## Adaptive estimates for the parameter of a mixture of two symmetric distributions.(English. Ukrainian original)Zbl 1232.62045

Theory Probab. Math. Stat. 82, 149-159 (2011); translation from Teor. Jmovirn. Mat. Stat. No. 82, 146-155.
Summary: A sample is observed from a mixture of two symmetric distributions that differ only by the location parameters. We use the method of estimating equations to estimate the unknown parameters of the components. The method works as follows: first, we construct an estimator for the optimal estimating functions; then we use it to construct adaptive estimators. We study the asymptotic behavior of the resulting estimators.

### MSC:

 62F12 Asymptotic properties of parametric estimators 62G05 Nonparametric estimation 62G20 Asymptotic properties of nonparametric inference
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### References:

 [1] R. Maĭboroda and O. Sugakova, Estimation of Euclidean parameters of a mixture of two symmetric distributions, Ukrain. Mat. Zh. 62 (2010), 945-953. (Ukrainian) · Zbl 1221.62044 [2] Laurent Bordes, Stéphane Mottelet, and Pierre Vandekerkhove, Semiparametric estimation of a two-component mixture model, Ann. Statist. 34 (2006), no. 3, 1204 – 1232. · Zbl 1112.62029 · doi:10.1214/009053606000000353 [3] David R. Hunter, Shaoli Wang, and Thomas P. Hettmansperger, Inference for mixtures of symmetric distributions, Ann. Statist. 35 (2007), no. 1, 224 – 251. · Zbl 1114.62035 · doi:10.1214/009053606000001118 [4] R. Maĭboroda, Estimation of mean positions and concentrations from observations of a two-component mixture of symmetric distributions, Teor. Ĭmovīr. Mat. Stat. 78 (2008), 132 – 140 (Ukrainian, with Ukrainian summary); English transl., Theory Probab. Math. Statist. 78 (2009), 147 – 156. · Zbl 1223.62021 [5] Jun Shao, Mathematical statistics, 2nd ed., Springer Texts in Statistics, Springer-Verlag, New York, 2003. · Zbl 1018.62001
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