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Likelihood ratio tests for monotone functions. (English) Zbl 1043.62037

Summary: 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.

MSC:

62G10 Nonparametric hypothesis testing
62G05 Nonparametric estimation
60J65 Brownian motion
62E20 Asymptotic distribution theory in statistics
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[1] 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. · Zbl 0066.38502
[2] Banerjee, M. (2000). Likelihood Ratio Inference in Regular and Nonregular Problems. Ph.D. dissertation, Univ. Washington.
[3] Banerjee, M. and Wellner, J. A. (2000). Likelihood ratio tests for monotone functions. Technical Report 377, Dept. Statistics, Univ. Washington. · Zbl 1043.62037
[4] Banerjee, M. and Wellner, J. A. (2001). Tests and confidence intervals for monotone functions: further developements. Technical report, Dept. Statistics, Univ. Washington. In preparation.
[5] Barlow, R. E., Bartholomew, Bremner, J. M. and Brunk, H. D. (1972). Statistical Inference under Order Restrictions Wiley, New York. · Zbl 0246.62038
[6] 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. · Zbl 0379.62026
[7] Brunk, H. D. (1970). Estimation of isotonic regression. Nonparametric Techniques in Statistical Inference (M. L. Puri, ed.) 177-195. Cambridge Univ. Press.
[8] Grenander, U. (1956). On the theory of mortality measurement, Part II. Skand. Actuar. 39 125- 153. · Zbl 0077.33715
[9] Groeneboom, P. (1983). The concave majorant of Brownian motion. Ann. Probab. 11 1016-1027. · Zbl 0523.60079
[10] 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. · Zbl 1373.62144
[11] 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. · Zbl 1043.62026
[12] Groeneboom, P. and Wellner, J. A. (1992). Information Bounds and Nonparametric Maximum Likelihood Estimation. Birkhäuser, Boston. · Zbl 0757.62017
[13] Groeneboom, P. and Wellner, J. A. (2001). Computing Chernoff’s distribution. J. Comput. Graph. Statist. 10 388-400. JSTOR: · Zbl 04567029
[14] Huang, J. and Wellner, J. A. (1995). Estimation of a monotone density or monotone hazard under random censoring. Scand. J. Statist. 22 3-33. · Zbl 0827.62032
[15] Huang, Y. and Zhang, C. H. (1994). Estimating a monotone density from censored observations. Ann. Statist. 22 1256-1274. · Zbl 0821.62016
[16] Kim, J. and Pollard, D. (1990). Cube root asymptotics. Ann. Statist. 18 191-219. · Zbl 0703.62063
[17] Leurgans, S. (1982). Asymptotic distributions of slope-of-greatest-convex-minorant estimators. Ann. Statist. 10 287-296. · Zbl 0484.62033
[18] Murphy, S. and Van der Vaart, A. W. (1997). Semiparametric likelihood ratio inference. Ann. Statist. 25 1471-1509. · Zbl 0928.62036
[19] Owen, A. (1995). Nonparametric likelihood confidence bands for a distribution function. J. Amer. Statist. Assoc. 90 516-521. JSTOR: · Zbl 0925.62170
[20] Prakasa Rao, B. L. S. (1969). Estimation of a unimodal density. Sankh\?ya. Ser. A 31 23-36. · Zbl 0181.45901
[21] Schick, A. and Yu, Q. (1999). Consistency of the GMLE with mixed case interval-censored data. Scand. J. Statist. 27 45-55. · Zbl 0938.62109
[22] 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. · Zbl 0862.60002
[23] 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.
[24] 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. · Zbl 1105.62372
[25] 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. umich.edu/ michaelw/. URL: · Zbl 0985.62076
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