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

Shurenkov, G.

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 \(N\to\infty\).

Reviewer: R. E. Maiboroda (Kyïv)

MSC:

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

Keywords:

post hoc change-point detection; asymptotic distribution; dynamic programming algorithm