Large Sample Change-Point Estimation when Distributions Are Unknown

Let $\textrm{X}=(\textrm{x}_1,\textrm{x}_2,\dots,\textrm{x}_n)$ be a sample consisting of $n$ independent observations in an arbitrary measurable space $\mathscr{X}$ such that the first $\theta$ observations have a distribution $F$ while the remaining $n-\theta$ ones follow $G\neq F$, the distributions $F$ and $G$ being unknown and quantities $n$ and $\theta$ large. In [A. A. Borovkov and Yu. Yu. Linke, Math. Methods Statist., 14 (2005), pp. 404–430] there were constructed estimators $\theta^*$ of the change-point $\theta$ that have proper error (i.e., such that $P_\theta\{|\theta^*-\theta|>k\}$ tends to zero as $k$ grows to infinity), under the assumption that we know a function $h$ for which the mean values of $h(\textrm{x}_j)$ under the distributions $F$ and $G$ are different from each other. Sequential procedures were also presented in that paper. In the present paper, we obtain similar results under a weakened form of the above assumption or even in its absence. One such weaker version assumes that we have functions $h_1,h_2,\ldots,h_l$ on $\mathscr{X}$ such that for at least one of them the mean values of $h_j(\textrm{x}_i)$ are different under $F$ and $G$. Another version does not assume the existence of known to us functions $h_j$, but allows the possibility of estimating the unknown distributions $F$ and $G$ from the initial and terminal segments of the sample $\textrm{X}$. Sequential procedures are also dealt with.