Lower bound estimation for a family of high-dimensional sparse covariance matrices. (English) Zbl 1531.62009
Summary: Lower bound estimation plays an important role for establishing the minimax risk. A key step in lower bound estimation is deriving a lower bound of the affinity between two probability measures. This paper provides a simple method to estimate the affinity between mixture probability measures. Then we apply the lower bound of the affinity to establish the minimax lower bound for a family of sparse covariance matrices, which contains Cai-Ren-Zhou’s theorem in [T. T. Cai et al., Electron. J. Stat. 10, No. 1, 1–59 (2016; Zbl 1331.62272)] as a special example.
MSC:
| 62C20 | Minimax procedures in statistical decision theory |
| 62H10 | Multivariate distribution of statistics |
| 62H12 | Estimation in multivariate analysis |
| 94A17 | Measures of information, entropy |
Keywords:
minimax risk; lower bound estimation; Kullback-Leibler divergence; affinity; mixture probability measure; sparse covariance matrixCitations:
Zbl 1331.62272References:
| [1] | Aston, J., Autin, F., Claeskens, G., Freyermuth, J. and Pouet, C., Minimax optimal procedures for testing the structure of multidimensional functions, Appl. Comput. Harmon. Anal.46(2) (2019) 288-311. · Zbl 1409.62027 |
| [2] | Avella-Medina, M., Battery, H., Fan, J. and Li, Q., Robust estimation of high-dimensional covariance and precision matrices, Biometrika105 (2018) 271-284. · Zbl 07072412 |
| [3] | Cai, Z. and Li, K., Maximum Kullback-Leibler distance for common distributions, J. Wuhan Univ. Natur. Sci. Ed.53(5) (2007) 513-517. · Zbl 1174.62415 |
| [4] | Cai, T., Ren, Z. and Zhou, H., Estimating structured high-dimensional covariance and precision matrices: Optimal rates and adaptive estimation, Electron. J. Stat.10(1) (2016) 1-59. · Zbl 1331.62272 |
| [5] | Cai, T. and Zhang, A., Rate-optimal perturbation bounds for singular subspaces with applications to high-dimensional statistics, Ann. Statist.46(1) (2018) 60-89. · Zbl 1395.62122 |
| [6] | Cai, T. and Zhou, H., Optimal rates of convergence for sparse covariance matrix estimation, Ann. Statist.40(5) (2012) 2389-2420. · Zbl 1373.62247 |
| [7] | Cover, T. and Thomas, J., Elements of Information Theory, 2nd edn. (Wiley-InterScience (John Wiley & Sons), 2006). · Zbl 1140.94001 |
| [8] | Jeyakumar, V. and Li, G., Exact second-order cone programming relaxations for some nonconvex minimax quadratic optimization problems, SIAM J. Optim.28(1) (2018) 760-787. · Zbl 1396.90064 |
| [9] | Meng, K., Yang, H., Yang, X. and Yu, C., Portfolio optimization under a minimax rule revisited, Optimization71(4) (2022) 877-905. · Zbl 1489.91237 |
| [10] | Tsybakov, A., Introduction to Nonparametric Estimation (Springer, 2009). · Zbl 1176.62032 |
| [11] | Wang, D. and Xu, J., Tight lower bound of sparse covariance matrix estimation in the local differential privacy model, Theoret. Comput. Sci.815 (2020) 47-59. · Zbl 1433.68129 |
| [12] | Xie, F., Cape, J., Priebe, C. and Xu, Y., Bayesian sparse spiked covariance model with a continuous matrix shrinkage prior, Bayesian Anal.17(4) (2022) 1193-1217. · Zbl 1531.62037 |
| [13] | Yao, X., A minimax method for finding saddle points of upper semi-differentiable locally Lipschitz continuous functional in Banach space and its convergence, J. Comput. Appl. Math.296 (2016) 528-549. · Zbl 1335.49052 |
| [14] | Zhang, Q., Park, C. and Chung, J., Minimax estimation of covariance and precision matrices for high-dimensional time series with long-memory, Statist. Probab. Lett.177 (2021) 109177. · Zbl 1474.62322 |
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