Quantitative Fourth Moment Theorem of Functions on the Markov Triple and Orthogonal Polynomials

In this paper, we consider a quantitative fourth moment theorem for functions (random variables) defined on the Markov triple E , μ , Γ , where μ is a probability measure and Γ is the carré du champ operator. A new technique is developed to derive the fourth moment bound in a normal approximation on the random variable of a general Markov diffusion generator, not necessarily belonging to a fixed eigenspace, while previous works deal with only random variables to belong to a fixed eigenspace. As this technique will be applied to the works studied by Bourguin et al. (2019), we obtain the new result in the case where the chaos grade of an eigenfunction of Markov diffusion generator is greater than two. Also, we introduce the chaos grade of a new notion, called the lower chaos grade, to find a better estimate than the previous one.

THE FOURTH HISTORICAL MOMENT IN SOCIAL RESEARCH ‘THE CRISIS OF REPRESENTATION’: A CHAPTER REVIEW

The historical moments of qualitative research reflect socially constructed quasi-historic conventions that remain crosscut and overlapping till the present. This progressive narrative is well represented and assessed in a historical overview by Denzin & Lincoln (2018) in their book “The Sage handbook of qualitative research” in the introduction “The discipline and practice of qualitative research”. Through a chapter review, this article particularly discusses the fourth moment of Quantitative research coined as “The crisis of representation”, which is believed to be the crossroads where social scientists remain entangled between the science and humanity perspective while conducting social research in order to forward social realities. This period of confusion simultaneously forwarded the multi-paradigm (positivism, postpositivism, and interpretivism), all of which have unique characteristics that are suitable for specific research. Thus, this paper sheds light on the overview of the crisis of representation and further explains the types of crises that occurred during this historical moment, including the crisis of representation, the crisis of legitimation, and a crisis of praxis. It is expected that apart from extending current literature this paper would support social scientists for selecting appropriate methods and paradigms as well as to justify their selection.

Dynamic Event-Triggered Adaptive Tracking Control for a Class of Unknown Stochastic Nonlinear Strict-Feedback Systems

In this paper, the dynamic event-triggered tracking control issue is studied for a class of unknown stochastic nonlinear systems with strict-feedback form. At first, neural networks (NNs) are used to approximate the unknown nonlinear functions. Then, a dynamic event-triggered controller (DETC) is designed through the adaptive backstepping method. Especially, the triggered threshold is dynamically adjusted. Compared with its corresponding static event-triggered mechanism (SETM), the dynamic event-triggered mechanism (DETM) can generate a larger execution interval and further save resources. Moreover, it is verified by two simulation examples that show that the closed-loop stochastic system signals are ultimately fourth moment semi-globally uniformly bounded (SGUUB).

R-estimators in GARCH models; asymptotics and applications

The quasi-maximum likelihood estimation is a commonly-used method for estimating the GARCH parameters. However, such estimators are sensitive to outliers and their asymptotic normality is proved under the finite fourth moment assumption on the underlying error distribution. In this paper, we propose a novel class of estimators of the GARCH parameters based on ranks of the residuals, called R-estimators, with the property that they are asymptotically normal under the existence of a finite 2 + δ moment of the errors and are highly efficient. We propose fast algorithm for computing the R-estimators. Both real data analysis and simulations show the superior performance of the proposed estimators under the heavy-tailed and asymmetric distributions.

Influence of digital technologies on higher education students in times of the Covid-19 pandemic and the repercussions on their study habits

In this article, a second analysis is carried out that consists of monitoring the study habits of a group of 17 students, during the period of their higher education. The study was carried out on students who correspond to the area of Computational Sciences and consists of making a comparative evaluation of the application of the study habits questionnaire to such a group of students in four moments of their career path. In the first, second and third moments, the students attended classes in a face-to-face modality and in the fourth moment the students attended due to COVID-19 pandemic situations in a virtual modality. Based on the second context, the analysis has been carried out. The interest in delving into study habits is to identify areas of opportunity and implement strategies that allow students to improve them and avoid vices that hinder enough to obtain better academic performance.

TEST FOR ZERO MEDIAN OF ERRORS IN AN ARMA–GARCH MODEL

Because the ARMA–GARCH model can generate data with some important properties such as skewness, heavy tails, and volatility persistence, it has become a benchmark model in analyzing financial and economic data. The commonly employed quasi maximum likelihood estimation (QMLE) requires a finite fourth moment for both errors and the sequence itself to ensure a normal limit. The self-weighted quasi maximum exponential likelihood estimation (SWQMELE) reduces the moment constraints by assuming that the errors and their absolute values have median zero and mean one, respectively. Therefore, it is necessary to test zero median of errors before applying the SWQMELE, as changing zero mean to zero median destroys the ARMA–GARCH structure. This paper develops an efficient empirical likelihood test without estimating the GARCH model but using the GARCH structure to reduce the moment effect. A simulation study confirms the effectiveness of the proposed test. The data analysis shows that some financial returns do not have zero median of errors, which cautions the use of the SWQMELE.

Multivariate Normal Approximation on the Wiener Space: New Bounds in the Convex Distance

We establish explicit bounds on the convex distance between the distribution of a vector of smooth functionals of a Gaussian field and that of a normal vector with a positive-definite covariance matrix. Our bounds are commensurate to the ones obtained by Nourdin et al. (Ann Inst Henri Poincaré Probab Stat 46(1):45–58, 2010) for the (smoother) 1-Wasserstein distance, and do not involve any additional logarithmic factor. One of the main tools exploited in our work is a recursive estimate on the convex distance recently obtained by Schulte and Yukich (Electron J Probab 24(130):1–42, 2019). We illustrate our abstract results in two different situations: (i) we prove a quantitative multivariate fourth moment theorem for vectors of multiple Wiener–Itô integrals, and (ii) we characterize the rate of convergence for the finite-dimensional distributions in the functional Breuer–Major theorem.