The Annals of Statistics

Estimation of a function under shape restrictions. Applications to reliability

L. Reboul

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Abstract

This paper deals with a nonparametric shape respecting estimation method for U-shaped or unimodal functions. A general upper bound for the nonasymptotic $\mathbb{L}_{1}$-risk of the estimator is given. The method is applied to the shape respecting estimation of several classical functions, among them typical intensity functions encountered in the reliability field. In each case, we derive from our upper bound the spatially adaptive property of our estimator with respect to the $\mathbb{L}_{1}$-metric: it approximately behaves as the best variable binwidth histogram of the function under estimation.

Article information

Source
Ann. Statist., Volume 33, Number 3 (2005), 1330-1356.

Dates
First available in Project Euclid: 1 July 2005

Permanent link to this document
https://projecteuclid.org/euclid.aos/1120224104

Digital Object Identifier
doi:10.1214/009053605000000138

Mathematical Reviews number (MathSciNet)
MR2195637

Zentralblatt MATH identifier
1072.62023

Subjects
Primary: 62G05: Estimation
Secondary: 62G07: Density estimation 62G08: Nonparametric regression 62N01: Censored data models 62N02: Estimation

Keywords
Variable binwidth histogram adaptive estimation hazard rate nonhomogeneous Poisson process data-driven estimator unimodal function U-shaped function

Citation

Reboul, L. Estimation of a function under shape restrictions. Applications to reliability. Ann. Statist. 33 (2005), no. 3, 1330--1356. doi:10.1214/009053605000000138. https://projecteuclid.org/euclid.aos/1120224104


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References

  • Barlow, R. E., Bartholomew, D. J., Bremner, J. M. and Brunk, H. D. (1972). Statistical Inference under Order Restrictions. Wiley, New York.
  • Barlow, R. E., Proschan, F. and Scheuer, E. M. (1972). A system debugging model. In Reliability Growth Symposium, Interim Note 22 Aberdeen Proving Ground 46--65. U.S. Army Materiel Systems Analysis Agency, Washington, DC.
  • Bartoszyński, R., Brown, B. W., McBride, C. M. and Thompson, J. R. (1981). Some nonparametric techniques for estimating the intensity function of a cancer related nonstationary Poisson process. Ann. Statist. 9 1050--1060.
  • Birgé, L. (1987). Robust estimation of unimodal densities. Technical report, Univ. Paris X--Nanterre.
  • Birgé, L. (1987). On the risk of histograms for estimating decreasing densities. Ann. Statist. 15 1013--1022.
  • Birgé, L. (1989). The Grenander estimator: A nonasymptotic approach. Ann. Statist. 17 1532--1549.
  • Birgé, L. (1997). Estimation of unimodal densities without smoothness assumptions. Ann. Statist. 25 970--981.
  • Brunk, H. D. (1970). Estimation of isotonic regression. In Nonparametric Techniques in Statistical Inference (M. L. Puri, ed.) 177--197. Cambridge Univ. Press, London.
  • Cencov, N. N. (1962). Evaluation of an unknown distribution density from observations. Soviet Math. Dokl. 3 1559--1562.
  • Clevenson, M. L. and Zidek, J. V. (1977). Bayes linear estimators of the intensity function of the nonstationary Poisson process. J. Amer. Statist. Assoc. 72 112--120.
  • Curioni, M. (1977). Estimation de la densité des processus de Poisson non homogènes. Ph.D. dissertation, Univ. Pierre et Marie Curie--Paris VI.
  • Devroye, L. and Györfi, L. (1985). Nonparametric Density Estimation: The $L_1$ View. Wiley, New York.
  • DeVore, R. A. and Lorentz, G. G. (1993). Constructive Approximation. Springer, Berlin.
  • Durot, C. (2002). Sharp asymptotics for isotonic regression. Probab. Theory Related Fields 122 222--240.
  • Grenander, U. (1956). On the theory of mortality measurement. II. Skand.-Aktuarietidskr. 39 125--153.
  • Groeneboom, P. (1985). Estimating a monotone density. In Proc. Berkeley Conference in Honor of Jerzy Neyman and Jack Kiefer (L. M. Le Cam and R. A. Olshen, eds.) 2 539--555. Wadsworth, Monterey, CA.
  • Groeneboom, P., Hooghiemstra, G. and Lopuhaä, H. P. (1999). Asymptotic normality of the $L_1$ error of the Grenander estimator. Ann. Statist. 27 1316--1347.
  • Kaplan, E. L. and Meier, P. (1958). Nonparametric estimation from incomplete observations. J. Amer. Statist. Assoc. 53 457--481.
  • Marron, J. S. and Padgett, W. J. (1987). Asymptotically optimal bandwidth selection for kernel density estimators from randomly right-censored samples. Ann. Statist. 15 1520--1535.
  • Massart, P. (1990). The tight constant in the Dvoretzky--Kiefer--Wolfowitz inequality. Ann. Probab. 18 1269--1283.
  • Müller, H.-G. and Wang, J.-L. (1990). Locally adaptive hazard smoothing. Probab. Theory Related Fields 85 523--538.
  • Müller, H.-G. and Wang, J.-L. (1994). Hazard rate estimation under random censoring with varying kernels and bandwidths. Biometrics 50 61--76.
  • Nelson, W. (1972). Theory and applications of hazard plotting for censured failure data. Technometrics 14 945--966.
  • Parzen, E. (1962). On estimation of a probability density function and mode. Ann. Math. Statist. 33 1065--1076.
  • Priestley, M. B. and Chao, M. T. (1972). Non-parametric function fitting. J. Roy. Statist. Soc. Ser. B 34 385--392.
  • Rudemo, M. (1982). Empirical choice of histograms and kernel density estimators. Scand. J. Statist. 9 65--78.
  • Shorack, G. R. and Wellner, J. A. (1986). Empirical Processes with Applications to Statistics. Wiley, New York.
  • Singpurwalla, N. D. and Wong, M. Y. (1983). Estimation of the failure rate. A survey of nonparametric methods. I. Non-Bayesian methods. Comm. Statist. Theory Methods 12 559--588.
  • Stone, C. J. (1984). An asymptotically optimal window selection rule for kernel density estimates. Ann. Statist. 12 1285--1297.
  • Tanner, M. A. and Wong, W. H. (1983). The estimation of the hazard function from randomly censored data by the kernel method. Ann. Statist. 11 989--993.
  • Wang, Y. (1995). The $L_1$ theory of estimation of monotone and unimodal densities. J. Nonparametr. Statist. 4 249--261.
  • Wegman, E. J. (1970). Maximum likelihood estimation of a unimodal density function. Ann. Math. Statist. 41 457--471.
  • Yandell, B. S. (1983). Nonparametric inference for rates with censored survival data. Ann. Statist. 11 1119--1135.