Aggregation of the fractional Brownian motion

In this paper we introduce fractional Brownian motion aggregated by a random order. We proved that this random process is self-similar with the same self-similarity index as in the initial fractional Browman motion. Sample paths of the newly proposed random process can be obtained from the sample paths of the initial fractional Brownian motion easily, so the newly proposed process can be useful in the statistics of random processes where empirical mvestigation of the estimators of self-similarity index is needed

Grupinio įverčio asimptotinio vietos invariatiškumo sritis

This paper is devoted to estimation of the tail index of heavy-tailed distributions. We consider the situation when initial i.i.d. data is perturbed by a constant type positive trend. Our main message is that such kind of trend should be removed before estimation of the tail by using group estimator because this estimator asymptotically is not location invariant in general

Extreme value statistics for evolving random networks

Our objective is to survey recent results concerning the evolution of random networks and related extreme value statistics, which are a subject of interest due to numerous applications. Our survey concerns the statistical methodology but not the structure of random networks. We focus on the problems arising in evolving networks mainly due to the heavy-tailed nature of node indices. Tail and extremal indices of the node influence characteristics like in-degrees, out-degrees, PageRanks, and Max-linear models arising in the evolving random networks are discussed. Related topics like preferential and clustering attachments, community detection, stationarity and dependence of graphs, information spreading, finding the most influential leading nodes and communities, and related methods are surveyed. This survey tries to propose possible solutions to unsolved problems, like testing the stationarity and dependence of random graphs using known results obtained for random sequences. We provide a discussion of unsolved or insufficiently developed problems like the distribution of triangle and circle counts in evolving networks, or the clustering attachment and the local dependence of the modularity, the impact of node or edge deletion at each step of evolution on extreme value statistics, among many others. Considering existing techniques of community detection, we pay attention to such related topics as coloring graphs and anomaly detection by machine learning algorithms based on extreme value theory. In order to understand how one can compute tail and extremal indices on random graphs, we provide a structured and comprehensive review of their estimators obtained for random sequences. Methods to calculate the PageRank and PageRank vector are shortly presented. This survey aims to provide a better understanding of the directions in which the study of random networks has been done and how extreme value analysis developed for random sequences can be applied to random networks

Estimator of the tail index for the Pareto law

By modifying Qi (2010) tail index estimator we introduce shift and scale free tail index estimator for the 3-parametric Pareto law. Asymptotic normality of the new estimator is derived. By applying de Haan and Peng (1998) comparison scheme for the tail index estimators it is obtained that new estimator outperforms Qi (2010) estimator when observations are independent and identically distributed Pareto random variables

Tail index estimation of PageRanks in evolving random graphs

Random graphs are subject to the heterogeneities of the distributions of node indices and their dependence structures. Superstar nodes to which a large proportion of nodes attach in the evolving graphs are considered. In the present paper, a statistical analysis of the extremal part of random graphs is considered. We used the extreme value theory regarding sums and maxima of non-stationary random length sequences to evaluate the tail index of the PageRanks and max-linear models of superstar nodes in the evolving graphs where existing nodes or edges can be deleted or not. The evolution is provided by a linear preferential attachment. Our approach is based on the analysis of maxima and sums of the node PageRanks over communities (block maxima and block sums), which can be independent or weakly dependent random variables. By an empirical study, it was found that tail indices of the block maxima and block sums are close to the minimum tail index of representative series extracted from the communities. The tail indices are estimated by data of simulated graph

www.elsevier.com/locate/jmva The increment ratio statistic

We introduce a new statistic written as a sum of certain ratios of second-order increments of partial sums process Sn = ∑n t=1 Xt of observations, which we call the increment ratio (IR) statistic. The IR statistic can be used for testing nonparametric hypotheses for d-integrated ( − 1 2 <d< 3 2) behavior of time series Xt, including short memory (d = 0), (stationary) long-memory (0 <d< 1 2) and unit roots (d = 1). If Sn behaves asymptotically as an (integrated) fractional Brownian motion with parameter H = d + 1 2, the IR statistic converges to a monotone function Λ(d) of d ∈ ( − 1 2, 3 2) as both the sample size N and the window parameter m increase so that N/m →∞. For Gaussian observations Xt, we obtain a rate of decay of the bias EIR−Λ(d) and a central limit theorem (N/m) 1/2 (IR − EIR) → N(0, σ2 (d)), in the region − 1 2 <d< 5 4. Graphs of the functions Λ(d) and σ(d) are included. A simulation study shows that the IR test for short memory (d = 0) against stationary long-memory alternatives (0 <d< 1 2) has good size and power properties and is robust against changes in mean, slowly varying trends and nonstationarities. We apply this statistic to sequences of squares of returns on financial assets and obtain a nuanced picture of the presence of long-memory in asse