arXiv subjects

Bikramjit Das

Publications and source records attributed to Bikramjit Das.

At least 19 recordsLinked to original sources

arXiv2026

Measuring Extreme Tail Association

Simultaneous occurrences of extreme events need not imply symmetric or reciprocal tail dependence. However, most existing measures of extremal dependence are inherently symmetric and hence often fail to capture directional influence in tail association. We introduce a rank-based measure of Extreme Tail Association (ETA) for bivariate data quantifying such directional influence of one variable on another in extreme tail regions. The proposed estimator is easily computable, consistent with its population counterpart, and asymptotically normal under mild conditions, allowing for statistical inference. We further develop a formal test for asymmetry in tail association based on a multiplier bootstrap procedure. The practical relevance of the methodology is illustrated using data on extreme price movements in major cryptocurrencies. Beyond providing a flexible tool for extremal association, the proposed framework offers a substantive argument for investigating causal relationships in extreme scenarios.

stat.ME↗

arXiv2025

Smart Timing for Mining: A Deep Learning Framework for Bitcoin Hardware ROI Prediction

Bitcoin mining hardware acquisition requires strategic timing due to volatile markets, rapid technological obsolescence, and protocol-driven revenue cycles. Despite mining's evolution into a capital-intensive industry, there is little guidance on when to purchase new Application-Specific Integrated Circuit (ASIC) hardware, and no prior computational frameworks address this decision problem. We address this gap by formulating hardware acquisition as a time series classification task, predicting whether purchasing ASIC machines yields profitable (Return on Investment (ROI) >= 1), marginal (0 < ROI < 1), or unprofitable (ROI <= 0) returns within one year. We propose MineROI-Net, an open-source Transformer-based architecture designed to capture multi-scale temporal patterns in mining profitability. Evaluated on data from 20 ASIC miners released between 2015 and 2024 across diverse market regimes, MineROI-Net outperforms recurrent, convolutional, and attention-based baselines, achieving 83.2% accuracy and 83.5% macro F1-score. The model demonstrates strong economic relevance, achieving 97.8% precision in detecting unprofitable periods and 81.5% precision in detecting profitable ones, while avoiding misclassifying profitable scenarios as unprofitable and vice versa. These results indicate that MineROI-Net offers a practical, data-driven tool for timing mining hardware acquisitions, potentially reducing financial risk in capital-intensive mining operations.

cs.LG↗

arXiv2025

Rethinking Federated Learning Over the Air: The Blessing of Scaling Up

Federated learning facilitates collaborative model training across multiple clients while preserving data privacy. However, its performance is often constrained by limited communication resources, particularly in systems supporting a large number of clients. To address this challenge, integrating over-the-air computations into the training process has emerged as a promising solution to alleviate communication bottlenecks. The system significantly increases the number of clients it can support in each communication round by transmitting intermediate parameters via analog signals rather than digital ones. This improvement, however, comes at the cost of channel-induced distortions, such as fading and noise, which affect the aggregated global parameters. To elucidate these effects, this paper develops a theoretical framework to analyze the performance of over-the-air federated learning in large-scale client scenarios. Our analysis reveals three key advantages of scaling up the number of participating clients: (1) Enhanced Privacy: The mutual information between a client's local gradient and the server's aggregated gradient diminishes, effectively reducing privacy leakage. (2) Mitigation of Channel Fading: The channel hardening effect eliminates the impact of small-scale fading in the noisy global gradient. (3) Improved Convergence: Reduced thermal noise and gradient estimation errors benefit the convergence rate. These findings solidify over-the-air model training as a viable approach for federated learning in networks with a large number of clients. The theoretical insights are further substantiated through extensive experimental evaluations.

cs.LG↗

arXiv2024

Robust Federated Learning Over the Air: Combating Heavy-Tailed Noise with Median Anchored Clipping

