Inference on the change point under a high dimensional sparse mean shift

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