×

Estimation of a convex function: Characterizations and asymptotic theory. (English) Zbl 1043.62027

Summary: 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 asymptotic distribution theory relies on the existence of an “invelope function” for integrated two-sided Brownian motion \(+t^4\) which is established in our companion paper ibid., 1620–1652, see the preceding entry, Zbl 1043.62025.

MSC:

62G07 Density estimation
62G08 Nonparametric regression and quantile regression
62E20 Asymptotic distribution theory in statistics

Citations:

Zbl 1043.62025
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Anevski, D. (1994). Estimating the derivative of a convexdensity. Technical Report 1994:8, Dept. Mathematical Statistics, Univ. Lund.
[2] Barlow, R. E., Bartholomew, R. J., Bremner, J. M. and Brunk, H. D. (1972). Statistical Inference under Order Restrictions. Wiley, New York. · Zbl 0246.62038
[3] Brunk, H. D. (1958). On the estimation of parameters restricted by inequalities. Ann. Math. Statist. 29 437-454. · Zbl 0087.34302
[4] Brunk, H. D. (1970). Estimation of isotonic regression. In Nonparametric Techniques in Statistical Inference (M. L. Puri, ed.) 177-195. Cambridge Univ. Press.
[5] Donoho, D. L. and Liu, R. C. (1987). Geometric rates of convergence, I. Technical Report 137, Dept. Statistics, Univ. Calfornia, Berkeley.
[6] Grenander, U. (1956). On the theory of mortality measurement, part II. Skandinavisk Aktuarietidskrift 39 125-153. · Zbl 0077.33715
[7] Groeneboom, P. (1985). Estimating a monotone density. In Proceedings of the Berkeley Conference in Honor of Jerzy Neyman and Jack Kiefer (L. M. Le Cam and R. A. Olshen, eds.) 2 539-554. IMS, Hayward, CA. · Zbl 1373.62144
[8] Groeneboom, P. (1988). Brownian motion with a parabolic drift and Airy functions. Probab. Theory Related Fields 81 79-109.
[9] Groeneboom, P. (1996). Inverse problems in statistics. Proceedings of the St. Flour Summer School in Probability. Lecture Notes in Math. 1648 67-164. Springer, Berlin. Groeneboom, P., Jongbloed, G. and Wellner, J. A. (2001a). A canonical process for estimation of convexfunctions: the ”invelope” of integrated Brownian motion +t4. Ann. Statist. 29 1620-1652. Groeneboom, P., Jongbloed, G. and Wellner, J. A. (2001b). Vertexdirection algorithms for computing nonparametric function estimates. Unpublished manuscript. · Zbl 0907.62042
[10] Groeneboom, P. and Wellner, J. A. (2000). Computing Chernoff’s distribution. J. Comput. Graph. Statist. 10 388-400. JSTOR: · Zbl 04567029
[11] Hampel, F. R. (1987). Design, modelling and analysis of some biological datasets. In Design, Data and Analysis, by Some Friends of Cuthbert Daniel (C. L. Mallows, ed.) 111-115. Wiley, New York.
[12] Hanson, D. L. and Pledger, G. (1976). Consistency in concave regression. Ann. Statist. 4 1038- 1050. · Zbl 0341.62034
[13] Hildreth, C. (1954). Point estimators of ordinates of concave functions. J. Amer. Statist. Assoc. 49 598-619. JSTOR: · Zbl 0056.38301
[14] Jongbloed, G. (1995). Three statistical inverse problems. Unpublished Ph.D. dissertation, Delft Univ. Technology.
[15] Jongbloed, G. (1998). The iterative convexminorant algorithm for nonparametric estimation. J. Comput. Graph. Statist. 7 310-321. JSTOR:
[16] Jongbloed, G. (2000). Minimaxlower bounds and moduli of continuity. Statist. Probab. Lett. 50 279-284. · Zbl 0965.60083
[17] Kim, J. and Pollard, D. (1990). Cube root asymptotics. Ann. Statist. 18 191-219. · Zbl 0703.62063
[18] Lavee, D., Safrie, U. N. and Meilijson, I. (1991). For how long do trans-Saharan migrants stop over at an oasis? Ornis Scandinavica 22 33-44.
[19] Mammen, E. (1991). Nonparametric regression under qualitative smoothness assumptions. Ann. Statist. 19 741-759. · Zbl 0737.62039
[20] 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.
[21] Prakasa Rao, B. L. S. (1969). Estimation of a unimodal density. Sankhy\?a Ser. A 31 23-36. · Zbl 0181.45901
[22] Robertson, T., Wright, F. T. and Dykstra, R. L. (1988). Order Restricted Statistical Inference. Wiley, New York. · Zbl 0645.62028
[23] Shorack, G. R. and Wellner, J. A. (1986). Empirical Processes with Applications to Statistics. Wiley, New York. · Zbl 1170.62365
[24] Van der Vaart, A. W. and Wellner, J. A. (1996). Weak Convergence and Empirical Processes. Springer, New York. · Zbl 0862.60002
[25] Van Eeden, C. (1956). Maximum likelihood estimation of ordered probabilities. Proc. Konink. Nederl. Akad. Wetensch. A 59 444-455. · Zbl 0086.12802
[26] Van Eeden, C. (1957). Maximum likelihood estimation of partially ordered probabilities, I. Proc. Konink. Nederl. Akad. Wetensch. A 60 128-136. · Zbl 0086.12803
[27] Wang, Y. (1994). The limiting distribution in concave regression. Preprint, Univ. Missouri, Columbia.
[28] Woodroofe, M. and Sun, J. (1993). A penalized maximum likelihood estimate of f 0+ when f is non-increasing. Statist. Sinica 3 501-515. · Zbl 0822.62020
[29] Wright, F. T. (1981). The asymptotic behavior of monotone regression estimates. Ann. Statist. 9 443-448. · Zbl 0471.62062
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.