Nonparametric functional estimation under order restrictions
Centre for Mathematical Sciences
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.
Density estimation; regression; monotonicity; convexity; deconvolution; kernel
smoothing; NPMLE; long range dependence; mixing; greatest convex minorant;