# zbMATH — the first resource for mathematics

Asymptotic normality of the $$L_1$$ error of the Grenander estimator. (English) Zbl 1105.62342
Summary: Groeneboom introduced a jump process that can be used (among other things) to study the asymptotic properties of the Grenander estimator of a monotone density. In this paper we derive the asymptotic normality of a suitably rescaled version of the $$L_1$$ error of the Grenander estimator, using properties of this jump process.

##### MSC:
 62G20 Asymptotic properties of nonparametric inference 62G07 Density estimation 62M05 Markov processes: estimation; hidden Markov models
Full Text:
##### References:
 [1] BIRGE, L. 1989. The Grenander estimator: a nonasymptotic approach. Ann. Statist. 17 \' 1532 1549. Z. [2] CSORGO, M. and HORVATH, L. 1988. Central limit theorems for L -norms of density estimators. \" ṕ Probab. Theory Related Fields 80 269 291. Z. · Zbl 0657.60026 [3] DEVROYE, L. and GYORFI, L. 1985. Nonparametric Density Estimation: the L View. Wiley, New \" 1 York. Z. · Zbl 0546.62015 [4] DEVROYE, L. 1987. A Course in Density Estimation. Birkhauser, Boston. \" Z. · Zbl 0617.62043 [5] DUROT, C. 1996. Sharp asymptotics for isotonic regression. Ph.D. thesis, Univ. Paris Sud, Orsay.Z. · Zbl 0992.60028 [6] GROENEBOOM, P. 1985. Estimating a monotone density. In Proceedings of the Berkeley ConferZ. ence in Honor of Jerzey Neyman and Jack Kiefer L. M. Le Cam and R. A. Olshen, eds. 2 539 555. Univ. California Press, Berkeley. Z. · Zbl 1373.62144 [7] GROENEBOOM, P. 1989. Brownian motion with a parabolic drift and Airy functions. Probab. Theory Related Fields 81 79 109. Z. [8] GROENEBOOM, P. and WELLNER, J. A. 1992. Information Bounds and Nonparametric Maximum Likelihood Estimation. Birkhauser, Boston. \" Z. · Zbl 0757.62017 [9] HOOGHIEMSTRA, G. and LOPUHAA, H. P. 1998. An extermal limit theorem for the argmax process öf Brownian motion minus a parabolic drift. Extremes 1 215 240. · Zbl 0928.60062 [10] IBRAGIMOV, I. A. and LINNIK Y. V. 1971. Independent and Stationary Sequences of Random Variables. Wolters-Noordhoff, Groningen. Z. · Zbl 0219.60027 [11] KIM, J. and POLLARD, D. 1990. Cube root asymptotics. Ann. Statist. 18 191 219. Z. · Zbl 0703.62063 [12] KOMLOS, J., MAJOR, P. and TUSNADY, G. 1975. An approximation of partial sums of independent \' ŔVs and the sample DF I. Z. Wahrsch. Verw. Gebiete 32 111 131. Z. · Zbl 0308.60029 [13] POLLARD, D. 1984. Convergence of Stochastic Processes. Springer, New York. Z. · Zbl 0544.60045 [14] PRAKASA, RAO, B. L. S. 1969. Estimation of a unimodal density. Sankhya Ser. A 31 23 36. Z. · Zbl 0181.45901 [15] ROGERS, L. C. G. and WILLIAMS, D. 1997. Diffusions, Markov Processes, and Martingales 1. Wiley, New York. Z. [16] WANG, Y. 1992. The L theory of estimation of monotone and unimodal densities. Ph.D. 1 dissertation, Univ. California, Berkeley.
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.