  We propose a new methodology for denoising, variance-stabilizing and
normalizing signals whose both mean and variance are parameterized by a single
unknown varying parameter, such as Poisson or scaled chi-squared. Key to our
methodology is the observation that the signed and square-rooted generalized
log-likelihood ratio test for the equality of the local means is approximately
and asymptotically distributed as standard normal under the null. We use these
test statistics within the Haar wavelet transform at each scale and location,
referring to them as the likelihood ratio Haar (LRH) coefficients of the data.
In the denoising algorithm, the LRH coefficients are used as thresholding
decision statistics, which enables the use of thresholds suitable for i.i.d.
Gaussian noise, despite the standard Haar coefficients of the signal being
heteroscedastic. In the variance-stabilizing and normalizing algorithm, the LRH
coefficients replace the standard Haar coefficients in the Haar basis
expansion. To the best of our knowledge, the variance-stabilizing and
normalizing properties of the generalized likelihood ratio test have not been
interpreted or exploited in this manner before. We prove the consistency of our
LRH smoother for Poisson counts with a near-parametric rate, and various
numerical experiments demonstrate the good practical performance of our
methodology.
