Vol 237, No 6 (2019)

In this study, we develop a clustering method for multivariate time series data. In practical situations, such problems can arise in finance, economics, control theory, and health science. First, we propose to use a simulation based approximation to the test statistic and develop a method to test if two multivariate time series are coming from same VAR process. Then, the testing method is extended to a group of multivariate time series objects. Finally, a new clustering algorithm is developed using the testing method. The proposed algorithm does not use a predetermined number of clusters and finds the best possible clustering from the data. Empirical studies are provided in this paper, and they establish the accuracy of the algorithm. Finally, as a practical example, the algorithm is implemented to identify activities of different persons from a real-life data obtained from single chest-mounted accelerometers worn by different individuals.

Adopting ideas of Katz (1963), Petrov (1965), Wang and Ahmad (2016), and Gabdullin, Makarenko, and Shevtsova (2016), we generalize the Rozovskii inequality (1974) which provides an estimate of the accuracy of the normal approximation to distribution of a sum of independent random variables in terms of the absolute value of the sum of truncated in a fixed point third-order moments and the sum of the second-order tails of random summands. The generalization is due to introduction of a truncation parameter and a weighting function from a set of functions originally introduced by Katz (1963). The obtained inequality does not assume finiteness of moments of random summands of order higher than the second and may be even sharper than the celebrated inequalities of Berry (1941), Esseen (1942, 1969), Katz (1963), Petrov (1965), and Wang & Ahmad (2016).

Extreme values are considered in samples with random size that have a mixed Poisson distribution that is generated by a doubly stochastic Poisson process. Some inequalities are proved relating the distributions and moments of extrema with those of the leading process (the mixing distribution). Limit theorems are proved for the distributions of max-compound Cox processes, and limit distributions are described. An important particular case of the negative binomial distribution of a sample size corresponding to the case where the Cox process is led by a gamma Lévy process is considered, explaining a possible genesis of tempered asymptotic models.

Five models of stochastic correlation are compared on the basis of the generated associations of Wiener processes. Associations are characterised by the copulas and their K-functions. A confidence domain for the randomly changing K-functions is build on the basis of simulated Wiener process pairs. The models are ordered by the magnitude of the domains of confidence bounds. Larger confidence domains or higher bounds represent higher correlation risk when the models are applied in mathematical finance.

Signal denoising methods based on the threshold processing of wavelet coefficients are widely used in various application areas. When applying these methods, it is usually assumed that the number of wavelet coefficients is fixed, and the noise distribution is Gaussian. Such a model has been well studied in the literature, and optimal threshold values have been calculated for different signal classes and loss functions. However, in some situations the sample size is not known in advance and is modeled by a random variable. In this paper, we consider a model with a random number of observations contaminated by a Gaussian noise, and study the behavior of the loss function based on the probabilities of errors in calculating wavelet coefficients for a growing sample size.

In this paper a finite-source retrial queuing system with collision of the customers is investigated by means of computer simulation. The server is not reliable; it is subjected to breakdowns, and the repairs depend on whether the state of the server is idle. The random variables used in the model are jointly independent and are exponential and gamma distributed. Two operation modes are considered in the case of busy breakdown. The novelty of the investigation is a comparison of the performance measures of these modes, and estimations obtained by the simulation are graphically illustrated showing the influence of the difference of the working modes on the performance measures such as mean and variance of the response time, mean and variance of the number of customers in the system, mean and variance of the sojourn time in the orbit, mean and variance of time time a customer spent in service.

This paper gives an overview of the existing approaches for modelling high quantiles of dependent spatial data and apply the methods to the bivariate circular case. We also adapt the bootstrap to the situation at hand.

Since any data set one might use is finite, the interest lies in estimating a continuous curve as its upper limit (or quantile function). This can be obtained by either a kernel type regression or a fitted parametric model. We also introduce a new, more realistic correction formula for a nonparametric method for estimating the pointwise maximum (called frontier in this setup [8]).

An additional common challenge in real-life applications is the dependence among subsequent observations. The theoretical results about the GPD limit of the exceedances beyond high threshold remain valid under mixing-type conditions, called D(u_n) in the extreme-value literature. However, if one intends to use the bootstrap-based reliability estimators, then they need to be adjusted — e.g., by the block-bootstrap approach in [15]. We estimate the reliability of the estimators by a suitable application of the m out of n bootstrap, which turned out to be suitable for high-quantile estimation.

We illustrate the introduced methods for simulated as well as real enzymes data.