arXiv subjects

Nadezhda Gribkova

Publications and source records attributed to Nadezhda Gribkova.

13 recordsLinked to original sources

arXiv2026

Asymptotic Expansion and Bounds for the Bias of Empirical Tail Value-at-Risk

Tail Value-at-Risk (TVaR) is a widely adopted risk measure playing a critically important role in both academic research and industry practice in insurance. In data applications, TVaR is often estimated using the empirical method, owing to its simplicity and nonparametric nature. The empirical TVaR has been explicitly advocated by regulatory authorities as a standard approach for computing TVaR. However, prior literature has pointed out that the empirical TVaR estimator is negatively biased, which can lead to a systemic underestimation of risk in finite-sample applications. This paper aims to deepen the understanding of the bias of the empirical TVaR estimator in two dimensions: its magnitude as well as the key distributional and structural determinants driving the severity of the bias. To this end, we derive a leading-term approximation for the bias based on its asymptotic expansion. The closed-form expression associated with the leading-term approximation enables us to obtain analytical insights into the structural properties governing the bias of the empirical TVaR estimator. To account for the discrepancy between the leading-term approximation and the true bias, we further derive an explicit upper bound for the bias. We validate the proposed bias analysis framework via simulations and demonstrate its practical relevance using real data.

math.ST↗

arXiv2025

Fundamentals of non-parametric statistical inference for integrated quantiles

We present a general non-parametric statistical inference theory for integrals of quantiles without assuming any specific sampling design or dependence structure. Technical considerations are accompanied by examples and discussions, including those pertaining to the bias of empirical estimators. To illustrate how the general results can be adapted to specific situations, we derive - at a stroke and under minimal conditions - consistency and asymptotic normality of the empirical tail-value-at-risk, Lorenz and Gini curves at any probability level in the case of the simple random sampling, thus facilitating a comparison of our results with what is already known in the literature. Results, notes and references concerning dependent (i.e., time series) data are also offered. As a by-product, our general results provide new and unified proofs of large-sample properties of a number of classical statistical estimators, such as trimmed means, and give additional insights into the origins of, and the reasons for, various necessary and sufficient conditions.

math.ST↗

arXiv2020

Functional Correlations in the Pursuit of Performance Assessment of Classifiers

In statistical classification and machine learning, as well as in social and other sciences, a number of measures of association have been proposed for assessing and comparing individual classifiers, raters, as well as their groups. In this paper, we introduce, justify, and explore several new measures of association, which we call CO-, ANTI- and COANTI-correlation coefficients, that we demonstrate to be powerful tools for classifying confusion matrices. We illustrate the performance of these new coefficients using a number of examples, from which we also conclude that the coefficients are new objects in the sense that they differ from those already in the literature.

stat.ML↗

arXiv2019

Cramér type large deviations for trimmed L-statistics

In this paper, we propose a new approach to the investigation of asymptotic properties of trimmed $L$-statistics and we apply it to the Cramér type large deviation problem. Our results can be compared with ones in Callaert et al.(1982) -- the first and, as far as we know, the single article, where some results on probabilities of large deviations for the trimmed $L$-statistics were obtained, but under some strict and unnatural conditions. Our approach is to approximate the trimmed $L$-statistic by a non-trimmed $L$-statistic (with smooth weight function) based on Winsorized random variables. Using this method, we establish the Cramér type large deviation results for the trimmed $L$-statistics under quite mild and natural conditions.

math.PR↗

arXiv2018

Assessing transfer functions in control systems

When dealing with control systems, it is useful and even necessary to assess the performance of underlying transfer functions. The functions may or may not be linear, may or may not be even monotonic. In addition, they may have structural breaks and other abberations that require monitoring and quantification to aid decision making. The present paper develops such a methodology, which is based on an index of increase that naturally arises as the solution to an optimization problem. We show theoretically and illustrate numerically that the empirical counterpart of the index needs to be used with great care and in-depth knowledge of the problem at hand in order to achieve desired large-sample properties, such as consistency.

math.ST↗

arXiv2018

Detecting intrusions in control systems: a rule of thumb, its justification and illustrations

Control systems are exposed to unintentional errors, deliberate intrusions, false data injection attacks, and various other disruptions. In this paper we propose, justify, and illustrate a rule of thumb for detecting, or confirming the absence of, such disruptions. To facilitate the use of the rule, we rigorously discuss background results that delineate the boundaries of the rule's applicability. We also discuss ways to further widen the applicability of the proposed intrusion-detection methodology.

stat.ME↗

arXiv2018

Estimating the index of increase via balancing deterministic and random data

We introduce and explore an empirical index of increase that works in both deterministic and random environments, thus allowing to assess monotonicity of functions that are prone to random measurement-errors. We prove consistency of the index and show how its rate of convergence is influenced by deterministic and random parts of the data. In particular, the obtained results suggest a frequency at which observations should be taken in order to reach any pre-specified level of estimation precision. We illustrate the index using data arising from purely deterministic and error-contaminated functions, which may or may not be monotonic.

