## Asymptotic of median estimates for many change-points.(Ukrainian, English)Zbl 1071.62045

Teor. Jmovirn. Mat. Stat. 70, 149-156 (2004); translation in Theory Probab. Math. Stat. 70, 167-176 (2005).
A sequence $$\zeta_1$$,…,$$\zeta_N$$ of independent r.v.’s is observed. The CDF of $$\zeta_j$$ can be $$F_1$$ or $$F_2$$ where $$F_i$$ are unknown CDFs with $$\operatorname{med}F_1\not=\operatorname{med}F_2$$. The (unobserved) sequence of numbers of these CDFs is called a trajectory $$h=(h_1,\dots,h_N)$$. The estimator $$\hat h$$ for $$h$$ is the minimizer of $J_{\widehat m}(h)=\sum_{i=1}^N (\pi_N(h_i,h_{i-1})+\phi_{\widehat m}(\zeta_i,h_i)),$ where $$\phi_m(x,1)=\mathbf{1}_{\{x>m\}}$$, $$\phi_m(x,2)=\mathbf{1}_{\{x<m\}}$$, $$\pi_N(g,l)=\pi_N\mathbf{1}_{\{g\not=l\}}$$, $$\pi_N$$ is a fixed sequence of numbers, $$\hat m=\text{med }(\zeta_1,\dots,\zeta_N)$$. Estimates of change points in $$h$$ are the change points in $$\hat h$$. The asymptotic distributions of these estimates are derived as $$N\to\infty$$.

### MSC:

 62G20 Asymptotic properties of nonparametric inference 62G05 Nonparametric estimation 62E20 Asymptotic distribution theory in statistics