Shrinkage regularization is an effective strategy to estimate the covariance matrix of multi-variate random vector in small sample scenarios. The purpose of this paper is to propose improved linear shrinkage estimators of covariance matrix as two types of Toeplitz-structured target matrices are respectively employed in the shrinkage procedure. Under Gaussian and non-Gaussian distributions, the corresponding shrinkage estimators are respectively obtained in closed form by unbiasedly estimating the unknown scalar quantities which involve the true covariance matrix. Compared with the existing estimators of same type, the proposed covariance estimators show a significant improvement on the mean squared error in numerical simulations. Moreover, example applications including portfolio risk estimation and classification of real data are provided for verifying the performance of proposed covariance estimators in small sample scenarios.

This paper tackles the problem of estimating the covariance matrix in large-dimension and small-sample-size scenarios. Inspired by the well-known linear shrinkage estimation, we propose a novel second-order Stein-type regularization strategy to generate well-conditioned covariance matrix estimators. We model the second-order Stein-type regularization as a quadratic polynomial concerning the sample covariance matrix and a given target matrix, representing the prior information of the actual covariance structure. To obtain available covariance matrix estimators, we choose the spherical and diagonal target matrices and develop unbiased estimates of the theoretical mean squared errors, which measure the distances between the actual covariance matrix and its estimators. We formulate the second-order Stein-type regularization as a convex optimization problem, resulting in the optimal second-order Stein-type estimators. Numerical simulations reveal that the proposed estimators can significantly lower the Frobenius losses compared with the existing Stein-type estimators. Moreover, a real data analysis in portfolio selection verifies the performance of the proposed estimators.

This paper investigates the inverse reinforcement learning (IRL) problem for fuzzy switched systems characterized by sojourn probabilities. Classical fuzzy Markov jump system approaches rely on transition probabilities, which are often difficult to estimate, whereas sojourn probabilities provide a more realistic and tractable description of switching behavior. To address the challenges of nonlinear dynamics, uncertain model residence, and the lack of known cost functions, a novel IRL framework is proposed. Different from existing methods, the framework (i) establishes IRL under sojourn-probability-based fuzzy switched systems, (ii) incorporates a dynamic learning rate mechanism that adapts to switching conditions, mitigating oscillations and accelerating convergence, and (iii) introduces constraint-based parameter updates together with a new penalty-matrix update strategy to ensure convergence and reduce computational complexity. Rigorous theoretical analysis demonstrates the stability and convergence of the proposed algorithms. Simulation study on a benchmark tunnel diode circuit confirm that the learner system can accurately reconstruct equivalent cost functions, reproduce expert trajectories, and achieve superior performance compared with conventional IRL methods. Note to Practitioners—This paper addresses the challenge of designing intelligent control strategies for nonlinear systems with stochastic switching–commonly seen in automated manufacturing, power grids, and transportation networks. Conventional Markov models require complete transition probabilities, which are difficult to obtain in practice. The proposed IRL framework instead uses sojourn probabilities to characterize operational behavior, enabling data-driven policy learning without full model knowledge. For practitioners, this approach allows the extraction of interpretable performance criteria directly from expert demonstrations or operational data. The adaptive learning-rate mechanism ensures faster and more stable convergence under abrupt switching, while the constraint-based parameter and penalty-matrix updates enhance robustness to uncertainty and noise. The framework can be implemented for optimizing control and maintenance policies in large-scale or networked automation systems and can be extended to cooperative multi-agent or cyber-physical systems where probabilistic information is incomplete.

The problem of estimating a large covariance matrix arises in various statistical applications. This paper develops new covariance matrix estimators based on shrinkage regularization. Individually, we consider two kinds of Toeplitz-structured target matrices as the data come from the complex Gaussian distribution. We derive the optimal tuning parameter under the mean squared error criterion in closed form by discovering the mathematical properties of the two target matrices. We get some vital moment properties of the complex Wishart distribution, then simplify the optimal tuning parameter. By unbiasedly estimating the unknown scalar quantities involved in the optimal tuning parameter, we propose two shrinkage estimators available in the large-dimensional setting. For verifying the performance of the proposed covariance matrix estimators, we provide some numerical simulations and applications to array signal processing compared to some existing estimators.