Consider the situation that we are estimating a $d$-dimensional density (with suitable regularity conditions) using kernel density estimation,
[Method1,conditional density estimation] We can proceed $d$ $1$-dim density estimation sequentially, i.e. $p(X_1),p(X_2\mid X_1),\cdots p(X_d\mid X_{d-1},\cdots X_1)$. In each step we can choose a $1$-dim optimal bandwidth.
[Method2, multivariate density estimation] Alternatively, we can estimate $p(X_1,\cdots X_d)$ directly as a $d$-dim density estimation problem. Thus, we can choose bandwidths for all $d$-coordinates simultaneously.(Of course not necessarily the same bandwidths in different coordinates.)
I have been told that, in kernel density estimation, the optimal bandwidth in each step of Method 1 will be different from optimal bandwidth chosen in each direction when the optimal bandwidth is chosen in a $d$-dim problem in Method 2(Say via Likelihood cross validation).
(1)Can anyone point to me a literature addressing such a problem? I found [Silverman] is not too useful in this question. My primary guess will be something about robustness, but I could be wrong.
(2)What is the major obstacle generalizing the univariate density estimation methods to higher dimension except for sparsity of the data?
(3)What is the well-adopted/popular choice of optimal bandwidth selection in a multivariate density estimation problem?
The second part of this post concerns the comparison between parametric(kernel density estimation) and nonparametric methods of density estimations(For example, Bayesian histogram).
It is generally a difficult problem to estimate the correct dimension of the density [1], but if now we already know the dimension $d$ of the data, then we can use some established methods to proceed our density estimation(Dirichlet or Gaussian process, for instances). And if our only concern now is precision (say MISE) of the estimation towards the density, then
(4)If we turn to Bayesian nonparametric methods of density estimation instead of the kernel density estimation (say Bayesian histogram), which one will perform better, under different cases with or without sparsity.Are there any existing literature that addressed the problem with some simulation studies?
Reference
[Silverman]Silverman, Bernard W. Density estimation for statistics and data analysis. Vol. 26. CRC press, 1986.
[1]Is there an accepted method to determine an approximate dimension for manifold learning
I am not an expert in the branch of density estimation so any reference and clarification will be of great help and appreciated. And this is mainly a ref-request as well as request for an overview.
