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Zhan, Tingting; Chevoneva, Inna; Iglewicz, Boris

Generalized weighted likelihood density estimators with application to finite mixture of exponential family distributions. (English) Zbl 1247.62113

Comput. Stat. Data Anal. 55, No. 1, 457-465 (2011).
Summary: The family of weighted likelihood estimators largely overlaps with minimum divergence estimators. They are robust to data contaminations compared to MLE. We define the class of generalized weighted likelihood estimators (GWLE), provide its influence function and discuss the efficiency requirements. We introduce a new truncated cubic-inverse weight, which is both first and second order efficient and more robust than previously reported weights. We also discuss new ways of selecting the smoothing bandwidth and weighted starting values for the iterative algorithm. The advantage of the truncated cubic-inverse weight is illustrated in a simulation study of three-component normal mixtures model with large overlaps and heavy contaminations. A real data example is also provided.

Cited in 2 Documents

MSC:

62G07 Density estimation
62G35 Nonparametric robustness
65C60 Computational problems in statistics (MSC2010)

Keywords:

finite normal mixture; generalized weighted likelihood estimator; influence function; smoothing bandwidth; truncated cubic-inverse weight; weighted starting value

Software:

R
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Full Text: DOI Link

References:

[1] Basu, A.; Harris, I. R.; Hjort, N. L.; Jones, M. C., Robust and efficient estimation by minimising a density power divergence, Biometrika, 85, 3, 549-559 (1998) · Zbl 0926.62021
[2] Basu, A.; Lindsay, B. G., Minimum disparity estimation for continuous models: efficiency, distributions and robustness, Annals of the Institute of Statistical Mathematics, 46, 4, 683-705 (1994) · Zbl 0821.62018
[3] Basu, A.; Lindsay, B. G., The iteratively reweighted estimating equation in minimum distance problems, Computational Statistics & Data Analysis, 45, 2, 105-124 (2004) · Zbl 1400.62074
[4] Beran, R., Minimum Hellinger distance estimates for parametric models, The Annals of Statistics, 5, 3, 445-463 (1977) · Zbl 0381.62028
[5] Beran, R., An efficient and robust adaptive estimator of location, The Annals of Statistics, 6, 2, 292-313 (1978) · Zbl 0378.62051
[6] Bhandari, S. K.; Basu, A.; Sarkar, S., Robust inference in parametric models using the family of generalized negative exponential dispatches, Australian & New Zealand Journal of Statistics, 48, 1, 95-114 (2006) · Zbl 1109.62015
[7] Cressie, N.; Read, T. R.C., Multinomial goodness-of-fit tests, Journal of the Royal Statistical Society. Series B. Methodological, 46, 3, 440-464 (1984) · Zbl 0571.62017
[8] Cuesta-Albertos, J. A.; Gordaliza, A.; Matrán, C., Trimmed \(k\)-means: an attempt to robustify quantizers, The Annals of Statistics, 25, 2, 553-576 (1997) · Zbl 0878.62045
[9] Fujisawa, H.; Eguchi, S., Robust estimation in the normal mixture model, Journal of Statistical Planning and Inference, 136, 11, 3989-4011 (2006) · Zbl 1104.62017
[10] Hampel, F. R.; Ronchetti, E. M.; Rousseeuw, P. J.; Stahel, W. A., (Robust Statistics: The Approach Based on Influence Functions. Robust Statistics: The Approach Based on Influence Functions, Wiley Series in Probability and Statistics (1985))
[11] Huber, P. J., (Robust Statistics. Robust Statistics, Wiley Series in Probability and Mathematical Statistics (1981), John Wiley & Sons Inc.: John Wiley & Sons Inc. New York) · Zbl 0536.62025
[12] Ishwaran, H.; James, L. F., Approximate dirichlet process computing in finite normal mixtures: smoothing and prior information, Journal of Computational and Graphical Statistics, 11, 3, 508-532 (2002)
[13] Ishwaran, H.; James, L. F.; Sun, J., Bayesian model selection in finite mixtures by marginal density decompositions, Journal of the American Statistical Association, 96, 456, 1316-1332 (2001) · Zbl 1051.62027
