arXiv subjects

Sidney I. Resnick

Publications and source records attributed to Sidney I. Resnick.

At least 19 recordsLinked to original sources

arXiv2025

Classification of Extremal Dependence in Financial Markets via Bootstrap Inference

Accurately identifying the extremal dependence structure in multivariate heavy-tailed data is a fundamental yet challenging task, particularly in financial applications. Following a recently proposed bootstrap-based testing procedure, we apply the methodology to absolute log returns of U.S. S&P 500 and Chinese A-share stocks over a time period well before the U.S. election in 2024. The procedure reveals more isolated clustering of dependent assets in the U.S. economy compared with China which exhibits different characteristics and a more interconnected pattern of extremal dependence. Cross-market analysis identifies strong extremal linkages in sectors such as materials, consumer staples and consumer discretionary, highlighting the effectiveness of the testing procedure for large-scale empirical applications.

math.ST↗

arXiv2023

2RV+HRV and Testing for Strong VS Full Dependence

Preferential attachment models of network growth are bivariate heavy tailed models for in- and out-degree with limit measures which either concentrate on a ray of positive slope from the origin or on all of the positive quadrant depending on whether the model includes reciprocity or not. Concentration on the ray is called full dependence. If there were a reliable way to distinguish full dependence from not-full, we would have guidance about which model to choose. This motivates investigating tests that distinguish between (i) full dependence; (ii) strong dependence (support of the limit measure is a proper subcone of the positive quadrant); (iii) weak dependence (limit measure concentrates on positive quadrant). We give two test statistics, analyze their asymptotically normal behavior under full and not-full dependence, and discuss applicability using bootstrap methods applied to simulated and real data.

math.ST↗

arXiv2021

Asymptotic Dependence of In- and Out-Degrees in a Preferential Attachment Model with Reciprocity

Reciprocity characterizes the information exchange between users in a network, and some empirical studies have revealed that social networks have a high proportion of reciprocal edges. Classical directed preferential attachment (PA) models, though generating scale-free networks, may give networks with low reciprocity. This points out one potential problem of fitting a classical PA model to a given network dataset with high reciprocity, and indicates alternative models need to be considered. We give one possible modification of the classical PA model by including another parameter which controls the probability of adding a reciprocated edge at each step. Asymptotic analyses suggest that large in- and out-degrees become fully dependent in this modified model, as a result of the additional reciprocated edges.

physics.soc-ph↗

arXiv2020

A Directed Preferential Attachment Model with Poisson Measurement

When modeling a directed social network, one choice is to use the traditional preferential attachment model, which generates power-law tail distributions. In a traditional directed preferential attachment, every new edge is added sequentially into the network. However, for real datasets, it is common to only have coarse timestamps available, which means several new edges are created at the same timestamp. Previous analyses on the evolution of social networks reveal that after reaching a stable phase, the growth of edge counts in a network follows a non-homogeneous Poisson process with a constant rate across the day but varying rates from day to day. Taking such empirical observations into account, we propose a modified preferential attachment model with Poisson measurement, and study its asymptotic behavior. This modified model is then fitted to real datasets, and we see it provides a better fit than the traditional one.

math.PR↗

arXiv2020

Extremes of Censored and Uncensored Lifetimes in Survival Data

The i.i.d. censoring model for survival analysis assumes two independent sequences of i.i.d. positive random variables, $(T_i^*)_{1\le i\le n}$ and $(U_i)_{1\le i\le n}$. The data consists of observations on the random sequence $\big(T_i=\min(T_i^*,U_i)$ together with accompanying censor indicators. Values of $T_i$ with $T_i^*\le U_i$ are said to be uncensored, those with $T_i^*> U_i$ are censored. We assume that the distributions of the $T_i^*$ and $U_i$ are in the domain of attraction of the Gumbel distribution and obtain the asymptotic distributions, as sample size $n\to\infty$, of the maximum values of the censored and uncensored lifetimes in the data, and of statistics related to them. These enable us to examine questions concerning the possible existence of cured individuals in the population.

