Wavelet Density Estimation

Wavelet based density estimators have gained in popularity due to their ability to approximate a large class of functions; adapting well to difficult situations such as when densities exhibit abrupt changes. The decision to work with wavelet density estimators brings along with it theoretical considerations (e.g. non-negativity, integrability) and empirical issues (e.g. computation of basis coefficients) that must be addressed in order to obtain a bona fide density. We present a new method to accurately estimate a non-negative, density which directly addresses many of the problems in practical wavelet density estimation. We cast the estimation procedure in a maximum likelihood framework that estimates the square root of the density √p; allowing us to obtain the natural non-negative density representation (√p)². Analysis of this method brings to light a remarkable theoretical connection with the Fisher information of the density and consequently leads to an efficient constrained optimization procedure to estimate the wavelet coefficients. (We acknowledge support from the National Science Foundation, NSF IIS-0307712.)
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