Tenth Jubilee Conference of the Euro-American Consortium for Promoting the Application of Mathematics in Technical and Natural Sciences, Albena, Bulgaria, June 20-25, 2018
The task of estimation of the tails of probability distributions having small samples seems to be still opened and almost unsolvable. The paper tries to make a step in filling this gap. In 2017 Jordanova et al. introduce six new characteristics of the heaviness of the tails of theoretical distributions. They rely on the probability to observe {\color{blue}mild or} extreme outliers. The main their advantage is that they always exist. This work presents some new properties of these characteristics. Using them six distribution sensitive estimators of the extremal index are defined. A brief simulation study compares their quality with the quality of Hill, t-Hill, Pickands and Deckers-Einmahl-de Haan estimators.
We are trying to give a mathematically correct definition of outliers. Our approach is based on the distance between two last order statistics and appears to be connected to the law of large numbers. Key words: outliers, law of large numbers, heavy tails, stability index.
A possibility to give strong mathematical definitions of outliers and heavy tailed distributions or their modification is discussed. Some alternatives for the notion of tail index are proposed. Key words: outliers, heavy tails, tail index.
The aim of the paper is to show that the presence of one possible type of outliers is not connected to that of heavy tails of the distribution. In contrary, typical situation for outliers appearance is the case of compact supported distributions. Key words: outliers; heavy tailed distributions; compact support
We define outliers as a set of observations which contradicts the proposed mathematical (statistical) model and we discuss the frequently observed types of the outliers. Further we explore what changes in the model have to be made in order to avoid the occurance of the outliers. We observe that some variants of the outliers lead to classical results in probability, such as the law of large numbers and the concept of heavy tailed distributions. Key words: outlier; the law of large numbers; heavy tailed distributions; model rejection.
Whether an extreme observation is an outlier or not, depends strongly on the corresponding tail behaviour of the underlying distribution. We develop an automatic, data-driven method to identify extreme tail behaviour that deviates from the intermediate and central characteristics. This allows for detecting extreme outliers or sets of extreme data that show less spread than the bulk of the data. To this end we extend a testing method proposed in Bhattacharya et al 2019 for the specific case of heavy tailed models, to all max-domains of attraction. Consequently we propose a tail-adjusted boxplot…
We study tail probabilities via some Gaussian approximations. Our results make refinements to large deviation theory. The proof builds on classical results by Bahadur and Rao. Binomial distributions and their tail probabilities are discussed in more detail.
Estimating the location and scale parameters is common in statistics, using, for instance, the well-known sample mean and standard deviation. However, inference can be contaminated by the presence of outliers if modeling is done with light-tailed distributions such as the normal distribution. In this paper, we study robustness to outliers in location-scale parameter models using both the Bayesian and frequentist approaches. We find sufficient conditions (e.g., on tail behavior of the model) to obtain whole robustness to outliers, in the sense that the impact of the outliers gradually decreases…
We review various inequalities for Mills' ratio (1 - Φ)/ϕ, where ϕand Φdenote the standard Gaussian density and distribution function, respectively. Elementary considerations involving finite continued fractions lead to a general approximation scheme which implies and refines several known bounds.
