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I have a sample $\boldsymbol{x}_i$ for $i$ in $1,\dots, n$, from a $d$ dimensional density $f(\boldsymbol{x})$ and I would like to estimate this unknown density. In addition I know that $f(\boldsymbol{x})$ is unimodal, but it can be skewed or fat tailed.

Given that $d$ is around 20, I think that using a standard kernel density estimator (KDE) is not an option (I can use a sample size $n$ of around $10^4$). In addition given that I know that $f(\boldsymbol{x})$ is unimodal, I don't think that all the flexibility of KDEs is needed in my case.

Hence I was looking for a parametric density estimator, that can be fitted reasonably fast. I've read a bit about multivariate skewed normal or student-t distributions but I would like to know if there are other options out there (maybe constrained KDEs or semi-parametric density estimators?).

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    $\begingroup$ I want to suggest something using an edgeworth expansion (i.e calculating higher order moments), but I'm also fairly sure this isn't a good idea. $\endgroup$ – GeorgeWilson Jun 1 '13 at 5:29
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    $\begingroup$ The problem with that approach is that the density can get negative in the tails, but it should do better than the normal in the "center" of the distribution. $\endgroup$ – Matteo Fasiolo Jun 5 '13 at 14:34
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    $\begingroup$ Presumably with a sample size of 10^4 there will be some real difficulties getting anything past accurate third order moments anyway? $\endgroup$ – GeorgeWilson Jun 6 '13 at 6:45
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    $\begingroup$ That's also true, maybe a solution would be to enforce a sparse structure on the higher moments. $\endgroup$ – Matteo Fasiolo Jun 6 '13 at 7:56
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Another potential approach is the 'nonparanormal' density estimator, which transforms each marginal distribution to standard Normal then estimates a correlation matrix. This is making a Gaussian copula assumption, which is relatively strong, but that's part of the high dimension/finite sample size tradeoff.

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Recently we have published a paper with a fast mode estimator for multidimensional unimodal distributions. https://doi.org/10.1016/j.spl.2019.108670 That is not the answer to this question, since we estimated the mode only, not the whole density. Besides, our algorithm needs some simple corrections to be applicable for the dimensionality $d=20$. However that shows that there do exist fast consistent estimators for the mode.

For semiparametric density estimators, you may see the paper by Hsu and Wu https://doi.org/10.1016/j.csda.2013.01.018 They use multivariate Box-Cox transform to fit the observed empirical distribution to a normal one. The main problem with the approach is that consistency is proved for (samples from a) lognormal dstribution only. Besides, the authors do not write out the complexity and consider samples of no more than 4096 points.

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