Approximation rates for finite mixtures of location-scale models

Finite mixture models provide useful methods for approximation and estimation of probability density functions. In the context of location-scale mixture models, we improve upon previous results to show that it is possible to quantitatively bound the ${\cal L}_{p}$ approximation error in terms of the number of mixture components $m$ for any $p\in\left(1,\infty\right)$. In particular, for approximation on $\mathbb{R}^{d}$ of targets $f_{0}\in{\cal W}^{s,p}$, the (fractional) Sobolev spaces with smoothness order $s\in\left(0,1\right]$, if $p<2$ then the error has size $O\left(m^{-\frac{s}{sq+d}}\right)$, while if $p\ge2$, the error has size $O\left(m^{-\frac{2s}{2\left(sq+d\right)}}\right)$, where $q=1/\left(p-1\right)$. We demonstrate that these results can be used to derive estimation rates for a location-scale mixture-based adaptive least-squares estimator.