Rate of convergence of the sample estimate of change-point in measurement error variance with possible difference in mean

Presenter: Iryna Sivak (University of Warwick)

Co-author: Alexander Kukush (Taras Shevchenko National University of Kyiv)

Abstract

We study a structural model of observations $$X_{k}=\zeta_{k}+(\mu_1+\varepsilon^1_{k})I_{\{k\le k_0\}}+(\mu_2+\varepsilon^2_{k}) I_{\{k > k_0\}}, \ k=1,2,...,T,$$ with change-point $k_0=k_0(T)$, where $k_0(T) =[\tau_0 T]$ and $\tau_0 \in (0,1)$ is true "ratio" of change-point. Here $\xi_{k}=\zeta+\mu_1 I_{\{k\le k_0\}}+\mu_2 I_{\{k>k_0\}}$ is a latent variable with possible change of mean measured with additive heteroscedastic error: $\varepsilon^1_k$ before the change-point and $\varepsilon^2_k$ after it. Particularly, this model describes change in error rate that occur due to range of measurements, which frequently appears in measuring instruments.

We assume that $\zeta_{k}, \ k= 1,...,T$ are i.i.d. random variables such that $E\zeta_{k}=0, \ Var(\zeta_{k})=\sigma_{\zeta}^2,$ and for each $i=1,2, \ \varepsilon^i_{k}, \ k= 1,...,T$ are i.i.d. errors with $E\varepsilon^i_{k}=0$ and $Var(\varepsilon^i_{k})=\sigma_i^2$, where both variances are known or estimated before, and different from each other. The estimator of $k_0$ is given by $$\displaystyle\widehat k=\underset{k}{\operatorname{argmin}} \ \left| \sigma_2^2-\frac 1 {T-k} \sum_{i=k+1}^T (X_i-\overline X^*)^2 - \sigma_1^2+\frac 1 k \sum_{i=1}^k (X_i-\overline X)^2 \right|,$$ where $\displaystyle \overline{X}=\frac 1 k \sum_{i=1}^k X_i$ and $\displaystyle \overline{X}^*=\frac 1 {T-k} \sum_{i=k+1}^T X_i$ are sample means before and after moment $k$ respectively.

Simulations show that, e.g., in case of Gaussian errors $\varepsilon^i_k$'s, the proposed estimator outperforms some classical quantile based CUSUM change-point estimator due to small difference in some sample quantiles before and after change-point. We show consistency of the estimator as $T\rightarrow\infty$ and give the rate of convergence.