Haar projection (${\Pi }_{V_J}f={\sum }_k⟨f,{\varphi }_{J,k}⟩{\varphi }_{J,k}$)

The Haar projection at level $J$ is the orthogonal projection onto the functions constant on level-$J$ dyadic intervals. It replaces a function by its mean on each cell. It can be expressed using level-$J$ Haar scaling functions, or using the constant function and Haar wavelets at levels below $J$.

In the Gaussian white noise model, estimate a Haar scaling function coefficient by $\int {\varphi }_{J,k}dY$. The resulting Haar projection estimator is an unbiased estimator with independent coefficient errors of variance $1/n$. On each cell it is the observed path increment divided by the cell length. This gives a finite-dimensional estimator without imposing smoothness on the drift.

The cellwise error of the Haar projection estimator in Gaussian white noise is $\sqrt{2^J/n}$ times a standard normal variable. Its supremum norm is therefore exactly that factor times the Gaussian maximum over $2^J$ cells. The Gaussian maximum bound without independence gives expected error at most $\sqrt{2^J(2J+2)\log 2/n}$. This concerns the stochastic error about the projection; approximation bias must be added when estimating the full drift.