The Annals of Statistics

Likelihood Ratio Tests for Monotone Functions

Moulinath Banerjee and Jon A. Wellner

Full-text: Open access


We study the problem of testing for equality at a fixed point in the setting of nonparametric estimation of a monotone function.The likelihood ratio test for this hypothesis is derived in the particular case of interval censoring (or current status data)and its limiting distribution is obtained. The limiting distribution is that of the integral of the difference of the squared slope processes corresponding to a canonical version of the problem involving Brownian motion $+t^2$ and greatest convex minorants thereof. Inversion of the family of tests yields pointwise confidence intervals for the unknown distribution function.We also study the behavior of the statistic under local and fixed alternatives.

Article information

Ann. Statist., Volume 29, Number 6 (2001), 1699-1731.

First available in Project Euclid: 5 March 2002

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62G05: Estimation
Secondary: 60G15: Gaussian processes 62E20: Asymptotic distribution theory

Asymptotic distribution Brownian motion constrained estimation fixed alternatives Gaussian process greatest convex minorant interval censoring Kullback-Leibler discrepancy least squares local alternatives likelihood ratio monotone function slope processes


Banerjee, Moulinath; Wellner, Jon A. Likelihood Ratio Tests for Monotone Functions. Ann. Statist. 29 (2001), no. 6, 1699--1731. doi:10.1214/aos/1015345959.

Export citation


  • Ayer, M., Brunk, H.D., Ewing, G.M., Reid, W.T. and Silverman, E. (1955). An empirical distribution function for sampling with incomplete information. Ann. Math. Statist. 26 641-647.
  • Banerjee, M. (2000). Likelihood Ratio Inference in Regular and Nonregular Problems. Ph.D. dissertation, Univ. Washington.
  • Banerjee, M. and Wellner, J. A. (2000). Likelihood ratio tests for monotone functions. Technical Report 377, Dept. Statistics, Univ. Washington.
  • Banerjee, M. and Wellner, J. A. (2001). Tests and confidence intervals for monotone functions: further developements. Technical report, Dept. Statistics, Univ. Washington. In preparation.
  • Barlow, R. E., Bartholomew, Bremner, J. M. and Brunk, H. D. (1972). Statistical Inference under Order Restrictions Wiley, New York.
  • Berk, R. H. and Jones, D. H. (1979). Goodness-of-fit test statistics that dominate the Kolmogorov statistics. Z. Wahrsch. Verw. Gebiete 47 47-59.
  • Brunk, H. D. (1970). Estimation of isotonic regression. Nonparametric Techniques in Statistical Inference (M. L. Puri, ed.) 177-195. Cambridge Univ. Press.
  • Grenander, U. (1956). On the theory of mortality measurement, Part II. Skand. Actuar. 39 125- 153.
  • Groeneboom, P. (1983). The concave majorant of Brownian motion. Ann. Probab. 11 1016-1027.
  • Groeneboom, P. (1985). Estimating a monotone density. In Proceedings of the Berkeley Conference in Honor of Jerzy Neyman and JackKiefer (L. M. LeCam and R. A. Olshen, eds.) 2 529-555. Wadsworth, Belmont, CA.
  • Groeneboom, P. (1988). Brownian motion with a parabolic drift and Airy functions. Probab. Theory Related Fields 81 79-109. Groeneboom, P., Jongbloed, G. and Wellner, J. A. (2000a). A canonical process for estimation of convex functions: the "invelope" of integrated Brownian motion +t4. Technical Report 369, Dept. Statistics, Univ. Washington. Groeneboom, P., Jongbloed, G. and Wellner, J. A. (2000b). Estimation of convex functions: characterizations and asymptotic theory. Technical Report 372, Dept. Statistics, Univ. Washington.
  • Groeneboom, P. and Wellner, J. A. (1992). Information Bounds and Nonparametric Maximum Likelihood Estimation. Birkh¨auser, Boston.
  • Groeneboom, P. and Wellner, J. A. (2001). Computing Chernoff's distribution. J. Comput. Graph. Statist. 10 388-400.
  • Huang, J. and Wellner, J. A. (1995). Estimation of a monotone density or monotone hazard under random censoring. Scand. J. Statist. 22 3-33.
  • Huang, Y. and Zhang, C. H. (1994). Estimating a monotone density from censored observations. Ann. Statist. 22 1256-1274.
  • Kim, J. and Pollard, D. (1990). Cube root asymptotics. Ann. Statist. 18 191-219.
  • Leurgans, S. (1982). Asymptotic distributions of slope-of-greatest-convex-minorant estimators. Ann. Statist. 10 287-296.
  • Murphy, S. and Van der Vaart, A. W. (1997). Semiparametric likelihood ratio inference. Ann. Statist. 25 1471-1509.
  • Owen, A. (1995). Nonparametric likelihood confidence bands for a distribution function. J. Amer. Statist. Assoc. 90 516-521.
  • Prakasa Rao, B. L. S. (1969). Estimation of a unimodal density. Sankh¯ya. Ser. A 31 23-36.
  • Schick, A. and Yu, Q. (1999). Consistency of the GMLE with mixed case interval-censored data. Scand. J. Statist. 27 45-55.
  • Van der Vaart, A. W. and Wellner, J. A. (1996). WeakConvergence and Empirical Processes. Springer, New York. Van Eeden, C. (1957a). Maximum likelihood estimation of partially or completely ordered parameters, I. Proc. K. Ned. Akad. Wet. 60; Indag. Math. 19 128-136. Van Eeden, C. (1957b). Maximum likelihood estimation of partially or completely ordered parameters, II. Proc. K. Ned. Akad. Wet. 60; Indag. Math. 19 201-211.
  • Wellner, J. A. (2001). Gaussian white noise models: a partial review and results for monotone functions. In IMS Lecture Notes and Monograph Series Volume in Honor of W. J. Hall. To appear.
  • Wellner, J. A. and Zhang, Y. (2000). Two estimators of the mean of a counting process with panel count data. Ann. Statist. 28 779-814.
  • Wu, W. B., Woodroofe, M. and Mentz, G. (2001). Isotonic regression: another look at the change point problem. Technical report, Univ. Michigan. Available at www.stat.lsa. michaelw/.