Asymptotic Efficient Estimation of the Change Point with Unknown Distributions

Abstract

Suppose $X_1,\cdots, X_n$ are distributed according to a probability measure under which $X_1,\cdots, X_n$ are independent, $X_1 \sim F_0$, for $i = 1,\cdots, \lbrack\theta_n n\rbrack$ and $X_i \sim F^{(n)}$ for $i = \lbrack\theta_nn\rbrack + 1, \cdots, n$ where $\lbrack x\rbrack$ denotes the integer part of $x$. In this paper we consider the asymptotic efficient estimation of $\theta_n$ when the distributions are not known. Our estimator is efficient in the sense that if $F^{(n)} = F_{\eta_n}, \eta_n \rightarrow 0$ and $\{F_\eta\}$ is a regular one-dimensional parametric family of distributions, then the estimator is asymptotically equivalent to the best regular estimator.