Wavelet Methods in Statistics with R pp 201-227 | Cite as

# Multiscale Variance Stabilization

Variance stabilization is one of the oldest techniques in statistics, see, for example, Barlett (1936). Suppose one has a random variable X, which has a distribution dependent on some parameter *θ* and that the variance of *X* is given by var(*X*) = *σ*^{2}(*θ*). If *X* was Gaussian with distribution *X* ~ *N*(*μ*, *σ*^{2}) and the parameter of interest is *μ*, then it is patently clear that the variance *σ*^{2} *does not* depend on *μ*. For many other random variables, this is not true. For example, if *Y* was distributed as a Poisson random variable, *Y* ~ Pois(λ), then the (only) parameter of interest is λ and the variance *σ*^{2}(λ) = λ. In other words, the variance depends directly on the parameter of interest, λ, which is also the mean. Hence, we often refer to a ‘mean-variance’ relationship, and for Poisson variables the variance is equal to the mean.

## Keywords

Haar Wavelet Piecewise Constant Variance Stabilization Poisson Random Variable Wavelet Shrinkage## Preview

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