Asymptotic efficient estimation of the change point with unknown distributions. (English) Zbl 0714.62027

Ritov, Y.

Ann. Stat. 18, No. 4, 1829-1839 (1990).

Summary: Suppose \(X_ 1,...,X_ n\) are distributed according to a probability measure under which \(X_ 1,...,X_ n\) are independent, \(X_ i\sim F_ 0\), for \(i=1,...,[\theta_ nn]\) and \(X_ i\sim F^{(n)}\) for \(i=[\theta_ nn]+1,...,n\), where [x] denotes the integer part of x.
We consider the asymptotic efficient estimation of \(\theta_ n\) when the distributions are not known. Our estimator is efficient in the sense that if \(F^{(n)}=F_{\eta_ n}\), \(\eta_ n\to 0\) and \(\{F_{\eta}\}\) is a regular one-dimensional parametric family of distributions, then the estimator is asymptotically equivalent to the best regular estimator.

MSC:

62G05	Nonparametric estimation
62G20	Asymptotic properties of nonparametric inference

Keywords:

limit of experiments; change point estimation; asymptotic efficient estimation; regular one-dimensional parametric family of distributions; best regular estimator