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 $n \to \infty$ ?*
> 
> 
> 



Consistent empirical estimation of Radon transform of a multivariate density function
$\newcommand\th\theta\newcommand\R{\mathbb R}$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.




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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 $100^m\ge10^{20}$.


