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Theoretical aspects of ill-posed problems in statistics (1991)

Carroll, R. J. ; Rooij, A. C. M. ; Ruymgaart, F. H.

Springer

Acta applicandae mathematicae 24 (1991), S. 113-140

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ISSN: 1572-9036

Keywords: Primary 62G05 ; secondary 65J10 ; Ill-posed problem ; operator inversion ; deconvolution ; biased sampling ; Wicksell's problem ; regression ; errors-in-variables ; mixtures ; empirical Radon transform

Source: Springer Online Journal Archives 1860-2000

Topics: Mathematics

Notes: 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.

Type of Medium: Electronic Resource

URL: http://dx.doi.org/10.1007/BF00046889

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On sequential density estimation (1976)

Carroll, R. J.

Springer

Probability theory and related fields 36 (1976), S. 137-151

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ISSN: 1432-2064

Source: Springer Online Journal Archives 1860-2000

Topics: Mathematics

Notes: Abstract We consider the problem of sequential estimation of a density function f at a point x 0 which may be known or unknown. Let T n be a sequence of estimators of x 0 . For two classes of density estimators f n , namely the kernel estimates and a recursive modification of these, we show that if N(d) is a sequence of integer-valued random variables and n(d) a sequence of constants with N(d)/n(d)→ 1 in probability as d → 0, then f N(d) (T N(d) -f(x 0) is asymptotically normally distributed (when properly normed). We also propose two new classes of stopping rules based on the ideas of fixed-width interval estimation and show that for these rules, N(d)/n(d) → 1 almost surely and EN(d)/n(d) → 1 as d → 0. One of the stopping rules is itself asymptotically normally distributed when properly normed and yields a confidence interval for f(x 0) of fixed-width and prescribed coverage probability.

Type of Medium: Electronic Resource

URL: http://dx.doi.org/10.1007/BF00533997

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