This paper investigates the mean-square stability of uncertain time-delay stochastic systems driven by G-Brownian motion, which are commonly referred to as G-SDDEs. To derive a new set of sufficient stability conditions, we employ the linear matrix inequality (LMI) method and construct a Lyapunov–Krasovskii function under the constraint of uncertainty bounds. The resulting sufficient condition does not require any specific assumptions on the G-function, making it more practical. Additionally, we provide numerical examples to demonstrate the validity and effectiveness of the proposed approach.
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.
This paper provides a method for enumerating signals impinging on an array of sensors based on the generalized Bayesian information criterion (GBIC). The proposed method motivates by a statistic for testing the sphericity of the covariance matrix when the sample size n is less than the dimension m. The statistic consists of the first four moments of sample eigenvalue distribution and relaxes the assumption of Gaussian distribution. We derive the asymptotical distribution of the statistic as m, n tends to infinity at the same ratio by random matrix theory and propose the expression of GBIC for determining the signal number. Numerical simulations demonstrate that the proposed method has a high probability of detection in both the Gaussian and the non-Gaussian noise, and performs better than other methods.
This paper addresses the issue of testing sphericity and identity of high-dimensional population covariance matrix when the data dimension exceeds the sample size. The central limit theorem of the first four moments of eigenvalues of sample covariance matrix is derived using random matrix theory for generally distributed populations. Further, some desirable asymptotic properties of the proposed test statistics are provided under the null hypothesis as data dimension and sample size both tend to infinity. Simulations show that the proposed tests have a greater power than existing methods for the spiked covariance model.
This paper discusses the problem of testing the sphericity of population covariance matrix when the sample size n and the dimensionality p both tend to infinity with p / n → c ∈ (0,1). A new test statistic is presented by utilizing an inequality, and asymptotic properties of the proposed statistic are derived for generally distributed population under the null hypothesis. Numerical simulations demonstrate that the proposed statistic has a significant improvement on test powers under the alternative hypothesis by comparing with some congeneric statistics.
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