Lectures on inverse problems. (English) Zbl 0907.62042

Dobrushin, R. (ed.) et al., Lectures on probability theory and statistics. Ecole d’été de probabilités de Saint-Flour XXIV – 1994. Lectures given at the summer school in Saint-Flour, France, July 7–23, 1994. Berlin: Springer. Lect. Notes Math. 1648, 67-164 (1996).
The author discusses inverse problems where one wants to estimate a distribution function or a functional of it in situations where random variables generated by this distribution function are only indirectly observable. In general, the distribution function is not estimable at the usual \(\sqrt n\)-rate in such cases whereas some smooth functionals of the model can be estimated at this rate. This has been earlier demonstrated in the case of estimation of unimodal densities or densities with monotone failure rate by the reviewer [Sankyā, Ser. A 31, 23-36 (1969; Zbl 0181.45901); Ann. Math. Stat. 41, 507-519 (1970; Zbl 0214.45903)].
Here the author studies the interval censoring problem which is also an inverse problem in the sense specified earlier. He uses methods from isotonic regression theory and convex optimization to compute the nonparametric maximum likelihood estimator (NPMLE) of the distribution function and obtains estimators of smooth functions of this distribution function which attain the \(\sqrt n\)-rate and are asymptotically efficient. Local asymptotic distribution theory of estimators is developed for one of the versions of interval censoring models.
For the entire collection see [Zbl 0855.00026].


62G05 Nonparametric estimation
62G20 Asymptotic properties of nonparametric inference
62E20 Asymptotic distribution theory in statistics