There are a number of mathematical formalisms of the term "outlier" in statistics, though there is no consensus on what the right notion ought to be. Accordingly, we try to give a consistent and robust definition for a specific type of outliers defined via order statistics. Our approach is based on ratios of partial sums of order statistics to investigate the tail behaviors of hypothetical and empirical distributions. We simulate our statistic on a set of distributions to mark potential outliers and use an algorithm to automatically select a cut-off point without the need of any further a prio…
University of Wisconsin-Madison（威斯康星大学麦迪逊分校）；University of California, San Diego（加州大学圣地亚哥分校）；Simons Institute for the Theory of Computing（西蒙斯计算理论研究所）
We study the fundamental task of outlier-robust mean estimation for heavy-tailed distributions in the presence of sparsity. Specifically, given a small number of corrupted samples from a high-dimensional heavy-tailed distribution whose mean $μ$ is guaranteed to be sparse, the goal is to efficiently compute a hypothesis that accurately approximates $μ$ with high probability. Prior work had obtained efficient algorithms for robust sparse mean estimation of light-tailed distributions. In this work, we give the first sample-efficient and polynomial-time robust sparse mean estimator for heavy-taile…
We revisit the outlier hypothesis testing framework of Li \emph{et al.} (TIT 2014) and derive fundamental limits for the optimal test. In outlier hypothesis testing, one is given multiple observed sequences, where most sequences are generated i.i.d. from a nominal distribution. The task is to discern the set of outlying sequences that are generated according to anomalous distributions. The nominal and anomalous distributions are \emph{unknown}. We consider the case of multiple outliers where the number of outliers is unknown and each outlier can follow a different anomalous distribution. Under…
We revisit the outlier hypothesis testing framework of Li \emph{et al.} (TIT 2014) and derive fundamental limits for the optimal test under the generalized Neyman-Pearson criterion. In outlier hypothesis testing, one is given multiple observed sequences, where most sequences are generated i.i.d. from a nominal distribution. The task is to discern the set of outlying sequences that are generated from anomalous distributions. The nominal and anomalous distributions are \emph{unknown}. We study the tradeoff among the probabilities of misclassification error, false alarm and false reject for tests…
This article discusses modelling of the tail of a multivariate distribution function by means of a large deviation principle (LDP), and its application to the estimation of the probability of a multivariate extreme event from a sample of n iid random vectors, with the probability bounded by powers of sample size with exponents below -1. One way to view classical tail limits is as limits of probability ratios. In contrast, the tail LDP provides asymptotic bounds or limits for log-probability ratios. After standardising the marginals to standard exponential, dependence is represented by a homoge…
In this paper we are concerned with a sample of asymptotically independent risks. Tail asymptotic probabilities for linear combinations of randomly weighted order statistics are approximated under various assumptions, where the individual tail behaviour has a crucial role. An application is provided for Log-Normal risks.
We find asymptotic formulas for error probabilities of two-fold Pearson goodness-of-fit test as functions of two critical levels. These results may be reformulated in terms of tails of two-dimensional distributions of the Bessel process. Necessary properties of the Infeld function are obtained.
Existing theory for multivariate extreme values focuses upon characterizations of the distributional tails when all components of a random vector, standardized to identical margins, grow at the same rate. In this paper, we consider the effect of allowing the components to grow at different rates, and characterize the link between these marginal growth rates and the multivariate tail probability decay rate. Our approach leads to a whole class of univariate regular variation conditions, in place of the single but multivariate regular variation conditions that underpin the current theories. These…
Cellwise outliers are likely to occur together with casewise outliers in modern data sets with relatively large dimension. Recent work has shown that traditional robust regression methods may fail for data sets in this paradigm. The proposed method, called three-step regression, proceeds as follows: first, it uses a consistent univariate filter to detect and eliminate extreme cellwise outliers; second, it applies a robust estimator of multivariate location and scatter to the filtered data to down-weight casewise outliers; third, it computes robust regression coefficients from the estimates obt…
Linear regression with normally distributed errors - including particular cases such as ANOVA, Student's t-test or location-scale inference - is a widely used statistical procedure. In this case the ordinary least squares estimator possesses remarkable properties but is very sensitive to outliers. Several robust alternatives have been proposed, but there is still significant room for improvement. This paper thus proposes an original method of estimation that offers the best efficiency simultaneously in the absence and the presence of outliers, both for the estimation of the regression coeffici…
We study Empirical Risk Minimizers (ERM) and Regularized Empirical Risk Minimizers (RERM) for regression problems with convex and $L$-Lipschitz loss functions. We consider a setting where $|\cO|$ malicious outliers contaminate the labels. In that case, under a local Bernstein condition, we show that the $L_2$-error rate is bounded by $ r_N + AL |\cO|/N$, where $N$ is the total number of observations, $r_N$ is the $L_2$-error rate in the non-contaminated setting and $A$ is a parameter coming from the local Bernstein condition. When $r_N$ is minimax-rate-optimal in a non-contaminated setting, th…
We propose a family of models that enable predictive estimation of time-varying extreme event probabilities in heavy-tailed and nonlinearly dependent time series. The models are a white noise process with conditionally log-Laplace stochastic volatility. In contrast to other, similar stochastic volatility formalisms, this process has analytic expressions for its conditional probabilistic structure that enable straightforward estimation of dynamically changing extreme event probabilities. The process and volatility are conditionally Pareto-tailed, with tail exponent given by the reciprocal of th…
