Communications for Statistical Applications and Methods

The Korean Statistical Society (한국통계학회)
권호 표지
DOI QR코드

DOI QR Code

Note on the Consistency of a Penalized Maximum Likelihood Estimate

벌점가능추정치의 일치성에 대하여

  • Ahn, Sung-Mahn (College of Business Administration, Kookmin University)
  • Published : 2009.07.31
Download PDF
⟨ Previous Next ⟩

Abstract

We prove the consistency of a penalized maximum likelihood estimate proposed by Ahn (2001). The PMLE not only avoids the well-known problem that the ordinary likelihood of the normal mixture model is unbounded for any given sample size, but also removes redundant components.

본 논문에서는 과대적합모형을 가정하여 성분의 수에 대한 벌점항목을 추가한 Ahn (2001)의 벌점가능도에 의한 추정치가 일치성을 달성하는 것에 대한 증명을 제시하였다. Wald (1949)의 조건을 만족하는 지를 보이기 위하여 과대적합모형를 활용하였다. 또한 모수가 한계점으로 수렴할 경우 우도값이 무한대로 간다는 문제점을 해결하였다.

Keywords

References

  1. Ahn, S. M. (2001). A penalized likelihood method for model complexity reduction in Gaussian Mixture density, The Korean Communications in Statistics, 8, 173-184
  2. Chen, J., Tan, X. and Zhang, R. (2008). Consistency of penalized MLE for normal mixtures in mean and variance, Statistica Sinica, 18, 443-465
  3. Cheng, R. C. H. and Liu, W. B. (2001). The consistency of estimators in finite mixtures models, Scandi-navian Journal of Statistics, 28, 603-616 https://doi.org/10.1111/1467-9469.00257
  4. Ciuperca. G., Ridolfi, A. and Idier, J. (2003). Penalized maximum likelihood estimator for normal mix-tures, Scandinavian Journal of Statistics, 30, 45-59 https://doi.org/10.1111/1467-9469.00317
  5. Cramer, H. (1946). Mathematical Methods of Statistics, Princeton University Press, Princeton
  6. Dempster, A. P., Laird, N. M. and Rubin, D. B. (1977). Maximum likelihood from incomplete data via the EM algorithm, Journal of the Royal Statistical Society. Series B, 39, 1-38
  7. Feng, Z. D. and McCulloch, C. E. (1996). Using bootstrap likelihood rations in finite mixture models, Joural of the Royal Statistical Society. Series B, 58, 609-617
  8. Hathaway, R. J. (1985). A constrained formulation of maximum likelihood estimation for normal mixture distributions, The Annals of Statistics, 13, 795-800 https://doi.org/10.1214/aos/1176349557
  9. Redner, R. (1981). Note on the consistency of the maximum likelihood estimate for nonidentifiable distri-butions, The Annals of Statistics, 9, 225-228 https://doi.org/10.1214/aos/1176345353
  10. Wald, A. (1949). Note on the consistency of the maximum likelihood estimate, The Annals of Mathematical Statistics, 20, 595-601 https://doi.org/10.1214/aoms/1177729952