Haar density estimator (${\hat{f}}_j(x)=\frac{2^j}{n}{\sum }_i{1}_{I_j(x)}(X_i)$)

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 $L{2}^{-j}+\sqrt{B{2}^j/n}$: 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.