• Bernoulli
  • Volume 19, Number 5A (2013), 1965-1999.

Testing monotonicity of a hazard: Asymptotic distribution theory

Piet Groeneboom and Geurt Jongbloed

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Two test statistics are introduced to test the null hypotheses that the sampling distribution has an increasing hazard rate on a specified interval $[0,a]$. These statistics are empirical $L_{1}$-type distances between the isotonic estimates, which use the monotonicity constraint, and either the empirical distribution function or the empirical cumulative hazard. They measure the excursions of the empirical estimates with respect to the isotonic estimates, owing to local non-monotonicity. Asymptotic normality of the test statistics, if the hazard is strictly increasing on $[0,a]$, is established under mild conditions. This is done by first approximating the global empirical distance by a distance with respect to the underlying distribution function. The resulting integral is treated as sum of increasingly many local integrals to which a central limit theorem can be applied. The behavior of the local integrals is determined by a canonical process, the difference between the stochastic process $x\mapsto W(x)+x^{2}$, where $W$ is standard two-sided Brownian motion, and its greatest convex minorant.

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Bernoulli, Volume 19, Number 5A (2013), 1965-1999.

First available in Project Euclid: 5 November 2013

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convex minorant failure rate global asymptotics Hungarian embedding


Groeneboom, Piet; Jongbloed, Geurt. Testing monotonicity of a hazard: Asymptotic distribution theory. Bernoulli 19 (2013), no. 5A, 1965--1999. doi:10.3150/12-BEJ437.

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  • [1] Dudley, R.M. and Norvaiša, R. (1999). Differentiability of Six Operators on Nonsmooth Functions and $p$-Variation. Lecture Notes in Math. 1703. Berlin: Springer. With the collaboration of Jinghua Qian.
  • [2] Durot, C. (2008). Testing convexity or concavity of a cumulated hazard rate. IEEE Transactions on Reliability 57 465–473.
  • [3] Gijbels, I. and Heckman, N. (2004). Nonparametric testing for a monotone hazard function via normalized spacings. J. Nonparametr. Stat. 16 463–477.
  • [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. (2011). Vertices of the least concave majorant of Brownian motion with parabolic drift. Electron. J. Probab. 16 2234–2258.
  • [7] Groeneboom, P. and Jongbloed, G. (2012). Smooth and non-smooth estimates of a monotone hazard. In From Probability to Statistics and Back: High-Domensional Models and Processes. IMS Collections 9 174–196. Beachwood, OH: IMS.
  • [8] Groeneboom, P. and Jongbloed, G. (2012). Isotonic ${L}_{2}$-projection test for local monotonicity of a hazard. J. Statist. Plann. Inference. 142 1644–1658.
  • [9] Groeneboom, P. and Temme, N.M. (2011). The tail of the maximum of Brownian motion minus a parabola. Electron. Commun. Probab. 16 458–466.
  • [10] Groeneboom, P. and Wellner, J.A. (1992). Information Bounds and Nonparametric Maximum Likelihood Estimation. DMV Seminar 19. Basel: Birkhäuser.
  • [11] Hall, P. and Van Keilegom, I. (2005). Testing for monotone increasing hazard rate. Ann. Statist. 33 1109–1137.
  • [12] Huang, Y.C. and Dudley, R.M. (2001). Speed of convergence of classical empirical processes in $p$-variation norm. Ann. Probab. 29 1625–1636.
  • [13] Ibragimov, I.A. and Linnik, Y.V. (1971). Independent and Stationary Sequences of Random Variables. Groningen: Wolters-Noordhoff. With a supplementary chapter by I.A. Ibragimov and V.V. Petrov, Translation from the Russian edited by J.F.C. Kingman.
  • [14] Komlós, J., Major, P. and Tusnády, G. (1975). An approximation of partial sums of independent $\mathrm{RV}$’s and the sample $\mathrm{DF}$. I. Z. Wahrsch. Verw. Gebiete 32 111–131.
  • [15] Kulikov, V.N. and Lopuhaä, H.P. (2008). Distribution of global measures of deviation between the empirical distribution function and its concave majorant. J. Theoret. Probab. 21 356–377.
  • [16] Loève, M. (1963). Probability Theory, 3rd ed. Princeton: Van Nostrand.
  • [17] Proschan, F. and Pyke, R. (1967). Tests for monotone failure rate. In Proc. Fifth Berkeley Sympos. Mathematical Statistics and Probability (Berkeley, Calif., 1965/66), Vol. III: Physical Sciences 293–312. Berkeley, CA: Univ. California Press.
  • [18] Qian, J. (1998). The $p$-variation of partial sum processes and the empirical process. Ann. Probab. 26 1370–1383.
  • [19] Robertson, T., Wright, F.T. and Dykstra, R.L. (1988). Order Restricted Statistical Inference. Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics. Chichester: Wiley.
  • [20] Rosenblatt, M. (1956). A central limit theorem and a strong mixing condition. Proc. Natl. Acad. Sci. USA 42 43–47.
  • [21] 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.