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I am working with 2D spatial data, $(X_i, Y_i),\; i=1, \cdots, N$.

My current research requires estimating the density of the distances between those data points in each of the two dimensions. So the underlying joint pdf that I am estimating is $f_{\Delta_{i,j} X, \Delta_{i,j} Y}(\Delta x, \Delta y)$, where $\Delta_{i,j} X=X_j - X_i$, for a subset of $i,j$ pairings.

I would like to impose the assumption that the deltas are radially symmetric (I have prior reasons for doing this relating to the data source). It occurs to me that when these data are sparse, this might improve the density estimate, providing the assumption is correct.

I therefore want to define and estimate the radial pdf $f_{\Delta_{i,j}R}(r)$, where $\Delta_{i,j}R$ is the Euclidean distance between $(X_i, Y_i)$ and $(X_j, Y_j)$.

However, I am not aware of any literature relating to the estimation of radial pdfs from spatial data. I originally intended to transform the 2D KDE using the Jacobian and integrate over the angular component, but the integral looks like it may have no analytic solution (at least not for Gaussian basis functions).

I can try to build a KDE 'directly' from the $\Delta_{i,j}R$ values, but I don't know what kind of basis function to use in this case, or how to select the bandwidth.

Any pointers greatly appreciated. Or indeed an explanation of why this is a terrible idea.

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