math.ST↗

arXiv2020

On a minimum distance procedure for threshold selection in tail analysis

Power-law distributions have been widely observed in different areas of scientific research. Practical estimation issues include how to select a threshold above which observations follow a power-law distribution and then how to estimate the power-law tail index. A minimum distance selection procedure (MDSP) is proposed in Clauset et al. (2009) and has been widely adopted in practice, especially in the analyses of social networks. However, theoretical justifications for this selection procedure remain scant. In this paper, we study the asymptotic behavior of the selected threshold and the corresponding power-law index given by the MDSP. We find that the MDSP tends to choose too high a threshold level and leads to Hill estimates with large variances and root mean squared errors for simulated data with Pareto-like tails.

math.ST↗

arXiv2019

Common Growth Patterns for Regional Social Networks: a Point Process Approach

Although recent research on social networks emphasizes microscopic dynamics such as retweets and social connectivity of an individual user, we focus on macroscopic growth dynamics of social network link formation. Rather than focusing on one particular dataset, we find invariant behavior in regional social networks that are geographically concentrated. Empirical findings suggest that the startup phase of a regional network can be modeled by a self-exciting point process. After the startup phase ends, the growth of the links can be modeled by a non-homogeneous Poisson process with constant rate across the day but varying rates from day to day, plus a nightly inactive period when local users are expected to be asleep. Conclusions are drawn based on analyzing four different datasets, three of which are regional and a non-regional one is included for contrast.

stat.AP↗

arXiv2018

Degree Growth Rates and Index Estimation in a Directed Preferential Attachment Model

Preferential attachment is widely used to model power-law behavior of degree distributions in both directed and undirected networks. In a directed preferential attachment model, despite the well-known marginal power-law degree distributions, not much investigation has been done on the joint behavior of the in- and out-degree growth. Also, statistical estimates of the marginal tail exponent of the power-law degree distribution often use the Hill estimator as one of the key summary statistics, even though no theoretical justification has been given. This paper focuses on convergence of the joint empirical measure for in- and out-degrees and proves the consistency of the Hill estimator. To do this, we first derive the asymptotic behavior of the joint degree sequences by embedding the in- and out-degrees of a fixed node into a pair of switched birth processes with immigration and then establish the convergence of the joint tail empirical measure. From these steps, the consistency of the Hill estimators is obtained.

math.PR↗

arXiv2017

Are Extreme Value Estimation Methods Useful for Network Data?

Preferential attachment is an appealing edge generating mechanism for modeling social networks. It provides both an intuitive description of network growth and an explanation for the observed power laws in degree distributions. However, there are often limitations in fitting parametric network models to data due to the complex nature of real-world networks. In this paper, we consider a semi-parametric estimation approach by looking at only the nodes with large in- or out-degrees of the network. This method examines the tail behavior of both the marginal and joint degree distributions and is based on extreme value theory. We compare it with the existing parametric approaches and demonstrate how it can provide more robust estimates of parameters associated with the network when the data are corrupted or when the model is misspecified.

stat.ME↗

arXiv2017

Fitting the Linear Preferential Attachment Model

Preferential attachment is an appealing mechanism for modeling power-law behavior of the degree distributions in directed social networks. In this paper, we consider methods for fitting a 5-parameter linear preferential model to network data under two data scenarios. In the case where full history of the network formation is given, we derive the maximum likelihood estimator of the parameters and show that it is strongly consistent and asymptotically normal. In the case where only a single-time snapshot of the network is available, we propose an estimation method which combines method of moments with an approximation to the likelihood. The resulting estimator is also strongly consistent and performs quite well compared to the MLE estimator. We illustrate both estimation procedures through simulated data, and explore the usage of this model in a real data example.

stat.ME↗

arXiv2017

Hidden Regular Variation under Full and Strong Asymptotic Dependence

Data exhibiting heavy-tails in one or more dimensions is often studied using the framework of regular variation. In a multivariate setting this requires identifying specific forms of dependence in the data; this means identifying that the data tends to concentrate along particular directions and does not cover the full space. This is observed in various data sets from finance, insurance, network traffic, social networks, etc. In this paper we discuss the notions of full and strong asymptotic dependence for bivariate data along with the idea of hidden regular variation in these cases. In a risk analysis setting, this leads to improved risk estimation accuracy when regular methods provide a zero estimate of risk. Analyses of both real and simulated data sets illustrate concepts of generation and detection of such models.

