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We investigate an algorithm named histogram transform ensembles (HTE) densityestimator whose effectiveness is supported by both solid theoretical analysisand significant experimental performance. On the theoretical side, bydecomposing the error term into approximation error and estimation error, weare able to conduct the following analysis: First of all, we establish theuniversal consistency under $L_1(\mu)$-norm. Secondly, under the assumptionthat the underlying density function resides in the H\"{o}lder space$C^{0,\alpha}$, we prove almost optimal convergence rates for both single andensemble density estimators under $L_1(\mu)$-norm and $L_{\infty}(\mu)$-normfor different tail distributions, whereas in contrast, for its subspace$C^{1,\alpha}$ consisting of smoother functions, almost optimal convergencerates can only be established for the ensembles and the lower bound of thesingle estimators illustrates the benefits of ensembles over single densityestimators. In the experiments, we first carry out simulations to illustratethat histogram transform ensembles surpass single histogram transforms, whichoffers powerful evidence to support the theoretical results in the space$C^{1,\alpha}$. Moreover, to further exert the experimental performances, wepropose an adaptive version of HTE and study the parameters by generatingseveral synthetic datasets with diversities in dimensions and distributions.Last but not least, real data experiments with other state-of-the-art densityestimators demonstrate the accuracy of the adaptive HTE algorithm.

Histogram Transform Ensembles for Density Estimation