The Annals of Statistics

Empirical likelihood ratio based confidence intervals for mixture proportions

Jing Qin

Full-text: Open access

Abstract

We consider the problem of estimating a mixture proportion using data from two different distributions as well as from a mixture of them. Under the model assumption that the log-likelihood ratio of the two densities is linear in the observations, we develop an empirical likelihood ratio based statistic for constructing confidence intervals for the mixture proportion. Under some regularity conditions, it is shown that this statistic converges to a chi-squared random variable. Simulation results indicate that the performance of this statistic is satisfactory. As a by-product, we give estimators for the two distribution functions. Connections with case-control studies and discrimination analysis are pointed out.

Article information

Source
Ann. Statist., Volume 27, Number 4 (1999), 1368-1384.

Dates
First available in Project Euclid: 4 April 2002

Permanent link to this document
https://projecteuclid.org/euclid.aos/1017938930

Digital Object Identifier
doi:10.1214/aos/1017938930

Mathematical Reviews number (MathSciNet)
MR1740107

Zentralblatt MATH identifier
0960.62048

Subjects
Primary: 62M10: Time series, auto-correlation, regression, etc. [See also 91B84]
Secondary: 62E20: Asymptotic distribution theory

Keywords
Case-control studies chi-squared distribution empirical likelihood exponential tilt models logistic function mixture models

Citation

Qin, Jing. Empirical likelihood ratio based confidence intervals for mixture proportions. Ann. Statist. 27 (1999), no. 4, 1368--1384. doi:10.1214/aos/1017938930. https://projecteuclid.org/euclid.aos/1017938930


Export citation

References

  • ANDERSON, J. A. 1979. Multivariate logistic compounds. Biometrika 66 17 26. Z.
  • BILLINGSLEY, P. 1968. Convergence of Probability Measures. Wiley, New York. Z.
  • BRESLOW, N. and DAY, N. E. 1980. Statistical Methods in Cancer Research: I. The Analysis of Case-Control Studies. IARC, Lyon. Z.
  • CHEN, S. X. and HALL, P. 1993. Smoothed empirical likelihood confidence intervals for quantiles. Ann. Statist. 21 1166 1181. Z.
  • COX, D. R. and SNELL, E. J. 1989. Analysis of Binary Data, 2nd ed. Chapman and Hall, London. Z.
  • DICICCIO, T. J., HALL, P. and ROMANO, J. P. 1989. Comparison of parametric and empirical likelihood. Biometrika 76 447 456. Z.
  • EFRON, B. 1975. The efficiency of logistic regression compared to discrimination analysis. J. Amer. Statist. Assoc. 70 892 897. Z.
  • EFRON, B. and TIBSHIRANI, R. 1996. Using specially designed exponential families for density estimation. Ann. Statist. 24 2431 2461. Z.
  • HALL, P. 1990. Pseudo-likelihood theory for empirical likelihood. Ann. Statist. 18 121 140. Z.
  • HALL, P. and LA SCALA, B. 1990. Methodology and algorithms of empirical likelihood. I. S. Review 58 109 127. Z.
  • HALL, P. and TITTERINGTON, D. M. 1984. Efficient nonparametric estimation of mixture proportions. J. Roy. Statist. Soc. Ser. B 46 465 473. Z.
  • HOSMER, D. W. 1973. A comparison of iterative maximum likelihood estimates of the parameters of a mixture of two normal distributions under three types of sample. Biometrics 29 761 770. Z.
  • LANCASTER, T. and IMBENS, G. 1996. Case-control studies with contaminated controls. J. Econometrics 71 145 160. Z.
  • MURRAY, G. D. and TITTERINGTON, D. M. 1978. Estimation problems with data from a mixture. J. Roy. Statist. Soc. Ser. C 27 325 334. Z. O'NEILL, T. J. 1980. The general distribution of the error rate of a classification procedure with applications to the logistic regression discrimination. J. Amer. Statist. Assoc. 75 154 160. Z.
  • OWEN, A. B. 1988. Empirical likelihood ratio confidence intervals for a single functional. Biometrika 75 237 249. Z.
  • OWEN, A. B. 1990. Empirical likelihood confidence regions. Ann. Statist. 18 90 120. Z.
  • PRENTICE, R. L. and PYKE, R. 1979. Logistic disease incidence models and case-control studies. Biometrika 66 403 411. Z.
  • PRESS, W. H., TEUKOLSKY, S. A., VETTERLING, W. T. and FLANNERY, B. P. 1992. Numerical Recipes in C, 2nd ed. Cambridge Univ. Press. Z.
  • QIN, J. 1993. Empirical likelihood in biased sample problems. Ann. Statist. 21 1182 1196. Z.
  • QIN, J. and LAWLESS, J. F. 1994. Empirical likelihood and general estimating equations. Ann. Statist. 22 300 325.
  • COLLEGE PARK, MARYLAND 20742 AND DEPARTMENT OF EPIDEMIOLOGY AND BIOSTATISTICS MEMORIAL SLOAN KETTERING CANCER CENTER 1275 YORK AVENUE, NEW YORK, NEW YORK 10021