These properties are interesting, and some of them are essential elements for the data-driven procedure because the value of a particular integral is independent of the scaling function, and has been computed precisely for positive wavelet kernels. In contrast, that integral is hard to compute for general wavelet kernels. It has been shown that general wavelet kernel-type estimators are asymptotically unbiased, and almost sure consistent. It is also shown that asymptotic normality holds for the positive wavelet estimators. Simulation results show that this method works better than the optimal window width method because the estimator can pick up peaks well.
Simulations also suggest the use of non-integer resolution level.
The author uses a linear combination of the estimators at two levels to achieve the similar results, and justifies the choice of the linear coefficients by using the triangular norm.