We introduce a new statistic written as a sum of certain ratios of second-order increments of partial sums process S n = n t=1 X t 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 X t , including short memory (d = 0), (stationary) long-memory (0 < d < 12 ) and unit roots (d = 1). If S n behaves asymptotically as an (integrated) fractional Brownian motion with parameter H = d + 1 2 , the IR statistic converges to a monotone function (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 X t , we obtain a rate of decay of the bias EIR− (d) and a central limit theorem (N/m) 1/2 (IR − EIR) → N(0, 2 (d)), in the region − 1 2 < d < 5 4 . Graphs of the functions (d) and (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 asset price volatility.
In the paper we propose some new class of functions which is used to construct tail index estimators. Functions from this new class is non-monotone in general, but presents a product of two monotone functions: the power function and the logarithmic function, which plays essential role in the classical Hill estimator. Introduced new estimators have better asymptotic performance comparing with the Hill estimator and other popular estimators over all range of the parameters present in the second order regular variation condition. Asymptotic normality of the introduced estimators is proved, and comparison (using asymptotic mean square error) with other estimators of the tail index is provided. Some preliminary simulation results are presented.
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 graphs.
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