The Korean Journal of Applied Statistics (응용통계연구)

The Korean Statistical Society (한국통계학회)
권호 표지
DOI QR코드

DOI QR Code

Minimum Density Power Divergence Estimation for Normal-Exponential Distribution

정규-지수분포에 대한 최소밀도함수승간격 추정법

  • Pak, Ro Jin (Department of Applied Statistics, Dankook University)
  • 박노진 (단국대학교 응용통계학과)
  • Received : 2014.01.24
  • Accepted : 2014.04.09
  • Published : 2014.06.30
Download PDF
⟨ Previous Next ⟩

Abstract

The minimum density power divergence estimation has been a popular topic in the field of robust estimation for since Basu et al. (1988). The minimum density power divergence estimator has strong robustness properties with the little loss in asymptotic efficiency relative to the maximum likelihood estimator under model conditions. However, a limitation in applying this estimation method is the algebraic difficulty on an integral involved in an estimation function. This paper considers a minimum density power divergence estimation method with approximated divergence avoiding such difficulty. As an example, we consider the normal-exponential convolution model introduced by Bolstad (2004). The estimated divergence in this case is too complicated; consequently, a Laplace approximation is employed to obtain a manageable form. Simulations and an empirical study show that the minimum density power divergence estimators based on an approximated estimated divergence for the normal-exponential model perform adequately in terms of bias and efficiency.

최소밀도함수승간격 추정법은 Baus 등 (1998)에 의해 처음 소개된 이후 많은 관심의 대상이 되었다. 최소밀도함수승간격 추정량은 우수한 로버스트 성질을 갖고 효율성도 최우추정량에 필적한 것으로 알려져 있다. 본 논문에서는 생물정보학에서 사용되는 노말-지수 분포에 근거한 추정량을 최소밀도함수승간격 추정법을 사용하여 구하는 방법을 다루고자 한다. 그런데 그 과정에서 간격을 적분을 통해 구하는 것이 매우 어려움으로 인해 직접적인 적분 대신 라플라스 근사를 시도할 것을 제안한다. 그 결과 추정량이 다소 효율성이 줄어들지만 로버스트 성질을 갖고 있음을 수학적 방법과 모의실험을 통하여 보였다.

Keywords

References

  1. Basu, A., Harris, I. R., Hjort, N. L. and Jones, M. C. (1998). Robust and efficient estimation by minimizing a density power divergence, Biometrika, 85, 549-559. https://doi.org/10.1093/biomet/85.3.549
  2. Bolstad, B. M. (2004). Low Level Analysis of High-density Oligonucleotide Array Data: Background, Nor-malization and Summarization, Dissertation, University of California-Berkeley.
  3. Byrd, R. H., Lu, P., Nocedal, J. and Zhu, C. (1995). A limited memory algorithm for bound constrained optimization, SIAM J. Sci Comput, 16, 1190-1208. https://doi.org/10.1137/0916069
  4. Hampel, F. R., Ronchetti, E. M., Rousseeuw, P. J. and Stahel, W. A. (1986). Robust Statistics: The Approach Based on Influence Functions, John Wiley & Sons, New York.
  5. Irizarry, R. A., Hobbs, B., Collin, F., Beazer-Barclay, Y. D., Antonellis, K. J., Scherf, U. and Speed, T. P. (2003). Exploration, normalization and summaries of high density oligonucleotide array probe level data, Biostatistics, 4, 249-264. https://doi.org/10.1093/biostatistics/4.2.249
  6. MacKay, D. J. C. (1998). Choice of basis for Laplace approximation, Mach Learn, 33, 77-86. https://doi.org/10.1023/A:1007558615313
  7. McGee, M. and Chen, Z. (2006). Parameter estimation for the exponential-normal convolution model for background correction of Affymetrix GeneChip data, Stat Appl Genet Molec Biol, 5:Article 24.
  8. Ritchie, M. E., Silver, J., Oshlack, A., Holmes, M., Diyagama, D., Holloway, A. and Smyth, G. K. (2007). A comparison of background correction methods for two-colour microarrays, Bioinformatics, 23, 2700-2707. https://doi.org/10.1093/bioinformatics/btm412
  9. Silver, J., Ritchie, M. E. and Smyth, G .K. (2009). Microarray background correction: maximum likelihood estimation for the normal-exponential convolution model, Biostatistics, 10, 352-363. https://doi.org/10.1093/biostatistics/kxn042