THEORETICAL ASPECTS OF ILL-POSED PROBLEMS IN STATISTICS Academic Article

abstract

• Ill-posed problems arise in a wide variety of practical statistical situations, ranging from biased sampling and Wicksell's problem in stereology to regression, errors-in-variables and empirical Bayes models. The common mathematics behind many of these problems is operator inversion. When this inverse is not continuous a regularization of the inverse is needed to construct approximate solutions. In the statistical literature, however, ill-posed problems are rather often solved in an ad hoc manner which obccures these common features. It is our purpose to place the concept of regularization within a general and unifying framework and to illustrate its power in a number of interesting statistical examples. We will focus on regularization in Hilbert spaces, using spectral theory and reduction to multiplication operators. A partial extension to a Banach function space is briefly considered. 1991 Kluwer Academic Publishers.

authors

• Carroll, Raymond

published proceedings

• ACTA APPLICANDAE MATHEMATICAE

author list (cited authors)

• CARROLL, R. J., VANROOIJ, A., & RUYMGAART, F. H.

citation count

• 18

complete list of authors

• CARROLL, RJ||VANROOIJ, ACM||RUYMGAART, FH

publication date

• August 1991

publisher

• Springer Nature  Publisher

keywords

• Biased Sampling
• Deconvolution
• Empirical Radon Transform
• Errors-in-variables
• Ill-posed Problem
• Mixtures
• Operator Inversion
• Regression
• Wicksell Problem

Digital Object Identifier (DOI)

• 10.1007/BF00046889

start page

• 113

end page

• 140

volume

• 24

issue

• 2

URL

• http%3A%2F%2Fdx.doi.org%2F10.1007%2Fbf00046889