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Likelihood ratio test for and against nonlinear inequality constraints. (English) Zbl 1105.62022
Summary: In applied statistics, a finite-dimensional parameter involved in the distribution function of the observed random variable is very often constrained by a number of nonlinear inequalities. This paper is devoted to studying the likelihood ratio test for and against the hypothesis that the parameter is restricted by some nonlinear inequalities. The asymptotic null distributions of the likelihood ratio statistics are derived by using the limits of the related optimization problems. The author also shows how to compute critical values for the tests.

MSC:
62F03 Parametric hypothesis testing
62F30 Parametric inference under constraints
62E20 Asymptotic distribution theory in statistics
Software:
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[1] Ahn H, Moon H, Kim S, Kodell RL (2002) A Newton-based approach for attributing tumor lethality in animal carcinogenicity studies. Comput Stat Data Anal 38:263–283 · Zbl 1079.62549
[2] Attouch H (1985) Variational convergence for function and operators. Pitman, London
[3] Bartholomew DJ (1959) A test of homogeneity for ordered alternatives. Biometrika 46:36–48 · Zbl 0087.14202
[4] Bazaraa MS, Shetty CM (1979) Nonlinear programming: theory and algorithms. Wiley, New York
[5] Billingsley P (1968) Convergence of probability measures. Wiley, New York · Zbl 0172.21201
[6] Chernoff H (1954) On the distribution of the likelihood ratios. Ann Math Statist 25:573–578 · Zbl 0056.37102
[7] Cramér H (1946) Mathematical methods of statistics. Princeton University Press, Princeton · Zbl 0063.01014
[8] El Barmi H, Dykstra R (1999) Likelihood ratio test against a set of inequality constraints. J Nonparametric Stat 11:233–250 · Zbl 1057.62505
[9] Feder P (1968) On the distribution of the likelihood ratios. Ann Math Stat 49:633–643
[10] Geyer CJ (1994) On the asymptotics of constrained M-estimation. Ann Stat 22:1993–2010 · Zbl 0829.62029
[11] Kudô A (1963) Multivariate analogue of the one-sided test. Biometrika 50:403–418 · Zbl 0121.13906
[12] Liu XS, Wang JD (2003) Testing for increasing convex order in several populations. Ann Inst Stat Math 55:121–136 · Zbl 1052.62023
[13] Mann HB, Wald A (1943) On stochastic limit and order relationships. Ann Math Stat 14:217–226 · Zbl 0063.03774
[14] Prakasa Rao BLS (1975) Tightness of probability measures generated by stochastic processes on metric spaces. Bull Inst Math Acad Sin 3:353–367 · Zbl 0331.60006
[15] Prakasa Rao BLS (1987) Asymptotic theory of statistical inference. Wiley, New York · Zbl 0604.62025
[16] Robertson T, Wegman EJ (1978) Likelihood ratio tests for order restrictions in exponential families. Ann Stat 6:485–505 · Zbl 0391.62016
[17] Self SG, Liang KY (1987) Asymptotic properties of maximum likelihood estimators and likelihood ratio tests under nonstandard conditions. J Amer Stat Assoc 82:605–610 · Zbl 0639.62020
[18] Shapiro A (1985) Asymptotic distribution of test statistics in the analysis of moment structures under inequality constraints. Biometrika 72:133–144 · Zbl 0596.62019
[19] Shapiro A (1988) Towards a unified theory of inequality constrained testing in multivariate analysis. Int Stat Rev 56:49–62 · Zbl 0661.62042
[20] Silvey SD (1959) The Lagrangian multiplier test. Ann Math Stat 30:389–407 · Zbl 0090.36302
[21] Vu H, Zhou S (1997) Generalization of likelihood ratio tests under nonstandard conditions. Ann Stat 25:897–916 · Zbl 0873.62022
[22] Wang J (1996) The asymptotics of least-squares estimators for constrained nonlinear regression. Ann Stat 24:1316–1326 · Zbl 0862.62057
[23] Wilks S (1938) The large sample distribution of the likelihood ratios for testing composite hypothesis. Ann Math Stat 9:60–62 · Zbl 0018.32003
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