## A canonical process for estimation of convex functions: the “invelope” of integrated Brownian motion $$+t^ 4$$.(English)Zbl 1043.62026

Summary: 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 greatest convex minorant of Brownian motion plus a parabolic drift in the problem of estimating monotone functions. An iterative cubic spline algorithm is introduced that solves the constrained least squares problem in the limit situation and some results, obtained by applying this algorithm, are shown to illustrate the theory.

### MSC:

 62G07 Density estimation 62G05 Nonparametric estimation 60G15 Gaussian processes 62E20 Asymptotic distribution theory in statistics 65C60 Computational problems in statistics (MSC2010) 62M09 Non-Markovian processes: estimation
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### References:

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