## Institute of Mathematical Statistics Collections

### Smooth and non-smooth estimates of a monotone hazard

#### Abstract

We discuss a number of estimates of the hazard under the assumption that the hazard is monotone on an interval $[0,a]$. The usual isotonic least squares estimators of the hazard are inconsistent at the boundary points $0$ and $a$. We use penalization to obtain uniformly consistent estimators. Moreover, we determine the optimal penalization constants, extending related work in this direction by [ Statist. Sinica 3 (1993) 501–515; Ann. Statist. 27 (1999) 338–360]. Two methods of obtaining smooth monotone estimates based on a non-smooth monotone estimator are discussed. One is based on kernel smoothing, the other on penalization.

#### Chapter information

Source
Banerjee, M., Bunea, F., Huang, J., Koltchinskii, V., and Maathuis, M. H., eds., From Probability to Statistics and Back: High-Dimensional Models and Processes -- A Festschrift in Honor of Jon A. Wellner, (Beachwood, Ohio, USA: Institute of Mathematical Statistics, 2013) , 174-196

Dates
First available in Project Euclid: 8 March 2013

https://projecteuclid.org/euclid.imsc/1362751187

Digital Object Identifier
doi:10.1214/12-IMSCOLL913

Subjects
Primary: 62G05: Estimation 62G05: Estimation
Secondary: 62E20: Asymptotic distribution theory

Rights

#### Citation

Groeneboom, Piet; Jongbloed, Geurt. Smooth and non-smooth estimates of a monotone hazard. From Probability to Statistics and Back: High-Dimensional Models and Processes -- A Festschrift in Honor of Jon A. Wellner, 174--196, Institute of Mathematical Statistics, Beachwood, Ohio, USA, 2013. doi:10.1214/12-IMSCOLL913. https://projecteuclid.org/euclid.imsc/1362751187.

#### References

• [1] Chernoff, H. (1964). Estimation of the Mode. Ann. Inst. Statist. Math. 16 31–41.
• [2] Durot, C. (2008). Testing convexity or concavity of a cumulated hazard rate. IEEE Trans. Rel. 57 465–473.
• [3] van Es, A. J., Jongbloed, G. and van Zuijlen, M. C. A. (1998). Isotonic inverse estimators for nonparametric deconvolution. Ann. Statist. 26 2395–2406.
• [4] Groeneboom, P. (1983). The concave majorant of Brownian motion. Ann. Probab. 11 1016–1027.
• [5] Groeneboom, P. (1989). Brownian motion with a parabolic drift and Airy functions. Probab. Theory Related Fields 81 79–109.
• [6] Groeneboom, P. and Jongbloed, G. (1995). Isotonic estimation and rates of convergence in Wicksell’s problem. Ann. Statist. 23 1518–1542.
• [7] Groeneboom, P. and Jongbloed, G. (2003). Density estimation in the uniform deconvolution model. Stat. Neerl. 57 136–157.
• [8] Groeneboom, P. and Jongbloed, G. (2010). Generalized continuous isotonic regression. Statist. Probab. Lett. 80 248–253.
• [9] Groeneboom, P. and Jongbloed, G. (2012). Testing monotonicity of a hazard: asymptotic distribution theory. Bernoulli. To appear.
• [10] Groeneboom, P. and Jongbloed, G. (2012). Isotonic $L_2$-projection test for local monotonicity of a hazard. J. Statist. Plann. Inference 142 1644–1658.
• [11] Groeneboom, P. and Wellner, J. A. (2001). Computing Chernoff’s distribution. J. Comput. Graph. Statist. 10 388–400.
• [12] Gijbels, I. and Heckman, N. (2004). Nonparametric testing for a monotone hazard function via normalized spacings. J. Nonparametr. Stat. 16 463–478.
• [13] Hall, P. and Keilegom, I. van (2005). Testing for monotone increasing hazard rate. Ann. Statist. 33 1109–1137.
• [14] Kiefer, J. and Wolfowitz, J. (1976). Asymptotically minimax estimation of concave and convex distribution functions. Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete 32 111–131.
• [15] Kim, J. and Pollard, D. (1990). Cube root asymptotics. Ann. Statist. 18 191–219.
• [16] Neumeyer, N. (2007). A note on uniform consistency of monotone function estimators. Statist. Probab. Lett. 77 693–703.
• [17] Pal, J. K. (2008). Spiking problem in monotone regression: Penalized residual sum of squares. Statist. Probab. Lett. 78 1548–1556.
• [18] Pal, J. K. (2009). End-point estimation for decreasing densities: asymptotic behaviour of the penalized likelihood ratio. Scand. J. Stat. 36 764–781.
• [19] Pal, J. K. and Woodroofe, M. (2006). On the distance between cumulative sum diagram and its greatest convex minorant for unequally spaced design points. Scand. J. Stat. 33 279–291.
• [20] Pal, J. K. and Woodroofe, M. (2007). Large sample properties of shape restricted regression estimators with smoothness adjustments. Statist. Sinica 17 1601–1616.
• [21] Prakasa Rao, B. L. S. (1970). Estimation for distributions with monotone failure rate. The Annals of Mathematical Statistics 41 507–519.
• [22] Proschan, F. and Pyke, R. (1967). Tests for monotone failure rate. Proceedings of the Fifth Berkeley Symposium on Mathematical Statistics and Probability 3 293–312.
• [23] Rice, J. and Rosenblatt, M. (1976). Estimation of the log survivor function and hazard function. Sankhya A 38 60–78.
• [24] Robertson, T., Wright, F. and Dykstra, R. (1972). Order Restricted Statistical Inference. John Wiley & Sons, New York.
• [25] Singpurwalla, N. D. and Wong, M. Y. (1983). Estimation of the failure rate—A survey of nonparametric methods, Part 1: Non-Bayesian methods. Comm. Statist. Theory Methods 12 559–588.
• [26] Tantiyaswasdikul, C. and Woodroofe, M. B. (1994). Isotonic smoothing splines under sequential designs. J. Statist. Plann. Inference 38 75–88.
• [27] Woodroofe, M. and Sun, J. (1993). A penalized likelihood estimate of $f(0+)$ when $f$ is nonincreasing. Statist. Sinica 3 501–515.
• [28] Woodroofe, M. and Sun, J. (1999). Testing uniformity versus a monotone density. Ann. Statist. 27 338–360.