Problems in Ill-posed Statistical Inference

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Proposal ID: DMS-0203942PI: Frits H. RuymgaartTitle: Problems in ill-posed statistical inferenceAbstractThere is a renewed interest from the part of statisticians in solving noisy integral equations. This kind of problem is ill-posed because the integral operator involved typically has an unbounded inverse and such models boil down to a generalization of traditional curve estimation. The purpose of this proposal is to pursue the general theory that is being developed into three directions. The first question is whether asymptotically efficient estimators of sufficiently smooth functionals of the input signal can be constructed despite the presence of the possibly unknown error density as an infinite dimensional nuisance parameter. The second question to be dealt with is the asymptotic distribution of the integrated squared error centered at its mean. Such precise asymptotic behavior of a global measure of accuracy has important applications to model checks. Finally a new method expedient for convolution equations with mathematical irregularities will be developed. The method is based on expansion in a wavelet basis coupled with inversion of the convolution operator in the time domain.Experimenters are often faced with the problem of recovering the input of a system when only the output is observable. Typically the observations will be blurred by measurement error and statistical procedures become pertinent. Examples include computer tomography employed in medical imaging, and Wicksell's problem in stereology where a transform of the particle size distribution is observed. Inverse heat conduction requires the recovery of the initial heat distribution (input) when the present one (output) is given. There are many more examples and related questions. For instance, can one estimate the total weight of a cable suspended at its endpoints, when only data regarding its shape are available? The construction of efficient estimators of such functionals of the input is one purpose of this research. A second question considered is precise information about the frequency distribution of the discrepancy between the estimated input and its expectation. Results can be useful in checking the validity of certain prior assumptions and could, for instance, be applied in recovering the luminosity distribution of the Milky Way. Finally a new method of input reconstruction in irregular cases will be developed using wavelets with potential application to image reconstruction.