Let $P$ be a "nice" distribution on $\mathbb R^m$ (e.g., multivariate Gaussian, etc.), with density $p$. Let $H := \{x \in \mathbb R^m \mid x^\top w = b\}$ be a hyperplane in $\mathbb R^m$ with unit-normal $w \in \mathbb R^m$. Finally, $R$ be the Radon transform of $p$ w.r.t $H$ by $$ R := \int_H p(x)\,ds(x), $$ where $ds(x)$ is the surface-area element on $H$. Finally, let $X_1,\ldots,X_n$ be an iid sample from $P$.

Question. Is there a simple statistical estimator $\widehat R_n := s(X_1,\ldots,X_n)$ which converges to $R$ in some sense ?

Let $P$ be a "nice" distribution on $\mathbb R^m$ (e.g., multivariate Gaussian, etc.), with density $p$. Let $H := \{x \in \mathbb R^m \mid x^\top w = b\}$ be a hyperplane in $\mathbb R^m$ with unit-normal $w \in \mathbb R^m$. Let $R$ be the Radon transform of $p$ w.r.t $H$ by $$ R := \int_H p(x)\,ds(x), $$ where $ds(x)$ is the surface-area element on $H$. Finally, let $X_1,\ldots,X_n$ be an iid sample from $P$.

Question. Is there a simple statistical estimator $\widehat R_n := s(X_1,\ldots,X_n)$ which converges to $R$ in the limit $n \to \infty$ ?