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Keywords
(13)
Asymptotic Distribution
Asymptotic Equivalence
Asymptotic Properties
Gaussian Distribution
Gaussian Random Variable
Likelihood Function
Likelihood Ratio Statistic
Limit Distribution
Maximum Likelihood Estimate
Nuisance Parameter
Parameter Space
Likelihood Ratio Test
Normalized Maximum Likelihood
Related Publications
(13)
On the Distribution of the Likelihood Ratio
Maximumlikelihood estimation in nonstandard conditions
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Variance components testing in the longitudinal mixed e ects model
Statistics in Human Genetics
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Asymptotic Properties of Maximum Likelihood Estimators and Likelihood Ratio Tests under Nonstandard Conditions
Asymptotic Properties of Maximum Likelihood Estimators and Likelihood Ratio Tests under Nonstandard Conditions,10.1080/01621459.1987.10478472,Journal
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Asymptotic Properties of Maximum Likelihood Estimators and Likelihood Ratio Tests under Nonstandard Conditions
(
Citations: 751
)
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Steven G. Self
,
KungYee Liang
Large sample properties of the
likelihood function
when the true parameter value may be on the boundary of the
parameter space
are described. Specifically, the
asymptotic distribution
of
maximum likelihood
estimators and
likelihood ratio
statistics are derived. These results generalize the work of Moran (1971), Chant (1974), and Chernoff (1954). Some of Chant's results are shown to be incorrect.The approach used in deriving these results follows from comments made by Moran and Chant. The problem is shown to be asymptotically equivalent to the problem of estimating the restricted mean of a multivariate
Gaussian distribution
from a sample of size 1. In this representation the
Gaussian random variable
corresponds to the limit of the normalized score statistic and the estimate of the mean corresponds to the limit of the
normalized maximum likelihood
estimator. Thus the limiting distribution of the
maximum likelihood
estimator is the same as the distribution of the projection of the
Gaussian random variable
onto the region of admissible values for the mean.A variety of examples is provided for which the limiting distributions of
likelihood ratio
statistics are mixtures of chisquared distributions. One example is provided with a
nuisance parameter
on the boundary for which the
asymptotic distribution
is not a mixture of chisquared distributions.
Journal:
Journal of The American Statistical Association  J AMER STATIST ASSN
, vol. 82, no. 398, pp. 605610, 1987
DOI:
10.1080/01621459.1987.10478472
Cumulative
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dx.doi.org
)
Citation Context
(219)
... Each comparison excluded a single variance component such that the LR was distributed as a 50:50 mixture of a
χ
2
distribution with 1 degree of freedom and point mass of zer
...
S Hong Lee
,
et al.
Estimating the proportion of variation in susceptibility to schizophre...
...). Mixture statistics of this form often arise in hypothesis testing where model parameters take values on the boundaries of the parameter space, and closedform expressions for mixing coefficients are difficult to obtain
...
Ben Murrell
,
et al.
Detecting Individual Sites Subject to Episodic Diversifying Selection
...415–420) for some general classes of large sample tests; see also Self and Liang (
1987
) and Drton (
2009
) for more details...
Martin Singull
,
et al.
More on the Kronecker Structured Covariance Matrix
...For more details, see,
1
5
7
10
11
16
17
19
, for example...
Reza Drikvandi
,
et al.
Testing variance components in balanced linear growth curve models
...For random effects, significance tests are conducted using a likelihood ratio test statistic where the sampling distribution of the test statistic is a mixture of two χ
^{2}
distributions, df = 0 and 1 (
Self & Liang, 1987
Stram & Lee, 1994
1995
;
Verbeke & Molenburghs, 2000
)...
Brennan R. Payne
,
et al.
The effects of print exposure on sentence processing and memory in old...
References
(7)
On asymptotic tests of composite hypotheses in nonstandard conditions
(
Citations: 40
)
D. CHANT
Journal:
Biometrika
, vol. 61, no. 2, pp. 291298, 1974
On the distribution of the likelihood ratio
(
Citations: 86
)
H. Cherno
Mathematical methods of statistics
(
Citations: 2301
)
Harald Cramer
Journal:
Psychnology Journal
, 1961
Theory of point estimation
(
Citations: 1171
)
E. L. Lehmann
Published in 1983.
Asymptotic Properties of Maximum Likelihood Estimates in the Mixed Model of the Analysis of Variance
(
Citations: 67
)
John J. Miller
Journal:
Annals of Statistics  ANN STATIST
, vol. 5, no. 1977, pp. 746762, 1977
Sort by:
Citations
(751)
Estimating the proportion of variation in susceptibility to schizophrenia captured by common SNPs
(
Citations: 1
)
S Hong Lee
,
Teresa R DeCandia
,
Stephan Ripke
,
Jian Yang
,
Patrick F Sullivan
,
Michael E Goddard
,
Matthew C Keller
,
Peter M Visscher
,
Naomi R Wray
Journal:
Nature Genetics  NAT GENET
, vol. 44, no. 3, pp. 247250, 2012
Detecting Individual Sites Subject to Episodic Diversifying Selection
Ben Murrell
,
Joel O. Wertheim
,
Sasha Moola
,
Thomas Weighill
,
Konrad Scheffler
,
Sergei L. Kosakovsky Pond
Journal:
PLOS Genetics  PLOS GENET
, vol. 8, no. 7, 2012
Comparing the Expected Misclassification Cost for Two Classifiers Based on Estimates From the Same Sample
James F. Troendle
,
Kai F. Yu
,
Peter H. Westfall
,
Gene Pennello
,
Enrique F. Schisterman
Published in 2012.
More on the Kronecker Structured Covariance Matrix
Martin Singull
,
M. Rauf Ahmad
,
Dietrich von Rosen
Journal:
Communications in Statisticstheory and Methods  COMMUN STATISTTHEOR METHOD
, vol. 41, no. 1314, pp. 25122523, 2012
Testing variance components in balanced linear growth curve models
Reza Drikvandi
,
Ahmad Khodadadi
,
Geert Verbeke
Journal:
Journal of Applied Statistics  J APPL STAT
, vol. 39, no. 3, pp. 563572, 2012