math.ST↗

arXiv2016

Multivariate Regular Variation of Discrete Mass Functions with Applications to Preferential Attachment Networks

Regular variation of a multivariate measure with a Lebesgue density implies the regular variation of its density provided the density satisfies some regularity conditions. Unlike the univariate case, the converse also requires regularity conditions. We extend these arguments to discrete mass functions and their associated measures using the concept that the the mass function can be embedded in a continuous density function. We give two different conditions, monotonicity and convergence on the unit sphere, both of which can make the discrete function embeddable. Our results are then applied to the preferential attachment network model, and we conclude that the joint mass function of in- and out-degree is embeddable and thus regularly varying.

math.PR↗

arXiv2015

Asymptotic Normality of In- and Out-Degree Counts in a Preferential Attachment Model

Preferential attachment in a directed scale-free graph is widely used to model the evolution of social networks. Statistical analyses of social networks often relies on node based data rather than conventional repeated sampling. For our directed edge model with preferential attachment, we prove asymptotic normality of node counts based on a martingale construction and a martingale central limit theorem. This helps justify estimation methods based on the statistics of node counts which have specified in-degree and out-degree.

math.PR↗

arXiv2013

Clustering of Markov chain exceedances

The tail chain of a Markov chain can be used to model the dependence between extreme observations. For a positive recurrent Markov chain, the tail chain aids in describing the limit of a sequence of point processes $\{N_n,n\geq1\}$, consisting of normalized observations plotted against scaled time points. Under fairly general conditions on extremal behaviour, $\{N_n\}$ converges to a cluster Poisson process. Our technique decomposes the sample path of the chain into i.i.d. regenerative cycles rather than using blocking argument typically employed in the context of stationarity with mixing.

math.ST↗

arXiv2013

Regularly Varying Measures on Metric Spaces: Hidden Regular Variation and Hidden Jumps

We develop a framework for regularly varying measures on complete separable metric spaces $\mathbb{S}$ with a closed cone $\mathbb{C}$ removed, extending material in Hult & Lindskog (2006), Das, Mitra & Resnick (2013). Our framework provides a flexible way to consider hidden regular variation and allows simultaneous regular variation properties to exist at different scales and provides potential for more accurate estimation of probabilities of risk regions. We apply our framework to iid random variables in $\mathbb{R}_+^\infty$ with marginal distributions having regularly varying tails and to càdlàg Lévy processes whose Lévy measures have regularly varying tails. In both cases, an infinite number of regular variation properties coexist distinguished by different scaling functions and state spaces.

math.PR↗

arXiv2011

Asymptotics of Markov Kernels and the Tail Chain

An asymptotic model for extreme behavior of certain Markov chains is the "tail chain". Generally taking the form of a multiplicative random walk, it is useful in deriving extremal characteristics such as point process limits. We place this model in a more general context, formulated in terms of extreme value theory for transition kernels, and extend it by formalizing the distinction between extreme and non-extreme states. We make the link between the update function and transition kernel forms considered in previous work, and we show that the tail chain model leads to a multivariate regular variation property of the finite-dimensional distributions under assumptions on the marginal tails alone.

math.PR↗

arXiv2011

Modeling Multiple Risks: Hidden Domain of Attraction

Hidden regular variation is a sub-model of multivariate regular variation and facilitates accurate estimation of joint tail probabilities. We generalize the model of hidden regular variation to what we call hidden domain of attraction. We exhibit examples that illustrate the need for a more general model and discuss detection and estimation techniques.

math.PR↗

arXiv2011

Conditioning on an extreme component: Model consistency with regular variation on cones

Multivariate extreme value theory assumes a multivariate domain of attraction condition for the distribution of a random vector. This necessitates that each component satisfies a marginal domain of attraction condition. An approximation of the joint distribution of a random vector obtained by conditioning on one of the components being extreme was developed by Heffernan and Tawn [12] and further studied by Heffernan and Resnick [11]. These papers left unresolved the consistency of different models obtained by conditioning on different components being extreme and we here provide clarification of this issue. We also clarify the relationship between these conditional distributions, multivariate extreme value theory and standard regular variation on cones of the form $[0,\infty]\times(0,\infty]$.

math.PR↗