arXiv subjects

Sidney I. Resnick

Publications and source records attributed to Sidney I. Resnick.

28 records · Page 2Linked to original sources

arXiv2010

On the superposition of heterogeneous traffic at large time scales

Various empirical and theoretical studies indicate that cumulative network traffic is a Gaussian process. However, depending on whether the intensity at which sessions are initiated is large or small relative to the session duration tail, Mikosch et al. (Ann Appl Probab, 12:23-68, 2002) and Kaj and Taqqu (Progress Probab, 60:383-427, 2008) have shown that traffic at large time scales can be approximated by either fractional Brownian motion (fBm) or stable Levy motion. We study distributional properties of cumulative traffic that consists of a finite number of independent streams and give an explanation of why Gaussian examples abound in practice but not stable Levy motion. We offer an explanation of how much vertical aggregation is needed for the Gaussian approximation to hold. Our results are expressed as limit theorems for a sequence of cumulative traffic processes whose session initiation intensities satisfy growth rates similar to those used in Mikosch et al. (Ann Appl Probab, 12:23-68, 2002).

math.PR↗

arXiv2010

When does the mean excess plot look linear?

In risk analysis, the mean excess plot is a commonly used exploratory plotting technique for confirming iid data is consistent with a generalized Pareto assumption for the underlying distribution, since in the presence of such a distribution thresholded data have a mean excess plot that is roughly linear. Does any other class of distributions share this linearity of the plot? Under some extra assumptions, we are able to conclude that only the generalized Pareto family has this property.

math.ST↗

arXiv2010

Hidden Regular Variation: Detection and Estimation

Hidden regular variation defines a subfamily of distributions satisfying multivariate regular variation on $\mathbb{E} = [0, \infty]^d \backslash \{(0,0, ..., 0) \} $ and models another regular variation on the sub-cone $\mathbb{E}^{(2)} = \mathbb{E} \backslash \cup_{i=1}^d \mathbb{L}_i$, where $\mathbb{L}_i$ is the $i$-th axis. We extend the concept of hidden regular variation to sub-cones of $\mathbb{E}^{(2)}$ as well. We suggest a procedure for detecting the presence of hidden regular variation, and if it exists, propose a method of estimating the limit measure exploiting its semi-parametric structure. We exhibit examples where hidden regular variation yields better estimates of probabilities of risk sets.

math.PR↗

arXiv2010

Modeling total expenditure on warranty claims

We approximate the distribution of total expenditure of a retail company over warranty claims incurred in a fixed period [0, T], say the following quarter. We consider two kinds of warranty policies, namely, the non-renewing free replacement warranty policy and the non-renewing pro-rata warranty policy. Our approximation holds under modest assumptions on the distribution of the sales process of the warranted item and the nature of arrivals of warranty claims. We propose a method of using historical data to statistically estimate the parameters of the approximate distribution. Our methodology is applied to the warranty claims data from a large car manufacturer for a single car model and model year.

q-fin.GN↗

arXiv2009

Aggregation of Risks and Asymptotic independence

We study the tail behavior of the distribution of the sum of asymptotically independent risks whose marginal distributions belong to the maximal domain of attraction of the Gumbel distribution. We impose conditions on the distribution of the risks $(X,Y)$ such that $P(X + Y > x) \sim (const)P (X > x)$. With the further assumption of non-negativity of the risks, the result is extended to more than two risks. We note a sufficient condition for a distribution to belong to both the maximal domain of attraction of the Gumbel distribution and the subexponential class. We provide examples of distributions which satisfy our assumptions. The examples include cases where the marginal distributions of $X$ and $Y$ are subexponential and also cases where they are not. In addition, the asymptotic behavior of linear combinations of such risks with positive coefficients is explored leading to an approximate solution of an optimization problem which is applied to portfolio design.

