Nonparametric Functional Estimation under Order Restrictions

Research output: ThesisDoctoral Thesis (compilation)


This thesis consists of three papers (Papers A-C) on problems in nonparametric functional estimation, in particular density and regression function estimation and deconvolution, under order assumptions. Pointwise limit distribution results are stated for the obtained estimators, which include isotonic regression estimates, nonparametric maximum likelihood estimates of monotone densities, estimates of convex regression and density functions and deconvolution estimates.

Paper A states a limit distribution formula for the greatest convex minorant mapping and its derivative, which is then applied to regression function and density function estimation under monotonicity or convexity restrictions, at points of continuity and under various smoothness assumptions on the unknown function. Also treated is isotonization of kernel estimates, with application to regression and density estimation. Paper B extends the results of Paper A to limit results at points of discontinuity of the unknown function. Paper C is concerned with deconvolution under order restrictions.

Our methods give a unified approach to regression and density function estimation with order restrictions, thereby restating many previously known results as special cases as well as obtaining new results, mainly for dependent data and/or discontinuous functions.


Research areas and keywords

Subject classification (UKÄ) – MANDATORY

  • Probability Theory and Statistics


  • monotonicity, convexity, deconvolution, kernel smoothing, NPMLE, long range dependence, greatest convex minorant, mixing, Density estimation, limit distribution., regression, Mathematics, Matematik
Original languageEnglish
Awarding Institution
Supervisors/Assistant supervisor
  • Ola Hössjer, Supervisor
Award date2000 Apr 7
  • Centre for Mathematical Sciences, Lund University
Print ISBNs91-628-4062-2
Publication statusPublished - 2000
Publication categoryResearch

Bibliographic note

Defence details Date: 2000-04-07 Time: 10:15 Place: Centre for Mathematical Sciences, Sölvegatan 18, room MH:A External reviewer(s) Name: Gill, Richard Title: Professor Affiliation: Universiteit Utrecht, Dept. of Mathematics, ---