Is there an equivalent to VC-dimension for density estimation as opposed to classification?
For distributions with finite support of size $d$, when the error metric is the $\ell_1$ distance, the analogue of VC dimension is exactly $d$. (In fact, it's pretty much the VC dimension -- since to estimate a distribution over $d$ in $\ell_1$ is equivalent to agnostically PAC-learning the concept class $2^{[d]}$).
For discrete distributions with infinite support, no PAC-type result can be given. Perhaps somewhat surprisingly, a fully empirical (and nearly optimal!) sample-dependent bound on the $\ell_1$ distance between the empirical and true distributions can be given, see:
https://arxiv.org/abs/2004.12680
(to appear in NIPS 2020).
For continuous distributions, the situation is more subtle. You basically can't estimate anything without some absolute continuity and smoothness assumptions; see, for example the impossibility results:
https://papers.ssrn.com/sol3/papers.cfm?abstract_id=139691
https://www.sciencedirect.com/science/article/abs/pii/S0167715298002466
https://www.jstor.org/stable/2242068?seq=1
What's typically done is that we define a class of distributions with certain smoothness properties (Lipschitz, Hölder, Sobolev, Besov) and then design an algorithm that achieves a certain rate for this class, and then argue that no algorithm can achieve a better rate (making our algorithm &quot;minimax&quot; for that class).