Том 237, № 5 (2019)

We define a new class of positive and measurable functions that are bounded by regularly varying functions (which were introduced by Karamata). We study integrals and Laplace transforms of these functions. We use the obtained results to study the tail of convolutions of distribution functions. The results are extended to functions that are bounded by O-regularly varying functions.

Signal partitioning or signal segmentation allows to perform data classification, prediction of signals’ behavior, and profound interpretation of obtained data. In accordance with the signal model some distribution characteristics of the window variance noise component are investigated. It was shown that when all true underlying signal components remain unchanged, the window variance noise component is gamma distributed. Applying window variance to multicomponent signal (i.e., pharmacokinetic curve), it was shown that the window variance allows to split visually the signal record into phases due to prevalent processes.

This paper is a review of recent results for large random matrices with heavy-tailed entries. First, we outline the development of and some classical results in random matrix theory. We focus on large sample covariance matrices, their limiting spectral distributions, and the asymptotic behavior of their largest and smallest eigenvalues and their eigenvectors. The limits significantly depend on the finite or infiniteness of the fourth moment of the entries of the random matrix. We compare the results for these two regimes which give rise to completely different asymptotic theories. Finally, the limits of the extreme eigenvalues of sample correlation matrices are examined.

In this paper a closed retrial queueing system is considered with a finite number of customers. If an arriving (primary or secondary) request finds the server busy, two modes are possible: the job is transferred to the orbit (no conflict) or the job under service is interrupted and both of them are transferred to the orbit (conflict). Jobs in the orbit can retry reaching the server after a random time. The unreliable case where the server is subject to breakdown is also investigated. These types of systems can be solved by numerical, asymptotical, and simulation methods. The novelty of the investigations is that it provides a new approach to an algorithmic solution for calculating the steady-state probabilities of the system. With the help of these probabilities the main performance measures can be computed. Several sample examples illustrate the effect of different parameters on the distribution on requests in the system.

One of the most popular models in survival analysis is the Cox proportional hazards model. This model has been widely used because of its simplicity. Despite its simplicity, the basic problem of the Cox model is its inability to enter unknown risk factors into the model. Some risk factors may affect the survival of a trial unit, but due to time and cost constraints, there is no possibility to measure all of these factors in the form of explanatory variables in the model. In many cases, measuring risk factors is not possible. For entering unknown risk factors into the model, a positive random variable, representative of unknown risk factors, is multiplied in the model. Then a new class of survival models, namely frailty models, is introduced. When survival data are spatially correlated, frailty models do not have good performance for analyzing the data. In these cases, a Gaussian random field would be used as frailty variable such that the spatial correlation of the data can be included in the model. But in some applications, the Gaussian assumption of spatial effects is not realistic. In this paper, we use a spatial survival model with Gaussian and non-Gaussian random effects. Considering the complexity of likelihood function for spatial survival models and the lack of closed form, the frequency approach is very time-consuming for estimation of the model parameters. We use the Bayesian approach and MCMC algorithms for estimating the model parameters. Next, the application of the model is shown in an analysis of a real dataset.

Kernel-type density or regression estimator does not produce a constant estimator variance over the domain. To correct this problem, K. Nishida and Y. Kanazawa (2011, 2015) proposed a variance-stabilizing (VS) local variable bandwidth for kernel regression estimators. K. Nishida (2017) proposed another strategy to construct VS local linear regression estimator using a convex combination of three skewing estimators proposed by Choi and Hall (1998). In this study, we show that variance stabilization can be accomplished by a Bayesian approach in the case of kernel density estimator using conjugate prior.

Animal activity is a descriptor that can be potentially used to track the health and well-being, which is obviously very important to improve the management process and productivity of farms. This paper deals with an animal behavior recognition problem using a chaos theory approach where we adopt such a technique for automatic classification of calves behavioral states. Two main mutually exclusive behaviors are of interest, namely, lying and standing/walking with six possible activities: feeding, drinking water, drinking milk, playing, rumination, and neutral. The time series generated by ear-tags with a 3D-accelerometer after a wavelet denoising transformation and frequency stabilization are treated as representations of the nonlinear dynamical system. The dynamical system of a certain animal state exhibits specific strange attractor in a phase space. A characterization of such an attractor is performed through metric, dynamic, and topological invariants including Lyapunov exponent, correlation dimension, length of a phase trajectory, sum of edges forming a convex hull and others. These measures are used as a feature vector for the subsequent classification. In a cross validation scheme, six classifiers are built on each training set, and the hyper-parameters are optimized using an inner validation set. The classifier that reaches the highest accuracy on the inner validation set is used to classify the outer validation set. It is shown that this approach can be useful at predicting activity states as an alternative methodology for the animal behavior state recognition problem with acceptable classification accuracy. Furthermore it is possible to include this procedure as part of an ensemble method in machine learning where a combination of different models is used.