Abstract
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.
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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
Attouch H (1985) Variational convergence for function and operators. Pitman, London
Bartholomew DJ (1959) A test of homogeneity for ordered alternatives. Biometrika 46:36–48
Bazaraa MS, Shetty CM (1979) Nonlinear programming: theory and algorithms. Wiley, New York
Billingsley P (1968) Convergence of probability measures. Wiley, New York
Chernoff H (1954) On the distribution of the likelihood ratios. Ann Math Statist 25:573–578
Cramér H (1946) Mathematical methods of statistics. Princeton University Press, Princeton
El Barmi H, Dykstra R (1999) Likelihood ratio test against a set of inequality constraints. J Nonparametric Stat 11:233–250
Feder P (1968) On the distribution of the likelihood ratios. Ann Math Stat 49:633–643
Geyer CJ (1994) On the asymptotics of constrained M-estimation. Ann Stat 22:1993–2010
Kudô A (1963) Multivariate analogue of the one-sided test. Biometrika 50:403–418
Liu XS, Wang JD (2003) Testing for increasing convex order in several populations. Ann Inst Stat Math 55:121–136
Mann HB, Wald A (1943) On stochastic limit and order relationships. Ann Math Stat 14:217–226
Prakasa Rao BLS (1975) Tightness of probability measures generated by stochastic processes on metric spaces. Bull Inst Math Acad Sin 3:353–367
Prakasa Rao BLS (1987) Asymptotic theory of statistical inference. Wiley, New York
Robertson T, Wegman EJ (1978) Likelihood ratio tests for order restrictions in exponential families. Ann Stat 6:485–505
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
Shapiro A (1985) Asymptotic distribution of test statistics in the analysis of moment structures under inequality constraints. Biometrika 72:133–144
Shapiro A (1988) Towards a unified theory of inequality constrained testing in multivariate analysis. Int Stat Rev 56:49–62
Silvey SD (1959) The Lagrangian multiplier test. Ann Math Stat 30:389–407
Vu H, Zhou S (1997) Generalization of likelihood ratio tests under nonstandard conditions. Ann Stat 25:897–916
Wang J (1996) The asymptotics of least-squares estimators for constrained nonlinear regression. Ann Stat 24:1316–1326
Wilks S (1938) The large sample distribution of the likelihood ratios for testing composite hypothesis. Ann Math Stat 9:60–62
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Liu, X. Likelihood Ratio Test for and Against Nonlinear Inequality Constraints. Metrika 65, 93–108 (2007). https://doi.org/10.1007/s00184-006-0062-y
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DOI: https://doi.org/10.1007/s00184-006-0062-y