Large Sample Change-Point Estimation when Distributions Are Unknown

Abstract

Let ${\rm X}=({\rm x}_1,{\rm x}_2,\ldots,{\rm x}_n)$ be a sample consisting of n independent observations in an arbitrary measurable space ${\cal X}$ 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 ${\bf P}_\theta\{|\theta^*-\theta|>k\}$ tends to zero as k grows to infinity), under the assumption that we know a function h for which the mean values of $h({\rm x}_j)$ 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 $\cX$ such that for at least one of them the mean values of $h_j({\rm x}_i)$ 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 ${\rm X}$. Sequential procedures are also dealt with.