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**An estimator for the parameters of concentrations of two-component mixtures with censored data.**
*(Ukrainian, English)*
Zbl 1199.62024

Teor. Jmovirn. Mat. Stat. 76, 150-159 (2007); translation in Theory Probab. Math. Stat. 76, 169-179 (2008).

The problem for a model of mixtures with varying concentrations is stated as follows: Assume that \(\mathcal X_1\) and \(\mathcal X_2\) are two populations with distribution functions of the life times \(H_1(t)\) and \(H_2(t)\), respectively. A statistician has a sample whose elements are divided into the groups \(Y_0,\dots,Y_{K+1}\) according to certain characteristics. It is known that objects of the group \(Y_0\) belong to \(\mathcal X_1\) (“positively” classified objects), while the objects of the group \(Y_{K+1}\) belong to \(\mathcal X_2\) (“negatively” classified objects). The observations of other groups are not classified and may belong to either \(\mathcal X_1\) or \(\mathcal X_2\). The probability that an object of the group \(j\) belongs to \(\mathcal X_1\) is denoted by \(\theta_j\). Then the distribution of the life times of objects of the group \(Y_j\) is given by \(F_j(t)=\theta_jH_1(t)+(1-\theta_j)H_2(t),\;t\geq0\). With the assumption \(\theta_0=1\) and \(\theta_{K+1}=0\) the problem is to estimate \(\theta_j,\;j=1,\dots,K\), from the sample. Mixtures whose distribution of the life times is given by the indicated expression are called mixtures with varying concentrations.

Let \(n_j\) denote the number of observations in the group \(Y_j\), \(n=n_0+\cdots+n_{K+1}\). Each object is characterized by an ordered pair of random variables \((T_{ij}, U_{ij})\), where \(T_{ij}\) is the life time of the object \(i\) of the group \(Y_j\), and where \(U_{ij}\) is the moment of its (possible) censoring, \(i=1,\dots,n_j,\;j=0,\dots,K+1\). In fact, a statistician observes the variables \(\min(T_{ij}, U_{ij})\) and \({\mathbb I}_{\{T{ij}<U_{ij}\}}\) in the case of samples with censoring, where \({\mathbb I}_{\{T{ij}<U_{ij}\}}\) is the indicator of the event that an object \(i\) of the group \(j\) is censored. Assume that, for fixed \(j\), \(T_{ij}\) have the distribution functions \(F_j(t)=P\{T_{ij}<t\}\) and \(U_{ij}\) have the distribution functions \(C_j(t)= P\{U_{ij}<t\},\;t\geq0,\) respectively. The distribution functions \(F_j(t)\) are of the indicated form with some (unknown) concentrations \(\theta _j\). The author finds estimates for the concentrations \(\theta_j\) of the components and proves that the estimators are strongly consistent and asymptotically normal.

Let \(n_j\) denote the number of observations in the group \(Y_j\), \(n=n_0+\cdots+n_{K+1}\). Each object is characterized by an ordered pair of random variables \((T_{ij}, U_{ij})\), where \(T_{ij}\) is the life time of the object \(i\) of the group \(Y_j\), and where \(U_{ij}\) is the moment of its (possible) censoring, \(i=1,\dots,n_j,\;j=0,\dots,K+1\). In fact, a statistician observes the variables \(\min(T_{ij}, U_{ij})\) and \({\mathbb I}_{\{T{ij}<U_{ij}\}}\) in the case of samples with censoring, where \({\mathbb I}_{\{T{ij}<U_{ij}\}}\) is the indicator of the event that an object \(i\) of the group \(j\) is censored. Assume that, for fixed \(j\), \(T_{ij}\) have the distribution functions \(F_j(t)=P\{T_{ij}<t\}\) and \(U_{ij}\) have the distribution functions \(C_j(t)= P\{U_{ij}<t\},\;t\geq0,\) respectively. The distribution functions \(F_j(t)\) are of the indicated form with some (unknown) concentrations \(\theta _j\). The author finds estimates for the concentrations \(\theta_j\) of the components and proves that the estimators are strongly consistent and asymptotically normal.

Reviewer: Mikhail P. Moklyachuk (Kyïv)