math.PR↗

arXiv2009

Extremal dependence analysis of network sessions

We refine a stimulating study by Sarvotham et al. [2005] which highlighted the influence of peak transmission rate on network burstiness. From TCP packet headers, we amalgamate packets into sessions where each session is characterized by a 5-tuple (S, D, R, Peak R, Initiation T)=(total payload, duration, average transmission rate, peak transmission rate, initiation time). After careful consideration, a new definition of peak rate is required. Unlike Sarvotham et al. [2005] who segmented sessions into two groups labelled alpha and beta, we segment into 10 sessions according to the empirical quantiles of the peak rate variable as a demonstration that the beta group is far from homogeneous. Our more refined segmentation reveals additional structure that is missed by segmentation into two groups. In each segment, we study the dependence structure of (S, D, R) and find that it varies across the groups. Furthermore, within each segment, session initiation times are well approximated by a Poisson process whereas this property does not hold for the data set taken as a whole. Therefore, we conclude that the peak rate level is important for understanding structure and for constructing accurate simulations of data in the wild. We outline a simple method of simulating network traffic based on our findings.

stat.AP↗

arXiv2009

Detecting a conditional extrme value model

In classical extreme value theory probabilities of extreme events are estimated assuming all the components of a random vector to be in a domain of attraction of an extreme value distribution. In contrast, the conditional extreme value model assumes a domain of attraction condition on a sub-collection of the components of a multivariate random vector. This model has been studied in \cite{heffernan:tawn:2004,heffernan:resnick:2007,das:resnick:2008a}. In this paper we propose three statistics which act as tools to detect this model in a bivariate set-up. In addition, the proposed statistics also help to distinguish between two forms of the limit measure that is obtained in the model.

math.ST↗

arXiv2007

Multivariate Regular Variation on Cones: Application to Extreme Values, Hidden Regular Variation and Conditioned Limit Laws

We attempt to bring some modest unity to three subareas of heavy tail analysis and extreme value theory: limit laws for componentwise maxima of iid random variables;hidden regular variation and asymptotic independence;conditioned limit laws when one component of a random vector is extreme. The common theme is multivariate regular variation on a cone and the three cases cited come from specifying the cones $[0,\infty]^d\setminus \{\boldsymbol 0\};(0,\infty]^d;$ and $[0,\infty]\times (0,\infty]$.

math.PR↗

arXiv2007

Limit laws for random vectors with an extreme component

Models based on assumptions of multivariate regular variation and hidden regular variation provide ways to describe a broad range of extremal dependence structures when marginal distributions are heavy tailed. Multivariate regular variation provides a rich description of extremal dependence in the case of asymptotic dependence, but fails to distinguish between exact independence and asymptotic independence. Hidden regular variation addresses this problem by requiring components of the random vector to be simultaneously large but on a smaller scale than the scale for the marginal distributions. In doing so, hidden regular variation typically restricts attention to that part of the probability space where all variables are simultaneously large. However, since under asymptotic independence the largest values do not occur in the same observation, the region where variables are simultaneously large may not be of primary interest. A different philosophy was offered in the paper of Heffernan and Tawn [J. R. Stat. Soc. Ser. B Stat. Methodol. 66 (2004) 497--546] which allows examination of distributional tails other than the joint tail. This approach used an asymptotic argument which conditions on one component of the random vector and finds the limiting conditional distribution of the remaining components as the conditioning variable becomes large. In this paper, we provide a thorough mathematical examination of the limiting arguments building on the orientation of Heffernan and Tawn [J. R. Stat. Soc. Ser. B Stat. Methodol. 66 (2004) 497--546]. We examine the conditions required for the assumptions made by the conditioning approach to hold, and highlight simililarities and differences between the new and established methods.

math.PR↗

arXiv2007

QQ plots, Random sets and data from a heavy tailed distribution

The QQ plot is a commonly used technique for informally deciding whether a univariate random sample of size n comes from a specified distribution F. The QQ plot graphs the sample quantiles against the theoretical quantiles of F and then a visual check is made to see whether or not the points are close to a straight line. For a location and scale family of distributions, the intercept and slope of the straight line provide estimates for the shift and scale parameters of the distribution respectively. Here we consider the set S_n of points forming the QQ plot as a random closed set in R^2. We show that under certain regularity conditions on the distribution F, S_n converges in probability to a closed, non-random set. In the heavy tailed case where 1-F is a regularly varying function, a similar result can be shown but a modification is necessary to provide a statistically sensible result since typically F is not completely known.

math.PR↗