## Estimation of a function under shape restrictions. Applications to reliability.(English)Zbl 1072.62023

Summary: 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.

### MSC:

 62G05 Nonparametric estimation 62G07 Density estimation 62M09 Non-Markovian processes: estimation 62N02 Estimation in survival analysis and censored data
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### References:

 [1] Barlow, R. E., Bartholomew, D. J., Bremner, J. M. and Brunk, H. D. (1972). Statistical Inference under Order Restrictions . Wiley, New York. · Zbl 0246.62038 [2] 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. · Zbl 0278.90035 [3] 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. JSTOR: · Zbl 0475.62084 [4] Birgé, L. (1987). Robust estimation of unimodal densities. Technical report, Univ. Paris X–Nanterre. · Zbl 0646.62033 [5] Birgé, L. (1987). On the risk of histograms for estimating decreasing densities. Ann. Statist. 15 1013–1022. JSTOR: · Zbl 0646.62033 [6] Birgé, L. (1989). The Grenander estimator: A nonasymptotic approach. Ann. Statist. 17 1532–1549. JSTOR: · Zbl 0703.62042 [7] Birgé, L. (1997). Estimation of unimodal densities without smoothness assumptions. Ann. Statist. 25 970–981. · Zbl 0888.62033 [8] Brunk, H. D. (1970). Estimation of isotonic regression. In Nonparametric Techniques in Statistical Inference (M. L. Puri, ed.) 177–197. Cambridge Univ. Press, London. [9] Cencov, N. N. (1962). Evaluation of an unknown distribution density from observations. Soviet Math. Dokl. 3 1559–1562. · Zbl 0133.11801 [10] 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. · Zbl 0366.62007 [11] Curioni, M. (1977). Estimation de la densité des processus de Poisson non homogènes. Ph.D. dissertation, Univ. Pierre et Marie Curie–Paris VI. [12] Devroye, L. and Györfi, L. (1985). Nonparametric Density Estimation : The $$L_1$$ View . Wiley, New York. · Zbl 0546.62015 [13] DeVore, R. A. and Lorentz, G. G. (1993). Constructive Approximation . Springer, Berlin. · Zbl 0797.41016 [14] Durot, C. (2002). Sharp asymptotics for isotonic regression. Probab. Theory Related Fields 122 222–240. · Zbl 0992.60028 [15] Grenander, U. (1956). On the theory of mortality measurement. II. Skand.-Aktuarietidskr. 39 125–153. · Zbl 0077.33715 [16] 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. · Zbl 1373.62144 [17] 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. · Zbl 1105.62342 [18] Kaplan, E. L. and Meier, P. (1958). Nonparametric estimation from incomplete observations. J. Amer. Statist. Assoc. 53 457–481. · Zbl 0089.14801 [19] 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. JSTOR: · Zbl 0657.62038 [20] Massart, P. (1990). The tight constant in the Dvoretzky–Kiefer–Wolfowitz inequality. Ann. Probab. 18 1269–1283. JSTOR: · Zbl 0713.62021 [21] Müller, H.-G. and Wang, J.-L. (1990). Locally adaptive hazard smoothing. Probab. Theory Related Fields 85 523–538. · Zbl 0677.62034 [22] Müller, H.-G. and Wang, J.-L. (1994). Hazard rate estimation under random censoring with varying kernels and bandwidths. Biometrics 50 61–76. · Zbl 0824.62097 [23] Nelson, W. (1972). Theory and applications of hazard plotting for censured failure data. Technometrics 14 945–966. [24] Parzen, E. (1962). On estimation of a probability density function and mode. Ann. Math. Statist. 33 1065–1076. · Zbl 0116.11302 [25] Priestley, M. B. and Chao, M. T. (1972). Non-parametric function fitting. J. Roy. Statist. Soc. Ser. B 34 385–392. · Zbl 0263.62044 [26] Rudemo, M. (1982). Empirical choice of histograms and kernel density estimators. Scand. J. Statist. 9 65–78. · Zbl 0501.62028 [27] Shorack, G. R. and Wellner, J. A. (1986). Empirical Processes with Applications to Statistics . Wiley, New York. · Zbl 1170.62365 [28] 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. · Zbl 0513.62050 [29] Stone, C. J. (1984). An asymptotically optimal window selection rule for kernel density estimates. Ann. Statist. 12 1285–1297. JSTOR: · Zbl 0599.62052 [30] 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. JSTOR: · Zbl 0546.62017 [31] Wang, Y. (1995). The $$L_1$$ theory of estimation of monotone and unimodal densities. J. Nonparametr. Statist. 4 249–261. · Zbl 1383.62103 [32] Wegman, E. J. (1970). Maximum likelihood estimation of a unimodal density function. Ann. Math. Statist. 41 457–471. · Zbl 0195.48601 [33] Yandell, B. S. (1983). Nonparametric inference for rates with censored survival data. Ann. Statist. 11 1119–1135. JSTOR: · Zbl 0598.62050
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