Corrections to LRT on Large Dimensional Covariance Matrix by RMT

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 chi-square 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.Comment: 25 pages, 2 figures and 3 table

Likelihood ratio tests for covariance matrices of high-dimensional normal distribution

a b s t r a c t 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 2 ð0; 1. The result for y¼ 1 is much different from the case for y 2 ð0; 1Þ. 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 asymptotic normal distribution under the assumption p=n 1 -y 1 2 ð0; 1 and p=n 2 -y 2 2 ð0; 1. The case for maxfy 1 ,y 2 g ¼ 1 is much different from the case max fy 1 ,y 2 g o 1

Assessing the Therapeutic Effect of 630 nm Light-emitting Diodes Irradiation on the Recovery of Exercise-induced Hand Muscle Fatigue with Surface Electromyogram

This paper aims to investigate the effect of light emitting diode therapy (LEDT) on exercise-induced hand muscle fatigue by measuring the surface electromyography (sEMG) of flexor digitorum superficialis. Ten healthy volunteers were randomly placed in the equal sized LEDT group and control group. All subjects performed a sustained fatiguing isometric contraction with the combination of four fingertips except thumb at 30% of maximal voluntary contraction (MVC) until exhaustion. The active LEDT or an identical passive rest therapy was then applied to flexor digitorum superficialis. Each subject was required to perform a re-fatigue task immediately after therapy which was the same as the pre-fatigue task. Average rectified value (ARV) and fractal dimension (FD) of sEMG were calculated. ARV and FD were significantly different between active LEDT and passive rest groups at 20%–50%, 70%–80%, and 100% of normalized contraction time (P \u3c 0.05 ). Compared to passive rest, active LEDT induced significantly smaller increase in ARV values and decrease in FD values, which shows that LEDT is effective on the recovery of muscle fatigue. Our preliminary results also suggest that ARV and FD are potential replacements of biochemical markers to assess the effects of LEDT on muscle fatigue