In random matrix theory, large random covariance matrices for a class of stationary processes have been applied to many high-dimensional statistical problems. This paper provides a central limit theorem for the linear spectral statistics of these matrices, by proving, after noticing that the variables in the matrix of observations have a same covariance structure as that in sample covariance matrix formed by Gaussian random variables, that the mean and covariance functions of the Gaussian limits depends on the spectral density of the underlying process. Some examples of linear and nonlinear stationary processes to be applied to the result are given.
Testing for serial correlation is fundamental in regression analysis, yet methods for matrix-type data remain limited. This paper develops a test for serial correlation within the trace regression framework. By extending empirical likelihood to matrix data, we propose an empirical log-likelihood ratio statistic. Its asymptotic distribution is derived under both the null and local alternatives. Some simulations are conducted to demonstrate its finite-sample performance. Finally, the proposed test is applied to straw-burning fire-point data, confirming its practical utility.
The development of new models using specialized forms of reliability functions has been a topic of interest among researchers for the past five decades. A significant amount of research also exists in the bivariate case, where various bivariate distributions have been constructed using different failure patterns. As far as bivariate quantile-based reliability analysis is concerned, research in this direction is in its early stages. In this paper, we propose a new distribution by assuming a linear form of quantile-based hazard gradients. This is a novel approach for developing bivariate distributions. We extensively study the properties and applications of this distribution and present several well-known bivariate distributions as members of this family. The marginal and joint density functions of the distribution exhibit various shapes, which enable it to model various types of bivariate lifetime data. The distributional characteristics based on ordinary moments and L-moments are analyzed, and the reliability properties of the distribution are examined. Despite being constructed using a linear form of quantile-based hazard gradients, the hazard gradients derived from the distribution function exhibit non-monotonic shapes, which further enhance the distribution's applicability in modeling bivariate data. The utility of the distribution in modeling is illustrated using real data examples.
In this paper, we study the distance between zeros of the second derivative and roots of the polynomial p(n)(z) = Pi(s)(l=1)(z - xi(l)) Pi(n+1-s)(j=1)(z - X-j), where X-1, ... , Xn+1-s are i.i.d. complex-valued random variables, and xi(1), ... , xi(s) are distinct, fixed deterministic values. By giving a study which is about Hoeffding's inequality for cross-product sum, we reveal that under certain conditions, with probability at least 1 - o(1), p(n)(z) has s zeros of the second derivative, w(1)((n)), ... , w(s)((n)), such that for 1 < l < s, w(l)((n)) is the unique zero of the second derivative of p(n) that is within a distance of O(n(-1)) of xi(l).
Quantiles are used to measure risk or return in risk management and investment decision-making, while the data available in finance are often of high-frequency. In this paper, we study the joint confidence regions for a finite number of quantiles under strong mixing high-frequency data by employing the blockwise empirical likelihood (EL) method and the blockwise adjust empirical likelihood (AEL) method. We firstly construct the blockwise EL ratio statistic and the blockwise AEL ratio statistic for a finite number of quantiles and prove their asymptotic properties. Then we conduct simulations which show that the confidence regions based on the AEL method perform better than the EL method and the normal approximation method. As an application of our theoretic results, we also give an empirical analysis for real data.
