Nonparametric estimation of a convex bathtub-shaped hazard function

Bernoulli

Volumn 15, Issue 4, 2009, Pages 1010-1035

  Jankowski, Hanna K ^a   Wellner, Jon A ^b  

^a YORK UNIVERSITY   (Canada)

^b UNIVERSITY OF WASHINGTON   (United States)

Author keywords

Antimode; Bathtub; Consistency; Convex; Failure rate; Force of mortality; Hazard rate; Invelope process; Limit distribution; Nonparametric estimation; U shaped

Indexed keywords

EID: 77649202534     PISSN: 13507265     EISSN: None     Source Type: Journal    
DOI: 10.3150/09-BEJ202     Document Type: Article

References (30)

1. Balabdaoui, F. (2007). Consistent estimation of a convex density at the origin. Math. Methods Statist. 16 77-95. MR2335091
2. Banerjee, M. (2008). Estimating monotone, unimodal and U-shaped failure rates using asymptotic pivots. Statist. Sinica 18 467-492. MR2411614
3. [3] Banerjee, M. and Wellner, J.A. (2001). Likelihood ratio tests for monotone functions. Ann. Statist. 29 1699-1731. MR1891743 (Pubitemid 33405990)
4. [4] Banerjee, M. and Wellner, J.A. (2005). Confidence intervals for current status data. Scand. J. Statist. 32 405-424. MR2204627 (Pubitemid 41251535)
5. Baraud, Y. and Birgé, L. (2009). Estimating the intensity of a random measure by histogram type estimators. Probab. Theory Related Fields 143 239-284. MR2449129
6. Brunel, E. and Comte, F. (2006). Adaptive nonparametric regression estimation in presence of right censoring. Math. Methods Statist. 15 233-255. MR2278288
7. Cai, T. and Low, M. (2007). Adaptive estimation and confidence intervals for convex functions and monotone functions. Technical report. Dept. Statistics, Univ. Pennsylvania.
8. Carolan, C. and Dykstra, R. (1999). Asymptotic behavior of the Grenander estimator at density flat regions. Canad. J. Statist. 27 557-566. MR 1745821
9. Gruppo di Lavoro MPS (2004). Catalogo parametrico dei terremoti italiani, versione 2004 (cpti04). INGV, Bologna. Available at http://emidius.mi.ingv.it/ CPTI. in Italian.
10. Grenander, U. (1956). On the theory of mortality measurement. II. Skand. Aktuarietidskr. 39 125-153. MR0093415
11. 4. Ann. Statist. 29 1620-1652. MR1891741
12. [12] Groeneboom, P., Jongbloed, G. and Wellner, J.A. (2001). Estimation of a convex function: Characterizations and asymptotic theory. Ann. Statist. 29 1653-1698. MR1891742 (Pubitemid 33405989)
13. Groeneboom, P., Jongbloed, G. and Wellner, J.A. (2008). The support reduction algorithm for computing non-parametric function estimates in mixture models. Scand. J. Statist. 35 385-399. MR2446726
14. [14] Haupt, E. and Schabe, H. (1997). The TTT transformation and a new bathtub distribution model. J. Statist. Plann. Inference 60 229-240. MR 1456628 (Pubitemid 127163919)
15. Jankowski, H. and Wellner, J.A. (2007). Nonparametric estimation of a convex bathtub-shaped hazard function. Technical Report 521. Univ. Washington, Department of Statistics.
16. Jankowski, H. and Wellner, J.A. (2009). Computation of nonparametric convex hazard estimators via profile methods. J. Nonparametr. Stat. 21 505-518.
17. Jankowski, H., Wang, X., McCauge, H. and Wellner, J. (2008). convexHaz: R functions for convex hazard rate estimation. R package version 0.2.
18. Jongbloed, G. (2000). Minimax lower bounds and moduli of continuity. Statist. Probab. Lett. 50 279-284. MR 1792307
19. La Rocca, L. (2008). Bayesian non-parametric estimation of smooth hazard rates for seismic hazard assessment. Scand. J. Statist. 35 524-539. MR2446733
20. Lai, C.D., Xie, M. and Murthy, N.P. (2001). Bathtub-shaped failure rate life distributions. In Advances in Reliability (N. Balakrishnan and C.R. Rao, eds.) Handbook of Statistics 20 69-104. Amsterdam: North-Holland Publishing. MR1861919
21. Politis, D.N. and Romano, J.P. (1994). Large sample confidence regions based on subsamples under minimal assumptions. Ann. Statist. 22 2031-2050. MR1329181
22. Politis, D.N., Romano, J.P. and Wolf, M. (1999). Subsampling. New York: Springer. MR1707286
23. Rajarshi, S. and Rajarshi, M.B. (1988). Bathtub distributions: A review. Comm. Statist. Theory Methods 17 2597-2621. MR0955351
24. Reboul, L. (2005). Estimation of a function under shape restrictions. Applications to reliability. Ann. Statist. 33 1330-1356. MR2195637
25. Reynaud-Bouret, P. (2003). Adaptive estimation of the intensity of inhomogeneous Poisson processes via concentration inequalities. Probab. Theory Related Fields 126 103-153. MR1981635
26. Robertson, T., Wright, F.T. and Dykstra, R.L. (1988). Order Restricted Statistical Inference. Chichester: Wiley. MR0961262
27. Rockafellar, R.T. (1970). Convex Analysis. Princeton Mathematical Series 28. Princeton, NJ: Princeton Univ. Press.
28. Shorack, G.R. and Wellner, J.A. (1986). Empirical Processes with Applications to Statistics. New York: Wiley. MR0838963
29. van der Vaart, A.W. and Wellner, J.A. (1996). Weak Convergence and Empirical Processes with Applications in Statistics. New York: Springer. MR1385671
30. Woodroofe, M. and Sun, J. (1993). A penalized maximum likelihood estimate of f(0+) when f is nonincreasing. Statist. Sinica 3 501-515. MR1243398