A canonical process for estimation of convex functions: the “invelope” of integrated Brownian motion + t

A canonical process for estimation of convex functions: the "invelope" of integrated Brownian motion + t^4

A Canonical Process for Estimation of Convex Functions: the “invelope” of Integrated

A process associated with integrated Brownian motion is introduced that characterizes the limit behavior of nonparametric least squares and maximum likelihood estimators of convex functions and convex densities, respectively. We call this process “the invelope” and show that it is an almost surely uniquely defined function of integrated Brownian motion. Its role is comparable to the role of the...

A canonical process for estimation of convex functions : the \ invelope

A process associated with integrated Brownian motion is introduced that characterizes the limit behavior of nonparametric least squares and maximum likelihood estimators of convex functions and convex densities, respectively. We call this process \the invelope" and show that it is an almost surely uniquely de ned function of integrated Brownian motion. Its role is comparable to the role of the ...

Estimation of a Convex Function: Characterizations and Asymptotic Theory

We study nonparametric estimation of convex regression and density functions by methods of least squares (in the regression and density cases) and maximum likelihood (in the density estimation case). We provide characterizations of these estimators, prove that they are consistent and establish their asymptotic distributions at a fixed point of positive curvature of the functions estimated. The ...

Estimation of a k−monotone density, part 3: limiting Gaussian versions of the problem; invelopes and envelopes

Let k be a positive integer. The limiting distribution of the nonparametric maximum likelihood estimator of a k−monotone density is given in terms of a smooth stochastic process Hk described as follows: (i) Hk is everywhere above (or below) Yk, the k− 1 fold integral of two-sided standard Brownian motion plus (k!/(2k)!)t2k when k is even (or odd). (ii) H(2k−2) k is convex. (iii) Hk touches Yk a...