In this paper, we give an explanation to the failure of two likelihood ratio procedures for testing about covariance matrices from Gaussian populations when the dimension is large compared to the sample size. Next, using recent central limit theorems for linear spectral statistics of sample covariance matrices and of random F-matrices, we propose necessary corrections for these LR tests to cope with high-dimensional effects. The asymptotic distributions of these corrected tests under the null are given. Simulations demonstrate that the corrected LR tests yield a realized size close to nominal level for both moderate p (around 20) and high dimension, while the traditional LR tests with χ 2 approximation fails.Another contribution from the paper is that for testing the equality between two covariance matrices, the proposed correction applies equally for non-Gaussian populations yielding a valid pseudo-likelihood ratio test.

N (6)-methyladenosine (m(6)A) is the most abundant and reversible internal modification ubiquitously occurring in eukaryotic mRNA, albeit the significant biological roles of m(6)A methylation have remained largely unclear. The well-known DNA and histone methylations play crucial roles in epigenetic modification of biologic processes in eukaryotes. Analogously, the dynamic and reversible m(6)A RNA modification, which is installed by methyltransferase (METTL3, METTL14, and WTAP), reversed by demethylases (FTO, ALKBH5) and mediated by m(6)A-binding proteins (YTHDF1-3, YTHDC1), may also have a profound impact on gene expression regulation. Recent discoveries of the distributions, functions, and mechanisms of m(6)A modification suggest that this methylation functionally modulates the eukaryotic transcriptome to influence mRNA transcription, splicing, nuclear export, localization, translation, and stability. This reversible mRNA methylation shed light on a new dimension of post-transcriptional regulation of gene expression at the RNA level. m(6)A methylation also plays significant and broad roles in various physiological processes, such as development, fertility, carcinogenesis, stemness, early mortality, meiosis and circadian cycle, and links to obesity, cancer, and other human diseases. This review mainly describes the current knowledge of m(6)A and perspectives on future investigations.

Skeletal muscle plays important roles in whole-body energy homeostasis. Excessive skeletal muscle lipid accumulation is associated with some metabolic diseases such as obesity and Type 2 Diabetes. The energy sensor AMPK (AMP-activated protein kinase) is a key regulator of skeletal muscle lipid metabolism, but the precise regulatory mechanism remains to be elucidated. Here, we provide a novel mechanism by which AMPK regulates skeletal muscle lipid accumulation through fat mass and obesity-associated protein (FTO)-dependent demethylation of N6-methyladenosine (m6A). We confirmed an inverse correlation between AMPK and skeletal muscle lipid content. Moreover, inhibition of AMPK enhanced lipid accumulation, while activation of AMPK reduced lipid accumulation in skeletal muscle cells. Notably, we found that mRNA m6A methylation levels were inversely correlated with lipid content in skeletal muscle. Furthermore, AMPK positively regulated the m6A methylation levels of mRNA, which could negatively regulate lipid accumulation in C2C12. At the molecular level, we demonstrated that AMPK regulated lipid accumulation in skeletal muscle cells by regulating FTO expression and FTO-dependent demethylation of m6A. Together, these results provide a novel regulatory mechanism of AMPK on lipid metabolism in skeletal muscle cells and suggest the possibility of controlling skeletal muscle lipid deposition by targeting AMPK or using m6A related drugs.

For a random sample of size n obtained from a p-variate normal population, the likelihood ratio test (LRT) for the covariance matrix equal to a given matrix is considered. By using the Selberg integral, we prove that the LRT statistic converges to a normal distribution under the assumption p/n → y ∈ (0, 1]. The result for y = 1 is much different from the case for y ∈ (0, 1). Another test is studied: given two sets of random observations of sample size n 1 and n 2 from two p-variate normal distributions, we study the LRT for testing the two normal distributions having equal covariance matrices. It is shown through a corollary of the Selberg integral that the LRT statistic has an aymptotic normal distribution under the assumption p/n 1 → y 1 ∈ (0, 1] and p/n 2 → y 2 ∈ (0, 1]. The case for max{y 1 , y 2 } = 1 is much different from the case max{y 1 , y 2 } < 1.

The multivariate nonlinear Granger causality developed by Bai et al. (2010) (Mathematics and Computers in simulation. 2010; 81: 5-17) plays an important role in detecting the dynamic interrelationships between two groups of variables. Following the idea of Hiemstra-Jones (HJ) test proposed by Hiemstra and Jones (1994) (Journal of Finance. 1994; 49(5): 1639-1664), they attempt to establish a central limit theorem (CLT) of their test statistic by applying the asymptotical property of multivariate U-statistic. However, Bai et al. (2016) (2016; arXiv: 1701.03992) revisit the HJ test and find that the test statistic given by HJ is NOT a function of U-statistics which implies that the CLT neither proposed by Hiemstra and Jones (1994) nor the one extended by Bai et al. (2010) is valid for statistical inference. In this paper, we re-estimate the probabilities and reestablish the CLT of the new test statistic. Numerical simulation shows that our new estimates are consistent and our new test performs decent size and power.