= Haar density estimator
{c}
{title2=$\widehat f_j(x)=\frac{2^j}{n}\sum_i\mathbf1_{I_j(x)}(X_i)$}
A Haar density estimator estimates the <Haar scaling function> coefficients by sample averages. It is exactly the <histogram> with cells of width $2^{-j}$. Its expectation is the <Haar projection> of the density. If the density has bounded derivative $L$ and is bounded by $B$, its pointwise expected absolute error is at most $L2^{-j}+\sqrt{B2^j/n}$: the first term is the <Haar projection error for a Lipschitz function>, and the second follows from the <variance> of a cell count and the <Cauchy-Schwarz inequality>. Choosing $2^j\asymp n^{1/3}$ gives error $O(n^{-1/3})$ while preserving nonnegativity and integral one.
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