Nonparametric change-point estimation. (English) Zbl 0637.62041

Carlstein, E.

Ann. Stat. 16, No. 1, 188-197 (1988).

Consider a sequence of independent random variables \(\{X_ i:\) \(1\leq i\leq n\}\) having cdf F for \(i\leq \theta n\) and cdf G otherwise. A class of strongly consistent estimators for the change-point \(\theta\in (0,1)\) is proposed. The estimators require no knowledge of the functional forms or parametric families of F and G. Furthermore, F and G need not differ in their means (or other measure of location). The only requirement is that F and G differ on a set of positive probability.
The proof of consistency provides rates of convergence and bounds on the error probability for the estimators. The estimators are applied to two well-known data sets, in both cases yielding results in close agreement with previous parametric analyses. A simulation study is conducted, showing that the estimators perform well even when F and G share the same mean, variance and skewness.

MSC:

62G05	Nonparametric estimation
60F15	Strong limit theorems

Keywords:

Cramér-von Mises statistic; Kolmogorov-Smirnov statistic; sequence of independent random variables; strongly consistent estimators; change- point; consistency; rates of convergence; bounds on the error probability; simulation study