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Generalization of likelihood ratio tests under nonstandard conditions. (English) Zbl 0873.62022
Summary: We analyze the statistic which is the difference in the values of an estimating function evaluated at its local maxima on two different subsets of the parameter space, assuming that the true parameter is in each subset, but possibly on the boundary. Our results extend known methods by covering a large class of estimation problems which allow sampling from nonidentically distributed random variables.
Specifically, the existence and consistency of the local maximum estimators and asymptotic properties of useful hypothesis tests are obtained under certain law of large numbers and central limit-type assumptions. Other models covered include those with general log-likelihoods and/or covariates. As an example, the large sample theory of two-way nested random variance components models with covariates is derived from our main results.

62F05 Asymptotic properties of parametric tests
62F12 Asymptotic properties of parametric estimators
62E20 Asymptotic distribution theory in statistics
62F03 Parametric hypothesis testing
62H15 Hypothesis testing in multivariate analysis
62J10 Analysis of variance and covariance (ANOVA)
Full Text: DOI
[1] BILLINGSLEY, P. 1968. Convergence of Probability Measures. Wiley, New York. Z. · Zbl 0172.21201
[2] CHANT, D. 1974. On asy mptotic tests of composite hy potheses in nonstandard conditions. Biometrika 61 291 298. Z. JSTOR: · Zbl 0289.62022
[3] CHERNOFF, H. 1954. On the distribution of the likelihood ratio. Ann. Math. Statist. 25 573 578. Z. · Zbl 0056.37102
[4] CROWDER, M. 1990. On some nonregular tests for a modified Weibull model. Biometrika 77 449 506. JSTOR:
[5] FAHRMEIR, L. and KAUFMANN, H. 1985. Consistency and asy mptotic normality of the maximum likelihood estimator in generalized linear models. Ann. Statist. 13 342 368. Z. · Zbl 0594.62058
[6] FEDER, P. I. 1968. On the distribution of the log likelihood ratio test statistic when the true parameter is ‘near’ the boundaries of the hy pothesis region. Ann. Math. Statist. 39 2044 2055. Z. · Zbl 0212.23002
[7] GEy ER, C. J. 1994. On the asy mptotics of constrained M-estimation. Ann. Statist. 22 1993 2010. Z. · Zbl 0829.62029
[8] GHITANY, M. E., MALLER, R. A. and ZHOU, S. 1994. Exponential mixture models with long-term survivors and covariates. J. Multivariate Anal. 49 218 241. Z. · Zbl 0805.62028
[9] MORAN, A. P. 1971. Maximum-likelihood estimation in non-standard conditions. Proc. Cambridge Philos. Soc. 70 441 450. Z. · Zbl 0224.62013
[10] SEARLE, S. R., CASELLA, G. and MCCULLOCH, C. E. 1992. Variance Components. Wiley, New York. Z. · Zbl 0850.62007
[11] SELF, S. G. and LIANG, K. Y. 1987. Asy mptotic properties of maximum likelihood estimators and likelihood ratio tests under nonstandard conditions. J. Amer. Statist. Assoc. 82 605 610. JSTOR: · Zbl 0639.62020
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