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

Sugakova, O. V.

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.

Reviewer: R.E.Maiboroda (Kyiv)

MSC:

62G05	Nonparametric estimation
62G20	Asymptotic properties of nonparametric inference

Keywords:

posterior change-point estimation; consistency