Leveraging over-the-air computations for model aggregation is an effective approach to cope with the communication bottleneck in federated edge learning. By exploiting the superposition properties of multi-access channels, this approach facilitates an integrated design of communication and computation, thereby enhancing system privacy while reducing implementation costs. However, the inherent electromagnetic interference in radio channels often exhibits heavy-tailed distributions, giving rise to exceptionally strong noise in globally aggregated gradients that can significantly deteriorate the training performance. To address this issue, we propose a novel gradient clipping method, termed Median Anchored Clipping (MAC), to combat the detrimental effects of heavy-tailed noise. We also derive analytical expressions for the convergence rate of model training with analog over-the-air federated learning under MAC, which quantitatively demonstrates the effect of MAC on training performance. Extensive experimental results show that the proposed MAC algorithm effectively mitigates the impact of heavy-tailed noise, hence substantially enhancing system robustness.

cs.LG↗

arXiv2024

Asymptotic independence in higher dimensions and its implications on risk management

In the study of extremes, the presence of asymptotic independence signifies that extreme events across multiple variables are probably less likely to occur together. Although well-understood in a bivariate context, the concept remains relatively unexplored when addressing the nuances of the joint occurrence of extremes in higher dimensions. In this paper, we propose a notion of mutual asymptotic independence to capture the behavior of joint extremes in dimensions larger than two and contrast it with the classical notion of (pairwise) asymptotic independence. Additionally, we define k-wise asymptotic independence, which captures the tail dependence between pairwise and mutual asymptotic independence. The concepts are compared using examples of Archimedean, Gaussian and Marshall-Olkin copulas, among others. Finally, we discuss the implications of these new notions of asymptotic independence on assessing the risk of complex systems under distributional ambiguity.

math.ST↗

arXiv2023

Measuring risk contagion in financial networks with CoVaR

The stability of a complex financial system may be assessed by measuring risk contagion between various financial institutions with relatively high exposure. We consider a financial network model using a bipartite graph of financial institutions (e.g., banks, investment companies, insurance firms) on one side and financial assets on the other. Following empirical evidence, returns from such risky assets are modeled by heavy-tailed distributions, whereas their joint dependence is characterized by copula models exhibiting a variety of tail dependence behavior. We consider CoVaR, a popular measure of risk contagion and study its asymptotic behavior under broad model assumptions. We further propose the Extreme CoVaR Index (ECI) for capturing the strength of risk contagion between risk entities in such networks, which is particularly useful for models exhibiting asymptotic independence. The results are illustrated by providing precise expressions of CoVaR and ECI when the dependence of the assets is modeled using two well-known multivariate dependence structures: the Gaussian copula and the Marshall-Olkin copula.

q-fin.RM↗

arXiv2023

Inference for heavy-tailed data with Gaussian dependence

We consider a model for multivariate data with heavy-tailed marginal distributions and a Gaussian dependence structure. The different marginals in the model are allowed to have non-identical tail behavior in contrast to most popular modeling paradigms for multivariate heavy-tail analysis. Despite being a practical choice, results on parameter estimation and inference under such models remain limited. In this article, consistent estimates for both marginal tail indices and the Gaussian correlation parameters for such models are provided and asymptotic normality of these estimators are established. The efficacy of the estimation methods are exhibited using extensive simulations and then they are applied to real data sets from insurance claims, internet traffic, and, online networks.

stat.ME↗

arXiv2023

On heavy-tailed risks under Gaussian copula: the effects of marginal transformation

In this paper, we compute multivariate tail risk probabilities where the marginal risks are heavy-tailed and the dependence structure is a Gaussian copula. The marginal heavy-tailed risks are modeled using regular variation which leads to a few interesting consequences. First, as the threshold increases, we note that the rate of decay of probabilities of tail sets vary depending on the type of tail sets considered and the Gaussian correlation matrix. Second, we discover that although any multivariate model with a Gaussian copula admits the so called asymptotic tail independence property, the joint tail behavior under heavier tailed marginal variables is structurally distinct from that under Gaussian marginal variables. The results obtained are illustrated using examples and simulations.

