Inferences on Covariance Matrix with Blockwise Correlation Structure

Bo Lei, Xinyan Fan, Wei Lan

Keywords: Blockwise Correlation Matrix Estimation; Ridge-type Ratio Criterion; Spectral Clustering; Covariance Matrix.

Abstract:

Utilizing the sample moments of variable means within groups, we develop a novel closed-form estimator for blockwise correlation matrix of $p$ variables.
When the block number and group memberships of the variables are known, we demonstrate the asymptotic normality of parameter estimators and establish the stochastic convergence rate of the estimated blockwise correlation matrix and corresponding estimated covariance matrix, under certain moment conditions.
The method ensures positive semi-definiteness of the estimated covariance matrix without requiring a predetermined variable order, and can be applicable for high-dimensional data.
Moreover, to estimate the number of blocks and recover their memberships, respectively,
we employ the ridge-type ratio criterion and spectral clustering, and establish their consistency. Based on this, we extend the aforementioned properties of the asymptotic normality and stochastic convergence rate to the scenario where the group memberships are unknown and the block number is given.
Extensive simulations and an empirical study of stock returns in the Chinese stock market are analyzed to illustrate the usefulness of our proposed methods.

Primary Area: probabilistic methods (Bayesian methods, variational inference, sampling, UQ, etc.)

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Submission Number: 10036