal P_0$. There are $K\ge 2$ independent data sources. Source $k$ contains i.i.d. observations $W_{k,1},\dots,W_{k,n_k}$ from law $Q_k$, where each $W_k$ takes values in $\mathcal W_k$ and is generated from $X$ through a known observation operator $T_k$ (possibly many-to-one, corresponding to missingness/coarsening/measurement error), so that $Q_k=T_k(P)$. The analyst observes only $\{W_{k,i}:1\le i\le n_k,\,1\le k\le K\}$, not $X$.

This setup follows Graham et al. (2024).

Assume the fusion model is defined by known alignment restrictions linking the source laws to a common target law $P$, for example conditional or marginal equalities expressible as

with $\mathcal A$ known. The observed-data model is thus

Let $\Psi:\mathcal P_0\to\Theta$ be the target functional, where $\Theta$ may be infinite-dimensional (for example a function space such as $L^2$ or $\ell^\infty$), and $\Psi$ may be nonregular (not pathwise differentiable at some or all $P$).