q-fin.RM↗

arXiv2023

Aggregating heavy-tailed random vectors: from finite sums to L\'evy processes

The tail behavior of aggregates of heavy-tailed random vectors is known to be determined by the so-called principle of "one large jump'', be it for finite sums, random sums, or, L\'evy processes. We establish that, in fact, a more general principle is at play. Assuming that the random vectors are multivariate regularly varying on various subcones of the positive quadrant, first we show that their aggregates are also multivariate regularly varying on these subcones. This allows us to approximate certain tail probabilities which were rendered asymptotically negligible under classical regular variation, despite the "one large jump'' asymptotics. We also discover that depending on the structure of the tail event of concern, the tail behavior of the aggregates may be characterized by more than a single large jump. Eventually, we illustrate a similar phenomenon for multivariate regularly varying L\'evy processes, establishing as well a relationship between multivariate regular variation of a L\'evy process and multivariate regular variation of its L\'evy measure on different subcones.

math.PR↗

arXiv2021

Asymptotic Behavior of Common Connections in Sparse Random Networks

Random network models generated using sparse exchangeable graphs have provided a mechanism to study a wide variety of complex real-life networks. In particular, these models help with investigating power-law properties of degree distributions, number of edges, and other relevant network metrics which support the scale-free structure of networks. Previous work on such graphs imposes a marginal assumption of univariate regular variation (e.g., power-law tail) on the bivariate generating graphex function. In this paper, we study sparse exchangeable graphs generated by graphex functions which are multivariate regularly varying. We also focus on a different metric for our study: the distribution of the number of common vertices (connections) shared by a pair of vertices. The number being high for a fixed pair is an indicator of the original pair of vertices being connected. We find that the distribution of number of common connections are regularly varying as well, where the tail indices of regular variation are governed by the type of graphex function used. Our results are verified on simulated graphs by estimating the relevant tail index parameters.

math.ST↗

arXiv2020

Tail probabilities of random linear functions of regularly varying random vectors

We provide a new extension of Breiman's Theorem on computing tail probabilities of a product of random variables to a multivariate setting. In particular, we give a complete characterization of regular variation on cones in $[0,\infty)^d$ under random linear transformations. This allows us to compute probabilities of a variety of tail events, which classical multivariate regularly varying models would report to be asymptotically negligible. We illustrate our findings with applications to risk assessment in financial systems and reinsurance markets under a bipartite network structure.

math.PR↗

arXiv2020

Risk concentration under second order regular variation

Measures of risk concentration and their asymptotic behavior for portfolios with heavy-tailed risk factors is of interest in risk management. Second order regular variation is a structural assumption often imposed on such risk factors to study their convergence rates. In this paper, we provide the asymptotic rate of convergence of the measure of risk concentration for a portfolio of heavy-tailed risk factors, when the portfolio admits the so-called second order regular variation property. Moreover, we explore the relationship between multivariate second order regular variation for a vector (e.g., risk factors) and the second order regular variation property for the sum of its components (e.g., the portfolio of risk factors). Results are illustrated with a variety of examples.

math.PR↗

arXiv2019

On the heavy-tail behavior of the distributionally robust newsvendor

Since the seminal work of Scarf (1958) [A min-max solution of an inventory problem, Studies in the Mathematical Theory of Inventory and Production, pages 201-209] on the newsvendor problem with ambiguity in the demand distribution, there has been a growing interest in the study of the distributionally robust newsvendor problem. The model is criticized at times for being overly conservative since the worst-case distribution is discrete with a few support points. However, it is the order quantity prescribed from the model that is of practical relevance. A simple calculation shows that the optimal order quantity in Scarf's model with known first and second moment is also optimal for a censored student-t distribution with parameter 2. In this paper, we generalize this "heavy-tail optimality" property of the distributionally robust newsvendor to an ambiguity set where information on the first and the $α$th moment is known, for any real number $α> 1$. We show that the optimal order quantity for the distributionally robust newsvendor problem is also optimal for a regularly varying distribution with roughly a power law tail with tail index $α$. We illustrate the usefulness of the model in the high service level regime with numerical experiments, by showing that when a standard distribution such as the exponential or lognormal distribution is contaminated with a heavy-tailed (regularly varying) distribution, the distributionally robust optimal order quantity outperforms the optimal order quantity of the original distribution, even with a small amount of contamination.