[14] Jeong, D.-B.; Sarkar, S., Negative exponential disparity based family of goodness-of-fit tests for multinomial models, Journal of Statistical Computation and Simulation, 65, 1, 43-61 (2000) · Zbl 1121.62535
[15] Jones, M. C.; Hjort, N. L.; Harris, I. R.; Basu, A., A comparison of related density-based minimum divergence estimators, Biometrika, 88, 3, 865-873 (2001) · Zbl 1180.62047
[16] Lindsay, B. G., Efficiency versus robustness: the case for minimum Hellinger distance and related methods, The Annals of Statistics, 22, 2, 1081-1114 (1994) · Zbl 0807.62030
[17] Markatou, M., Mixture models, robustness, and the weighted likelihood methodology, Biometrics, 56, 2, 483-486 (2000) · Zbl 1060.62511
[18] Markatou, M.; Basu, A.; Lindsay, B. G., Weighted likelihood equations with bootstrap root search, Journal of the American Statistical Association, 93, 442, 740-750 (1998) · Zbl 0918.62046
[19] Parr, W. C., Minimum distance estimation: a bibliography, Communications in Staitstics — Theory and Methods, 10, 12, 1205-1224 (1981) · Zbl 0458.62035
[20] R Development Core Team. 2008. R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing, Vienna, Austria. ISBN 3-900051-07-0.; R Development Core Team. 2008. R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing, Vienna, Austria. ISBN 3-900051-07-0.
[21] Rao, C. R., Efficient estimates and optimum inference procedures in large samples, Journal of the Royal Statistical Society. Series B (Methodological), 24, 1, 46-72 (1962) · Zbl 0138.13103
[22] Richardson, S.; Green, P. J., On bayesian analysis of mixtures with an unknown number of components, Journal of the Royal Statistical Society. Series B (Methodological), 59, 4, 731-792 (1997) · Zbl 0891.62020
[23] Roeder, K., A graphical technique for determining the number of components in a mixture of normals, Journal of the American Statistical Association, 89, 426, 487-495 (1994) · Zbl 0798.62004
[24] Scott, D. W., Multivariate Density Estimation: Theory, Practice, and Visualization (1992), Wiley · Zbl 0850.62006
[25] Scott, D. W.; Terrell, G. R., Biased and unbiased cross-validation in density estimation, Journal of the American Statistical Association, 82, 400, 1131-1146 (1987) · Zbl 0648.62037
[26] Sheather, S. J.; Jones, M. C., A reliable data-based bandwidth selection method for kernel density estimation, Journal of the Royal Statistical Society. Series B (Methodological), 53, 3, 683-690 (1991) · Zbl 0800.62219
[27] Silverman, B. W., Density Estimation (1986), Chapman and Hall: Chapman and Hall London · Zbl 0617.62042
[28] Stephens, M., Bayesian analysis of mixture models with an unknown number of components — an alternative to reversible jump methods, The Annals of Statistics, 28, 1, 40-74 (2000) · Zbl 1106.62316
[29] Turner, D. A.; West, M., Bayesian analysis of mixtures applied to post-synaptic potential fluctuations, Journal of Neuroscience Methods, 47, 1-2, 1-21 (1993)
[30] Vincent, L.; Soille, P., Watersheds in digital spaces: an efficient algorithm based on immersion simulations, IEEE PAMI, 13, 6, 583-598 (1991)
[31] West, M., Hierarchical mixture models in neurological transmission analysis, Journal of the American Statistical Association, 92, 438, 587-606 (1997) · Zbl 0889.62095
[32] Woodward, W. A.; Parr, W. C.; Schucany, W. R.; Lindsey, H., A comparison of minimum distance and maximum likelihood estimation of a mixture proportion, Journal of the American Statistical Association, 79, 387, 590-598 (1984) · Zbl 0547.62017
[33] Zhang, G.; Ezura, Y.; Chervoneva, I.; Robinson, P. S.; Beason, D. P.; Carine, E. T.; Soslowsky, L. J.; Iozzo, R. V.; Birk1, D. E., Decorin regulates assembly of collagen fibrils and acquisition of biomechanical properties during tendon development, Journal of Cellular Biochemistry, 98, 1436-1449 (2006)
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