A test for the homogeneity of mixtures with varying concentrations (Q2722141)

Let \(O_{1},\dots,O_{N}\) be the observed objects which belong to one of the populations \(M.\) The number of populations, to which the object \(O\) belongs, is noticed by \(\text{ind}(O).\) For each object \(O_{j}\) some characteristic (random) \(\xi_{j}\) is observed. The distribution \(\xi_{j}\) may depend on \(\text{ind}(O_{j}):\) NEWLINE\[NEWLINEP\{\xi_{j}<x |\text{ind}(O_{j})=k\}=H_{k}(x).NEWLINE\]NEWLINE The functions \(H_{k}(x)\) are unknown. The value \(\text{ind}(O_{j})\) is also unknown, but the probability is known that the object belongs to the given population \(w_{k}(j)=P\{\text{ind}(O_{j})=k\}.\)NEWLINENEWLINENEWLINEThis work is devoted to the construction of a criterion for verifying the hypothesis \({\mathbf H_{0}}:\) \(H_{1}=H_{2}=...=H_{M}\) against the alternative hypothesis \({\mathbf H_{1}}:\) there exist \(k, m, x\) such that \(H_{k}(x)\neq H_{m}(x).\) A generalized Kolmogorov-Smirnov test for verifying the hypothesis of the homogeneity of the sample against the alternative sample from a mixture with varying concentrations is constructed. Asymptotic formulas and nonasymptotic upper bounds for probabilities of errors of first and second type are obtained.