zbMATH — the first resource for mathematics

Minimum Hellinger distance estimation in a nonparametric mixture model. (English) Zbl 1156.62024
Summary: We investigate the estimation problem of the mixture proportion \(\lambda \) in a nonparametric mixture model of the form \(\lambda F(x)+(1-\lambda )G(x)\) using the minimum Hellinger distance approach, where \(F\) and \(G\) are two unknown distributions. We assume that data from the distributions \(F\) and \(G\) as well as from the mixture distribution \(\lambda F+(1-\lambda )G\) are available. We construct a minimum Hellinger distance estimator of \(\lambda \) and study its asymptotic properties. The proposed estimator is chosen to minimize the Hellinger distance between a parametric mixture model and a nonparametric density estimator. We also develop a maximum likelihood estimator of \(\lambda \). Theoretical properties such as the existence, strong consistency, asymptotic normality and asymptotic efficiency of the proposed estimators are investigated. Robustness properties of the proposed estimator are studied using a Monte Carlo study. Two real data examples are also analyzed.

62G05 Nonparametric estimation
62F12 Asymptotic properties of parametric estimators
62G07 Density estimation
65C05 Monte Carlo methods
62F10 Point estimation
62F35 Robustness and adaptive procedures (parametric inference)
PDF BibTeX Cite
Full Text: DOI
[1] Anderson, J.A., Multivariate logistic compounds, Biometrika, 66, 17-26, (1979) · Zbl 0399.62029
[2] Beran, R., Minimum Hellinger distance estimators for parametric models, Ann. statist., 5, 445-463, (1977) · Zbl 0381.62028
[3] Chen, J., Optimal rate of convergence in finite mixture models, Ann. statist., 23, 221-234, (1992)
[4] Chen, J., Penalized likelihood-ratio test for finite mixture models with multinomial observations, Canad. J. statist., 26, 583-599, (1998) · Zbl 1066.62509
[5] Communications in Statistics, 1976. Special Issue on Remote Sensing, vol. A5, no. 12.
[6] Cordero-Braña, O.I., 1994. Minimum Hellinger distance estimation for finite mixture models. Unpublished Ph.D. dissertation, Utah State University.
[7] Cutler, A.; Cordero-Braña, O.I., Minimum Hellinger distance estimation for finite mixture models, J. amer. statist. assoc., 91, 1716-1723, (1996) · Zbl 0881.62035
[8] Devroye, L.P.; Gyorfi, L., Nonparametric density estimation: the \(l_1\) view, (1985), Wiley New York
[9] Devroye, L.P.; Wagner, T.J., The \(L_1\) convergence of kernel density estimates, Ann. statist., 7, 1136-1139, (1979) · Zbl 0423.62031
[10] Donoho, D.L.; Liu, R.C., The “automatic” robustness of minimum distance functionals, Ann. statist., 16, 552-586, (1988) · Zbl 0684.62030
[11] Fernholz, L., 1983. Von Mises calculus for statistical functionals. Lecture Notes in Statistics, vol. 19, Springer, New York. · Zbl 0525.62031
[12] Hall, P., On the nonparametric estimation of mixture proportions, J. roy. statist. soc. B, 43, 147-156, (1981) · Zbl 0472.62052
[13] Hall, P., Orthogonal series distribution function estimation, with applications, J. roy. statist. soc. B, 45, 81-88, (1983) · Zbl 0533.62036
[14] Hall, P.; Titterington, D.M., Efficient nonparametric estimation of mixture proportions, J. roy. statist. soc. B, 46, 465-473, (1984) · Zbl 0586.62050
[15] Hampel, F., A general qualitative definition of robustness, Ann. math. statist., 42, 1887-1896, (1971) · Zbl 0229.62041
[16] Hosmer, D.W., A comparison of iterative maximum likelihood estimates of the parameters of a mixture of two normal distributions under three types of samples, Biometrics, 29, 761-770, (1973)
[17] Karlis, D.; Xekalaki, E., Minimum Hellinger distance estimation for Poisson mixtures, Comput. statis. data anal., 29, 81-103, (1998) · Zbl 1042.62515
[18] Karunamuni, R.J., Wu, J., 2007. Minimum Hellinger distance estimation in a nonparametric mixture model. Technical Report, University of Alberta. · Zbl 1156.62024
[19] Lindsay, B.G., Efficiency versus robustness the case for minimum Hellinger distance and related methods, Ann. statist., 22, 1081-1114, (1994) · Zbl 0807.62030
[20] Lindsay, B.G., 1995. Mixture models: theory, geometry and applications. In: NSF-CBMS Regional Conference Series in Probability and Statistics, IMS, Hayward, CA, USA. · Zbl 1163.62326
[21] Lu, Z.; Hui, Y.V.; Lee, A.H., Minimum Hellinger distance estimation for finite mixtures of Poisson regression models and its applications, Biometrics, 59, 1016-1026, (2003) · Zbl 1274.62171
[22] Mclachlan, G.J.; Peel, D., Finite mixture models, (2000), Wiley New York · Zbl 0963.62061
[23] Qin, J., Empirical likelihood ratio based confidence intervals for mixture proportions, Ann. statist., 27, 1368-1384, (1999) · Zbl 0960.62048
[24] Rousseeuw, P.J.; Croux, C., Alternatives to the Median absolute deviation, J. amer. statist. assoc., 88, 1273-1283, (1993) · Zbl 0792.62025
[25] Scott, D.W., Multivariate density estimation: theory, practice and visualization, (1992), Wiley New York · Zbl 0850.62006
[26] Scott, D.W., Parametric statistical modeling by minimum integrated square error, Technometrics, 43, 274-285, (2001)
[27] Simpson, D.G., Minimum Hellinger distance estimation for the analysis of count data, J. amer. statist. assoc., 82, 802-807, (1987) · Zbl 0633.62029
[28] Sriram, T.N.; Vidyashankar, A.N., Minimum Hellinger distance estimation for supercritical galton – watson processes, Statist. probab. lett., 50, 331-342, (2000) · Zbl 0968.62060
[29] Tamura, R.N.; Boos, D.D., Minimum Hellinger distance estimation for multivariate location and covariance, J. amer. statist. assoc., 81, 223-229, (1986) · Zbl 0601.62051
[30] Titterington, D.M., Minimum distance nonparametric estimation of mixture proportions, J. roy. statist. soc. B, 45, 37-46, (1983) · Zbl 0563.62027
[31] Titterington, D.M.; Smith, A.F.M.; Makov, U.E., Statistical analysis of finite mixture distributions, (1985), Wiley New York · Zbl 0646.62013
[32] Woo, M.-J.; Sriram, T.N., Robust estimation of mixture complexity, J. amer. statist. assoc., 101, 1475-1486, (2006) · Zbl 1171.62322
[33] Woodward, W.A.; Whitney, P.; Eslinger, P.W., Minimum Hellinger distance estimation of mixture proportions, J. statist. plann. inference, 48, 303-319, (1995) · Zbl 0844.62021
[34] Yang, S., Minimum Hellinger distance estimation of parameter in the random censorship model, Ann. statist., 19, 579-602, (1991) · Zbl 0735.62036
[35] Ying, Z., Minimum Hellinger-type distance estimation for censored data, Ann. statist., 20, 1361-1390, (1992) · Zbl 0772.62014
[36] Zhang, B., An EM algorithm for a semiparametric finite mixture model, J. statist. comput. simul., 72, 791-802, (2002) · Zbl 1015.62029
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.