math.ST↗

arXiv2017

Weighted allocations, their concomitant-based estimators, and asymptotics

Various members of the class of weighted insurance premiums and risk capital allocation rules have been researched from a number of perspectives. Corresponding formulas in the case of parametric families of distributions have been derived, and they have played a pivotal role when establishing parametric statistical inference in the area. Non-parametric inference results have also been derived in special cases such as the tail conditional expectation, distortion risk measure, and several members of the class of weighted premiums. For weighted allocation rules, however, non-parametric inference results have not yet been adequately developed. In the present paper, therefore, we put forward empirical estimators for the weighted allocation rules and establish their consistency and asymptotic normality under practically sound conditions. Intricate statistical considerations rely on the theory of induced order statistics, known as concomitants.

math.ST↗

arXiv2017

Statistical foundations for assessing the difference between the classical and weighted-Gini betas

The `beta' is one of the key quantities in the capital asset pricing model (CAPM). In statistical language, the beta can be viewed as the slope of the regression line fitted to financial returns on the market against the returns on the asset under consideration. The insurance counterpart of CAPM, called the weighted insurance pricing model (WIPM), gives rise to the so-called weighted-Gini beta. The aforementioned two betas may or may not coincide, depending on the form of the underlying regression function, and this has profound implications when designing portfolios and allocating risk capital. To facilitate these tasks, in this paper we develop large-sample statistical inference results that, in a straightforward fashion, imply confidence intervals for, and hypothesis tests about, the equality of the two betas.

math.ST↗

arXiv2016

Cramér type moderate deviations for trimmed L-statistics

We establish Cramér type moderate deviation (MD}) results for heavy trimmed L-statistics; we obtain our results under a very mild smoothness condition on the inversion $F^{-1}$ ($F$ is the underlying distribution of i.i.d. observations) near two points, where trimming occurs, we assume also some smoothness of weights of the L-statistic. Our results complement previous work on Cramér type large deviations (LD) for trimmed L-statistics by Gribkova (2016) and Callaert et al. (1982).

math.PR↗

arXiv2016

Cramér type moderate deviations for intermediate trimmed means

In this article we establish Cramér type moderate deviation results for (intermediate) trimmed means $T_n=n^{-1} \sum_{i=k_n+1}^{n-m_n}X_{i:n}$, where $X_{i:n}$ -- the order statistics corresponding to the first $n$ observations of a~sequence $X_1,X_2,\dots $ of i.i.d random variables with $df$ $F$. We consider two cases of intermediate and heavy trimming. In the former case, when $\max(α_n,β_n)\to 0$ ($α_n=k_n/n$, $β_n=m_n/n$) and $\min(k_n,m_n)\to\infty$ as $n\to\infty$, we obtain our results under a~natural moment condition and a~mild condition on the rate at which $α_n$ and $β_n$ tend to zero. In the latter case we do not impose any moment conditions on $F$, instead, we require some smoothness of $F^{-1}$ in an~open set containing the limit points of the trimming sequences $α_n$, $1-β_n$.

math.PR↗

arXiv2011

The Empirical Edgeworth Expansion for a Studentized Trimmed Mean

We establish the validity of the empirical Edgeworth expansion (EE) for a studentized trimmed mean, under the sole condition that the underlying distribution function of the observations satisfies a local smoothness condition near the two quantiles where the trimming occurs. A simple explicit formula for the N^{-1/2} term (correcting for skewness and bias; N being the sample size) of the EE is given. In particular our result supplements previous work by P. Hall and A.R. Padmanabhan, On the bootstrap and the trimmed mean}, J. of Multivariate Analysis, v. 41 (1992), pp. 132-153. and H. Putter and W.R. van Zwet, Empirical Edgeworth expansions for symmetric statistics, Ann. Statist., v. 26 (1998), pp. 1540-1569. The proof is based on a U-statistic type approximation and also uses a version of Bahadur's representation for sample quantiles.

math.ST↗

arXiv2011

On the Bahadur - Kiefer Representation for Intermediate Sample Quantiles

We investigate a Bahadur-Kiefer type representation for the p-th empirical quantile corresponding to a sample of n i.i.d. random variables, when 0<p<1 is a sequence which, in particular, may tend to 0 or 1, i.e. we consider the case of intermediate sample quantiles. We obtain an 'in probability' version of the Bahadur -- Kiefer type representation for a $\kn$-th order statistic when $r_n=\kn\wedge (n-k_n)\to \infty$ under some mild regularity conditions, and an 'almost sure' version under additional assumption that $\log n/r_n\to 0$, $\nty$. A representation for the sum of order statistics laying between the population p-quantile and the corresponding empirical quantile is also established.

math.PR↗