Nonparametric Change-Point Estimation

Abstract

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.