Sankhya: The Indian Journal of Statistics

2001, Volume 63, Series A, Pt. 3, pp. 309--327

WAVELET SHRINKAGE FOR NATURAL EXPONENTIAL FAMILIES WITH CUBIC VARIANCE FUNCTIONS

By

ANESTIS ANTONIADIS, *University of Joseph Fourier, France*

PANAGIOTIS BESBEAS, *University of Kent at Canterbury, England*

and

THEOFANIS SAPATINAS, *University of Cyprus, Cyprus*

*SUMMARY.* Wavelet shrinkage estimation has
been found to be a powerful tool for the nonparametric estimation
of spatially variable phenomena. Most work in this area to date
has concentrated primarily on the use of wavelet shrinkage
techniques in the nonparametric regression context where the data
are modelled as observations of a signal corrupted with additive
Gaussian noise. Limited work for applications involving data which
are actually counts such as Poisson or Bernoulli data has also
been considered. Recently, Antoniadis and Sapatinas (2001) have
developed a wavelet shrinkage methodology for obtaining and
assessing smooth estimates for complicated data such as those
arising from a (univariate) natural exponential family with
quadratic variance function (the variance is, at most, a quadratic
function of the mean) studied by Morris (1982, 1983). The
Gaussian, Poisson, gamma, binomial, negative binomial and
generalized hyperbolic secant distributions are the only members
of this family.

In this article we show that, subject to certain modifications, the wavelet shrinkage methodology of Antoniadis and Sapatinas (2001) can be extended to the case where the data arise from a (univariate) natural exponential family with cubic variance function (the variance is, at most, a cubic function of the mean) studied by Letac and Mora (1990). Twelve different distributions are the only members of this family. The first six appear in Morris (1982, 1983); most of the other six appear as distributions of the first passage times in the literature, the inverse Gaussian distribution being the most famous example. As an illustration of the proposed wavelet shrinkage methodology, a simulation study for inverse Gaussian data has been conducted.

*AMS (1991) subject classification. *62G05, 62G08.

*Key words and phrases.* Cross-validation mean squared error,
diagonal shrinkage estimation, inverse Gaussian distribution,
modulation estimators, natural exponential families, nonparametric
regression, smoothing, wavelet shrinkage estimation.