Strong consistency of the maximum likelihood estimator for finite mixtures of location–scale distributions when the scale parameters are exponentially small

Bernoulli

Volumn 12, Issue 6, 2006, Pages 1003-1017

  Tanaka, Kentaro ^a   Takemura, Akimichi ^b  

^a TOKYO INSTITUTE OF TECHNOLOGY   (Japan)

^b UNIVERSITY OF TOKYO   (Japan)

Author keywords

Consistency; Maximum likelihood estimator; Mixture distribution

Indexed keywords

EID: 60249101399     PISSN: 13507265     EISSN: None     Source Type: Journal    
DOI: 10.3150/bj/1165269148     Document Type: Article

References (12)

1. Billingsley, P. (1995) Probability and Measure, 3rd edn. New York: Wiley.
2. Everitt, B.S. and Hand, D.J. (1981) Finite Mixture Distributions. London: Chapman & Hall.
3. Geman, S. and Hwang, C.-R. (1982) Nonparametric maximum likelihood estimation by the method of sieves. Ann. Statist., 10, 401-414.
4. Genovese, C.R. and Wasserman, L. (2000) Rates of convergence for the Gaussian mixture sieve. Ann. Statist, 28, 1105-1127.
5. Ghosal S. and van der Vaart, A.W. (2001) Entropies and rates of convergence for maximum likelihood and Bayes estimation for mixtures of normal densities. Ann. Statist., 29, 1233-1263.
6. Grenander, U. (1981) Abstract Inference. New York: Wiley.
7. Hathaway, R.J. (1985) A constrained formulation of maximum-likelihood estimation for normal mixture distributions. Ann. Statist., 13, 795-800.
8. McLachlan, G.J. and Peel, D. (2000) Finite Mixture Models. New York: Wiley.
9. Redner, R. (1981) Note on the consistency of the maximum likelihood estimate for nonidentifiable distributions. Ann. Statist., 9, 225-228.
10. Tanaka, K. and Takemura, A. (2005) Strong consistency of MLE for finite uniform mixtures when the scale parameters are exponentially small. Ann. Inst. Statist. Math., 57, 1-19.
11. Tanaka, K. and Takemura, A. (2006) Strong consistency of MLE for finite mixtures of location-scale distributions when the scale parameters are exponentially small. http://arXiv.org/abs/math.ST/0605148.
12. Wald, A. (1949) Note on the consistency of the maximum likelihood estimate. Ann. Math. Statist, 20, 595-601.