Topological entropy is one of the most difficult entropies to be used to analyze the DNA sequences, due to the finite sample and high-dimensionality problems. In order to overcome these problems, a generalized topological entropy is introduced. The relationship between the topological entropy and the generalized topological entropy is compared, which shows the topological entropy is a special case of the generalized entropy. As an application the generalized topological entropy in introns, exons and promoter regions was computed, respectively. The results indicate that the entropy of introns is higher than that of exons, and the entropy of the exons is higher than that of the promoter regions for each chromosome, which suggest that DNA sequence of the promoter regions is more regular than the exons and introns.

Next-generation sequencing technology has made it possible to detect rare genetic variants associated with complex human traits. In recent literature, various methods specifically designed for rare variants are proposed. These tests can be broadly classified into burden and nonburden tests. In this paper, we take advantage of the burden and nonburden tests, and consider the common effect and the individual deviations from the common effect. To achieve robustness, we use two methods of combining p-values, Fisher’s method and the minimum-p method. In rare variant association studies, to improve the power of the tests, we explore the advantage of the extreme phenotype sampling. At first, we dichotomize the continuous phenotypes before analysis, and the two extremes are treated as two different groups representing a dichotomous phenotype. We next compare the powers of several methods based on extreme phenotype sampling and random sampling. Extensive simulation studies show that our proposed methods by using extreme phenotype sampling are the most powerful or very close to the most powerful one in various settings of true models when the same sample size is used.

In the current global context of economic integration, unexpected events have an important influence in the financial field. In 2020, the “COVID-19” outbreak triggered financial turmoil throughout the whole country and even in the global market. In the wake of this era, how to sum up past developments and predict future development through change-point detection is particularly important. In this paper, four methods for detecting change-points are presented: the likelihood ratio method, least squares method, CUSUM method, and local comparison method. Considering that Bernstein polynomials have worked well in density function approximation, the multi-dimensional Bernstein polynomials are presented. The study applies multiple change-point detection methods to determine the most suitable degree of freedom $m_j$ for multi-dimensional Bernstein models, after which various rewriting expressions can be obtained. Next, “COVID-19” data and money supply data are used for change-point detection with good results. Then, we focus on conducting change-point testing on the S&P 500 index and SSE 50 index, indicating strong symmetry when major crisis events occur. All analyses indicate that change-point detection plays an important role in identifying major crisis events and financial shocks.

In periods of dramatic stock price volatility, the identification of change points in stock price time series is important for analyzing the structural changes in financial market data, as well as for risk prevention and control in the financial market. As their residuals follow a generalized error distribution, the problem of estimating the change point parameters of the $\beta$-ARCH model is solved by combining the Kalman filtering method and the Bayes method innovatively, and we give a method for parameter estimation of the Bayes factors for the occurrences of change points, the expected values of the change point positions, and the variance of the change points. By detecting the change points of the price of eight stocks with a high number of limit up and limit down changes occurring in the observation period, the following conclusions are obtained: (1) Change point detection using the $\beta$-ARCH model based on the Bayes method is effective. (2) For different values of $\beta$, this research study finds that based on the classical ARCH model (i.e., $\beta =1$) of the change point parameter, the results are relatively optimal. (3) The accuracy of change point detection can be improved by correcting stock short-term effects by using the Kalman filtering method.