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Кафедра теорії ймовірностей,

статистики

та актуарної математики

Механіко-математичний

факультет

prob.stat.act@gmail.com, probability@univ.kiev.ua

Tel/Fax: +38 (044) 259 03 92

### TR 0106U005864-S08

*Sugakova Olena*

Density estimation by observations with admixture**Abstract:**

A semiparametric two-component mixture model is considered, in which the distribution of one (primary) component is unknown and assumed symmetric. The distribution of the other component (admixture) is known. Kernel type estimates for the density of the primary component are considered. Asymptotic normality of the estimates is demonstrated.

link... (Final version is published in Theory of Stochastic Processes (2010), 16(32) no.1, 103-110)### TR 0106U005864-MS08

*Maiboroda Rostyslav, Sugakova Olena*

Generalized estimating equations for symmetric distributions observed with admixture**Abstract:**

A semiparametric two-component mixture model is considered, in which the distribution of one (primary) component is unknown and assumed symmetric. The distribution of the other component (admixture) is known. Generalized estimating equations are constructed for the estimation of the mixture proportion and the location parameter of the primary component. Asymptotic normality of the estimates is demonstrated and the lower bound for the asymptotic covariance matrix is obtained. An adaptive estimation technique is proposed to obtain the estimates with nearly optimal asymptotic variances.

*Keywords:*adaptive estimating equation, asymptotic normality, finite mixture model, symmetric distribution.

link... (Final version is published in Communications in Statistics - Theory and Methods, (2011) 40 no. 1, 96-116)### TR0106U005864-MS09

*Maiboroda Rostyslav, Sugakova Olena*

Nonparametric density estimation for symmetric distributions observed with admixture**Abstract:**

A semiparametric two-component mixture model is considered, in which the distribution of one (primary) component is unknown and assumed symmetric. The distribution of the other component (admixture) is known. We consider three estimates for the pdf of primary component: a naive one, a symmetrized naive estimate and a symmetrized estimate with adaptive weights. Asymptotic behavior and small sample performance of the estimates are investigated. Some rules of thumb for bandwidth selection are discussed.

*Keywords:*asymptotic normality, finite mixture model, symmetric distribution, kernel density estimate, rule of thumb, MISE

link... (Final version is to be published in Metrika)### TR 0106U005864-MS10

*Maiboroda Rostyslav, Sugakova Olena*

Generalized estimating equations for mixtures with varying concentrations**Abstract:**

A finite mixture model is considered, in which the mixing probabilities (concentrations of components in the mixture) can vary from observation to observation. These probabilities are known. A parametric model is assumed for one (first) component distribution while the distributions of the other components are completely unknown.

Generalized estimating equations are proposed for the estimation of an unknown parameter of the first component. Asymptotic normality of obtained estimates is demonstrated and the strict lower bound for the dispersion matrices of such estimates is derived. Adaptive estimation technique is discussed which allows to obtain the estimates with dispersion near to this lower bound.

link...### TR 0106U005864-MS11

*Maiboroda Rostyslav, Sugakova Olena*

Statistics of mixtures with varying concentrations with application to DNA microarray data analysis**Abstract:**

A finite mixture model is considered in which the mixing probabilities vary from observation to observation. Estimation of mixture components distributions, functional moments and densities is discussed. Tests are proposed for testing hypotheses on the moments. An application to the analysis of DNA microarrays data is considered.

link... (Submited to*Communications in Statistics - Theory and Methods*)