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We study a plug in least squares estimator for the change point parameter where change is in the mean of a high dimensional random vector under subgaussian or subexponential distributions. We obtain sufficient conditions under which this estimator possesses sufficient adaptivity against plug in estimates of mean parameters in order to yield an optimal rate of convergence $O_p(\xi^{-2})$ in the integer scale. This rate is preserved while allowing high dimensionality as well as a potentially diminishing jump size $\xi,$ provided $s\log (p\vee T)=o(\surd(Tl_T))$ or $s\log^{3/2}(p\vee T)=o(\surd(Tl_T))$ in the subgaussian and subexponential cases, respectively. Here $s,p,T$ and $l_T$ represent a sparsity parameter, model dimension, sampling period and the separation of the change point from its parametric boundary. Moreover, since the rate of convergence is free of $s,p$ and logarithmic terms of $T,$ it allows the existence of limiting distributions. These distributions are then derived as the {\it argmax} of a two sided negative drift Brownian motion or a two sided negative drift random walk under vanishing and non-vanishing jump size regimes, respectively. Thereby allowing inference of the change point parameter in the high dimensional setting. Feasible algorithms for implementation of the proposed methodology are provided. Theoretical results are supported with monte-carlo simulations.

We develop a projected least squares estimator for the change point parameter in a high dimensional time series model with a potential change point. Importantly we work under the setup where the jump size may be near the boundary of the region of detectability. The proposed methodology yields an optimal rate of convergence despite high dimensionality of the assumed model and a potentially diminishing jump size. The limiting distribution of this estimate is derived, thereby allowing construction of a confidence interval for the location of the change point. A secondary near optimal estimate is proposed which is required for the implementation of the optimal projected least squares estimate. The prestep estimation procedure is designed to also agnostically detect the case where no change point exists, thereby removing the need to pretest for the existence of a change point for the implementation of the inference methodology. Our results are presented under a general positive definite spatial dependence setup, assuming no special structure on this dependence. The proposed methodology is designed to be highly scalable, and applicable to very large data. Theoretical results regarding detection and estimation consistency and the limiting distribution are numerically supported via monte carlo simulations.

Binary segmentation, which is sequential in nature is thus far the most widely used method for identifying multiple change points in statistical models. Here we propose a top down methodology called arbitrary segmentation that proceeds in a conceptually reverse manner. We begin with an arbitrary superset of the parametric space of the change points, and locate unknown change points by suitably filtering this space down. Critically, we reframe the problem as that of variable selection in the change point parameters, this enables the filtering down process to be achieved in a single step with the aid of an $\ell_0$ regularization, thus avoiding the sequentiality of binary segmentation. We study this method under a high dimensional multiple change point linear regression model and show that rates convergence of the error in the regression and change point estimates are near optimal. We propose a simulated annealing (SA) approach to implement a key finite state space discrete optimization that arises in our method. Theoretical results are numerically supported via simulations. The proposed method is shown to possess the ability to agnostically detect the `no change' scenario. Furthermore, its computational complexity is of order $O(Np^2)$+SA, where SA is the cost of a SA optimization on a $N$(no. of change points) dimensional grid. Thus, the proposed methodology is significantly more computationally efficient than existing approaches. Finally, our theoretical results are obtained under weaker model conditions than those assumed in the current literature.

We propose a two step algorithm based on $\ell_1/\ell_0$ regularization for the detection and estimation of parameters of a high dimensional change point regression model and provide the corresponding rates of convergence for the change point as well as the regression parameter estimates. Importantly, the computational cost of our estimator is only $2\cdotp$Lasso$(n,p)$, where Lasso$(n,p)$ represents the computational burden of one Lasso optimization in a model of size $(n,p)$. In comparison, existing grid search based approaches to this problem require a computational cost of at least $n\cdot {\rm Lasso}(n,p)$ optimizations. Additionally, the proposed method is shown to be able to consistently detect the case of `no change', i.e., where no finite change point exists in the model. We work under a subgaussian random design where the underlying assumptions in our study are milder than those currently assumed in the high dimensional change point regression literature. We allow the true change point parameter $\tau_0$ to possibly move to the boundaries of its parametric space, and the jump size $\|\beta_0-\gamma_0\|_2$ to possibly diverge as $n$ increases. We then characterize the corresponding effects on the rates of convergence of the change point and regression estimates. In particular, we show that, while an increasing jump size may have a beneficial effect on the change point estimate, however the optimal rate of regression parameter estimates are preserved only upto a certain rate of the increasing jump size. This behavior in the rate of regression parameter estimates is unique to high dimensional change point regression models only. Simulations are performed to empirically evaluate performance of the proposed estimators. The methodology is applied to community level socio-economic data of the U.S., collected from the 1990 U.S. census and other sources.

The purpose of this study is to provide a new methodology of how one can consistently estimate a change-point in time series data. In contrast with previous studies, the suggested methodology employs only the empirical spectral density and its first moment. This is accomplished when both the means and variances before and after the unidentified time point are unknown. Then, the well-known Gauss-Newton algorithm is applied to estimate and provide asymptotic results for the parameters involved. Simulations carried out under different distributions, sizes and unknown time points confirm the validity and accuracy of the methodology. The real-world example considered in the paper illustrates the robustness of the methodology in the presence of even extreme outliers., Comment: 24 pages, 2 figures

We derive exact computable expressions for the asymptotic distribution of the change-point mle when a change in the mean occurred at an unknown point of a sequence of time-ordered independent Gaussian random variables. The derivation, which assumes that nuisance parameters such as the amount of change and variance are known, is based on ladder heights of Gaussian random walks hitting the half-line. We then show that the exact distribution easily extends to the distribution of the change-point mle when a change occurs in the mean vector of a multivariate Gaussian process. We perform simulations to examine the accuracy of the derived distribution when nuisance parameters have to be estimated as well as robustness of the derived distribution to deviations from Gaussianity. Through simulations, we also compare it with the well-known conditional distribution of the mle, which may be interpreted as a Bayesian solution to the change-point problem. Finally, we apply the derived methodology to monthly averages of water discharges of the Nacetinsky creek, Germany., Comment: Published in at http://dx.doi.org/10.1214/09-AOAS294 the Annals of Applied Statistics (http://www.imstat.org/aoas/) by the Institute of Mathematical Statistics (http://www.imstat.org)