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.


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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[1] Brunk, H. D. (1970). Estimation of isotonic regression. In Nonparametric Techniques in Statistical Inference (M. L. Puri, ed.). Cambridge Univ. Press.
[2] Dieudonn√©, J. (1969). Foundations of Modern Analysis. Academic Press, New York. · Zbl 0176.00502
[3] Groeneboom, P. (1985). Estimating a monotone density. In Proceedings of the Berkeley Conference in Honor of Jerzy Neyman and Jack Kiefer (L. M. LeCam and R. A. Olshen, eds.) 2 Wadsworth, Hayward, CA. · Zbl 1373.62144
[4] Groeneboom, P. (1988). Brownian motion with a parabolic drift and Airy functions. Probab. Theory Related Fields 81 79-109.
[5] Groeneboom, P., Jongbloed, G. and Wellner, J. A. (1999). Integrated Brownian motion conditioned to be positive. Ann. Probab. 27 1283-1303. Groeneboom, P., Jongbloed, G. and Wellner, J. A. (2001a). Estimation of convex functions: characterizations and asymptotic theory. Ann. Statist. 29 1653-1698. Groeneboom, P., Jongbloed, G. and Wellner, J. A. (2001b). Computation of nonparametric function estimators via vertex direction algorithms. Unpublished manuscript. · Zbl 0983.60078
[6] Groeneboom, P. and Wellner, J. A. (2001). Computing Chernoff’s distribution. J. Comput. Graph. Statist. 10 388-400. JSTOR: · Zbl 04567029
[7] Jongbloed, G. (1995). Three Statistical Inverse Problems. Ph.D. dissertation, Delft Univ. Technology.
[8] Mammen, E. (1991). Nonparametric regression under qualitative smoothness assumptions. Ann. Statist. 19 741-759. · Zbl 0737.62039
[9] Meyer, M. C. (1997). Shape restricted inference with applications to nonparametric regression, smooth nonparametric function estimation, and density estimation. Ph.D. dissertation, Dept. Statistics, Univ. Michigan, Ann Arbor.
[10] Prakasa Rao, B. L. S. (1969). Estimation of a unimodal density. Sankhy\?a. Ser. A 31 23-36. · Zbl 0181.45901
[11] Press, W. H., Teukolsky, S. A., Vetterling, W. T. and Flannery, B. P. (1992). Numerical Recipes in C. Cambridge Univ. Press. · Zbl 0778.65003
[12] Rogers, L. C. G. and Williams, D. (1994). Diffusions, Markov Processes and Martingales 1, 2nd ed. Wiley, New York. · Zbl 0826.60002
[13] Sinai, Y. G. (1992). Statistics of shocks in solutions of inviscid Burgers equation. Comm. Math. Phys. 148 601-621. · Zbl 0755.60105
[14] Wang, Y. (1994). The limiting distribution in concave regression. Preprint, Univ. Missouri, Columbia.
[15] Wright, S. J. (1997). Primal-Dual Interior Point Methods. SIAM, Philadelphia. · Zbl 0863.65031
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