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We study a plug in least squares estimator for the change point parameterwhere change is in the mean of a high dimensional random vector undersubgaussian or subexponential distributions. We obtain sufficient conditionsunder which this estimator possesses sufficient adaptivity against plug inestimates 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 allowinghigh dimensionality as well as a potentially diminishing jump size $\xi,$provided $s\log (p\vee T)=o(\surd(Tl_T))$ or $s\log^{3/2}(p\veeT)=o(\surd(Tl_T))$ 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 parametricboundary. Moreover, since the rate of convergence is free of $s,p$ andlogarithmic terms of $T,$ it allows the existence of limiting distributions.These distributions are then derived as the {\it argmax} of a two sidednegative drift Brownian motion or a two sided negative drift random walk undervanishing and non-vanishing jump size regimes, respectively. Thereby allowinginference of the change point parameter in the high dimensional setting.Feasible algorithms for implementation of the proposed methodology areprovided. Theoretical results are supported with monte-carlo simulations.

Inference on the change point in high dimensional time series models via plug in least squares