Minimax estimation of functionals in sparse vector model with correlated observations

2024/07/20 by Yuhao Wang, Wang, Yuhao, Pengkun Yang +3
Mathematics · #FOS: Mathematics #Statistical Methods and Inference #Statistics Theory (math.ST)

paper · pdf · doi:10.48550/arxiv.2407.14778

openalex publication_date 2024/07/20 · openalex created_date 2024/09/26 · openalex updated_date 2026/07/28

Abstract

We consider the observations of an unknown s-sparse vector \boldsymbol θ corrupted by Gaussian noise with zero mean and unknown covariance matrix \boldsymbol Σ. We propose minimax optimal methods of estimating the ℓ₂ norm of \boldsymbol θ and testing the hypothesis H₀: \boldsymbol θ=0 against sparse alternatives when only partial information about \boldsymbol Σ is available, such as an upper bound on its Frobenius norm and the values of its diagonal entries to within an unknown scaling factor. We show that the minimax rates of the estimation and testing are leveraged not by the dimension of the problem but by the value of the Frobenius norm of \boldsymbol Σ.

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