# Consistent empirical estimation of Radon transform of a multivariate density function

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 $$n \to \infty$$ ?

$$\newcommand\th\theta\newcommand\R{\mathbb R}$$Suppose that we have a parametric setting, that is, the unknown distribution $$P$$ belongs to a known parametric family $$(P_\th)$$ of distributions parameterized by a sufficiently low-dimensional parameter $$\th$$; this may be the case if $$P$$ is Gaussian. Then you can get (say) a maximum likelihood estimate (MLE) $$\hat\th_n$$ of $$\th$$ and estimate $$R$$ by $$\hat R_n:=\int_H p_{\hat\th_n}\,ds,$$ where $$p_{\hat\th_n}$$ is the density of $$P_{\hat\th_n}$$.
The MLE is usually consistent, so that you will have $$\hat\th_n\to\th$$ in probability (as $$n\to\infty$$). If now $$p_\th$$ is continuous on $$\th$$ uniformly on compact subsets of $$\R^m$$, you will get $$p_{\hat\th_n}\to p_\th$$ uniformly on compact subsets of $$\R^m$$. If, moreover, you can control the tails of the densities $$p_\th$$ so as to have their local uniform integrability (as you would have in the Gaussian case), then you will end up with the desired conclusion $$\hat R_n\to R$$ in probability.
If the setting is nonparametric but the dimension $$m$$ is small, then you can use nonparametric (say kernel) estimators of the unknown density $$p$$, instead of the parametric estimators. However, then, naturally, the convergence $$\hat R_n\to R$$ will be substantially more problematic.
If the setting is completely nonparametric and the dimension $$m$$ is not small (say $$m\ge10$$), then the situation will of course be even worse. Here you will need the sample size $$n$$ to be at least as big as something like $$100^m\ge10^{20}$$.