## A search for change points in a flow of independent observations.(English. Ukrainian original)Zbl 0921.62037

Theory Probab. Math. Stat. 181-186 (1997); translation from Teor. Jmovirn. Mat. Stat. 55, 167-172 (1996).
An algorithm for posterior change-point estimation is proposed for the case of many change-points in the sample. The observations $$\xi_1$$,…,$$\xi_N$$ are independent elements of some space $$\mathcal X$$, $$\{ F_i,\;i=1,\dots, K\}$$ is the set of their possible distributions $$\Pr\{\xi_i\in A\}=F_{h_j^0}(A)$$, where $$h_j^0=c_i$$ if $$\vartheta_i N< j<\vartheta_{i+1}N$$, and $$0=\vartheta_0<\vartheta_1<\dots<\vartheta_R<\vartheta_{R+1}=1$$ are the change-points of the observations. Denote by $$\eta_k$$ a r.v. with the distribution $$F_i$$. For the construction of the algorithm a sequence of numbers $$\pi_N\to\infty$$ and a function $$\phi:{\mathcal X}\times\{1,\dots,K\}\mapsto R$$ are used such that $\kappa=\inf_{1\leq i,k\leq K, i\not=k} (E\phi(\eta_i,k)- E\phi(\eta_i,i))>0$ and $$E(\phi(\eta_i,h_i)-E\phi(\eta_i,h_i))^2<\infty$$. The sequence $$\hat h=(\hat h_1,\dots, \hat h_N)$$ which minimizes $J(h)=\sum_{i=1}^N \pi_N(1-\delta(h_i,h_{i-1}))+\phi(\xi_i,h_i)$ is used as an estimator for $$h^0$$, and $$\hat \vartheta_i$$ which are the change-points of $$\hat h$$ are used to estimate $$\vartheta_i$$. It is proved that if $$\pi_N=CN^{\beta}$$, where $$0.5<\beta<1$$, then $$| \hat\vartheta_i-\vartheta_i| =O(N^{\beta-1)}$$ in probability.

### MSC:

 62G05 Nonparametric estimation 62G20 Asymptotic properties of nonparametric inference

### Keywords:

posterior change-point estimation; consistency