Least square estimate for parameters of concentrations of varying mixture. I: The consistency (Q2771527)

The model under consideration deals with observations \(\xi_1,\dots,\xi_n\) that are independent \(X\)-valued elements with distribution \(P\{\xi_j\in A\}=\sum_{k=1}^M w_j^k H_k(A)\), \(A\subset X\), where \(M\) is the number of elements in the mixture, \((X,B)\) is a measurable set, \(H_k\) is the unknown distribution of the \(k\)-th component and the concentration of the \(k\)-th component during the \(j\)-th observation \(w_j^k=w_j^k(\theta)\), \(\theta\in \Theta\), is assumed to have a known form but depending on the unknown value of the parameter \(\theta\) to be estimated. The author proposes a generalized least squares estimate for \(\theta\), proves that this estimate is consistent under some additional conditions, and discusses examples of applications.