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.

MSC:

 62N01 Censored data models 62F12 Asymptotic properties of parametric estimators 62G05 Nonparametric estimation
Full Text: