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Estimating the probability density function using empirical moments is quite popular. Is there any advantage to using a different polynomial basis than the usual $1$,$x$, $x^2$ ... etc? For the moment I am interested in distributions on $\mathbb{R}+$.

If there are any advantage, are they used in practice ?

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  • $\begingroup$ It may be easier to compute the coordinates in orthogonal bases than in other ones, such as $1,x,x^2,\dots$. Orthogonal bases are indeed used for density estimation; see e.g. arxiv.org/abs/1505.00275 (I don't know if that qualifies as practice, though). $\endgroup$ – Iosif Pinelis Feb 25 '18 at 22:58

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