Information bounds and nonparametric maximum likelihood estimation. (English) Zbl 0757.62017

DMV Seminar. 19. Basel: Birkhäuser Verlag. viii, 126 p. (1992).
The monograph consists of two parts which are based on lectures of the authors.
In Part I, a sketch of the theory of information lower bounds for nonparametric and semiparametric models is given. First, the concepts of score function, tangent set and tangent space are introduced, and also several convolution and asymptotic minimax theorems are stated in finite and infinite dimensional parameter spaces. Second, the differentiability theorem of implicitly defined functions due to A. van der Vaart [see Ann. Stat. 19, No. 1, 178-204 (1991; Zbl 0732.62035), and “Statistical estimation in large parameter spaces” (1988; Zbl 0629.62035)] is shown to be important to apply the convolution theorem, and its applications are also given.
In Part II, the theory of nonparametric maximum likelihood (NPML) estimation is stated in inverse problems. First, the characterization of NPML estimators is given in the interval censoring problem, and also the deconvolution problem is discussed. Second, the properties of the EM algorithm and the iterative convex minorant algorithm are illustrated in the interval censoring problem. Further, the consistency of the NPML estimator is proved in the case of interval censoring, and also the distribution theory is developed.


62G05 Nonparametric estimation
62-02 Research exposition (monographs, survey articles) pertaining to statistics
62G20 Asymptotic properties of nonparametric inference
62E20 Asymptotic distribution theory in statistics
62F12 Asymptotic properties of parametric estimators