math.ST↗

arXiv2019

Growth of Common Friends in a Preferential Attachment Model

The number of common friends (or connections) in a graph is a commonly used measure of proximity between two nodes. Such measures are used in link prediction algorithms and recommendation systems in large online social networks. We obtain the rate of growth of the number of common friends in a linear preferential attachment model. We apply our result to develop an estimate for the number of common friends. We also observe a phase transition in the limiting behavior of the number of common friends; depending on the range of the parameters of the model, the growth is either power-law, or, logarithmic, or static with the size of the graph.

math.PR↗

arXiv2018

Hidden regular variation, copula models, and the limit behavior of conditional excess risk measures

Risk measures like Marginal Expected Shortfall and Marginal Mean Excess quantify conditional risk and in particular, aid in the understanding of systemic risk. In many such scenarios, models exhibiting heavy tails in the margins and asymptotic tail independence in the joint behavior play a fundamental role. The notion of hidden regular variation has the advantage that it models both properties: asymptotic tail independence as well as heavy tails. An alternative approach to addressing these features is via copulas. First, we elicit connections between hidden regular variation and the behavior of tail copula parameters extending previous works in this area. Then we study the asymptotic behavior of the aforementioned conditional excess risk measures; first under hidden regular variation and then under restrictions on the tail copula parameters, not necessarily assuming hidden regular variation. We provide a broad variety of examples of models admitting heavy tails and asymptotic tail independence along with hidden regular variation and with the appropriate limit behavior for the risk measures of interest.

math.PR↗

arXiv2017

Risk contagion under regular variation and asymptotic tail independence

Risk contagion concerns any entity dealing with large scale risks. Suppose (X,Y) denotes a risk vector pertaining to two components in some system. A relevant measurement of risk contagion would be to quantify the amount of influence of high values of Y on X. This can be measured in a variety of ways. In this paper, we study two such measures: the quantity E[max(X-t,0)|Y > t] called Marginal Mean Excess (MME) as well as the related quantity E[X|Y > t] called Marginal Expected Shortfall (MES). Both quantities are indicators of risk contagion and useful in various applications ranging from finance, insurance and systemic risk to environmental and climate risk. We work under the assumptions of multivariate regular variation, hidden regular variation and asymptotic tail independence for the risk vector (X,Y). Many broad and useful model classes satisfy these assumptions. We present several examples and derive the asymptotic behavior of both MME and MES as the threshold t tends to infinity. We observe that although we assume asymptotic tail independence in the models, MME and MES converge to 1 under very general conditions; this reflects that the underlying weak dependence in the model still remains significant. Besides the consistency of the empirical estimators, we introduce an extrapolation method based on extreme value theory to estimate both MME and MES for high thresholds t where little data are available. We show that these estimators are consistent and illustrate our methodology in both simulated and real data sets.

math.ST↗

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↗

arXiv2017

Heavy-tailed random walks, buffered queues and hidden large deviations

It is well-known that large deviations of random walks driven by independent and identically distributed heavy-tailed random variables are governed by the so-called principle of one large jump. We note that further subtleties hold for such random walks in the large deviation scale which we call hidden large deviation. We apply this idea in the context of queueing processes with heavy-tailed service times and study approximations of severe congestion times for (buffered) queues. We conclude with simulated examples to verify our results.

math.PR↗