Basic univariate and bivariate statistics for symbolic data: a critical review
Some proofs of the problems of the basic statistics proposed for numeric symbolic data.
stat.ME↗
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Statistical inference for exponential functionals of Lévy processes
In this paper, we consider the exponential functional \(A_{\infty}=\int_0^\infty e^{-ξ_s}ds\) of a L{é}vy process \(ξ_s\) and aim to estimate the characteristics of \(ξ_{s}\) from the distribution of \(A_{\infty}\). We present a new approach, which allows to statistically infer on the L{é}vy triplet of \(ξ_{t}\), and study the theoretical properties of the proposed estimators. The suggested algorithms are illustrated with numerical simulations.
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Co-jumps and Markov counting systems in random environments
We provide transition rates for Markov counting systems subject to correlated environmental noises motivated by multi-strain disease models. Such noises induce simultaneous counts, which can help model infinitesimal count correlation (regardless of whether such correlation is due to correlated noises).
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Teaching precursors to data science in introductory and second courses in statistics
Statistics students need to develop the capacity to make sense of the staggering amount of information collected in our increasingly data-centered world. Data science is an important part of modern statistics, but our introductory and second statistics courses often neglect this fact. This paper discusses ways to provide a practical foundation for students to learn to "compute with data" as defined by Nolan and Temple Lang (2010), as well as develop "data habits of mind" (Finzer, 2013). We describe how introductory and second courses can integrate two key precursors to data science: the use of reproducible analysis tools and access to large databases. By introducing students to commonplace tools for data management, visualization, and reproducible analysis in data science and applying these to real-world scenarios, we prepare them to think statistically in the era of big data.
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On the Measurement of Economic Tail Risk
This paper attempts to provide a decision-theoretic foundation for the measurement of economic tail risk, which is not only closely related to utility theory but also relevant to statistical model uncertainty. The main result is that the only risk measures that satisfy a set of economic axioms for the Choquet expected utility and the statistical property of elicitability (i.e. there exists an objective function such that minimizing the expected objective function yields the risk measure) are the mean functional and the median shortfall, which is the median of tail loss distribution. Elicitability is important for backtesting. We also extend the result to address model uncertainty by incorporating multiple scenarios. As an application, we argue that median shortfall is a better alternative than expected shortfall for setting capital requirements in Basel Accords.
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A New Approach to Inference in Multi-Survey Studies with Unknown Population Size
We investigate a Poisson sampling design in the presence of unknown selection probabilities when applied to a population of unknown size for multiple sampling occasions. The fixed-population model is adopted and extended upon for inference. The complete minimal sufficient statistic is derived for the sampling model parameters and fixed-population parameter vector. The Rao-Blackwell version of population quantity estimators is detailed. An application is applied to an emprical population. The extended inferential framework is found to have much potential and utility for empirical studies.
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The phenotypic expansion and its boundaries
The development of sport performances in the future is a subject of myth and disagreement among experts. As arguments favoring and opposing such methodology were discussed, other publications empirically showed that the past development of performances followed a non linear trend. Other works, while deeply exploring the conditions leading to world records, highlighted that performance is tied to the economical and geopolitical context. Here we investigated the following human boundaries: development of performances with time in Olympic and non-Olympic events, development of sport performances with aging among humans and others species (greyhounds, thoroughbreds, mice). Development of performances from a broader point of view (demography & lifespan) in a specific sub-system centered on primary energy was also investigated. We show that the physiological developments are limited with time. Three major and direct determinants of sport performance are age, technology and climatic conditions (temperature). However, all observed developments are related to the international context including the efficient use of primary energies. This last parameter is a major indirect propeller of performance development. We show that when physiological and societal performance indicators such as lifespan and population density depend on primary energies, the energy source, competition and mobility are key parameters for achieving long term sustainable trajectories. Otherwise, the vast majority (98.7%) of the studied trajectories reaches 0 before 15 generations, due to the consumption of fossil energy and a low mobility rate. This led us to consider that in the present turbulent economical context and given the upcoming energy crisis, societal and physical performances are not expected to grow continuously.
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Inhomogeneous K-function for germ-grain models
In this paper, we propose a generalization to germ-grain models of the inhomogeneous K-function of Point Processes. We apply them to a sample of images of peripheral blood smears obtained from patients with Sickle Cell Disease, in order to decide whether the sample belongs to the thin, thick or morphological region.
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On Renyi entropy convergence of the max domain of attraction
In this paper, we prove that the Renyi entropy of linearly normalized partial maxima of independent and identically distributed random variables is convergent to the corresponding limit Renyi entropy when the linearly normalized partial maxima converges to some nondegenerate random variable.
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Null hypothesis significance tests: A mix-up of two different theories, the basis for widespread confusion and numerous misinterpretations
Null hypothesis statistical significance tests (NHST) are widely used in quantitative research in the empirical sciences including scientometrics. Nevertheless, since their introduction nearly a century ago significance tests have been controversial. Many researchers are not aware of the numerous criticisms raised against NHST. As practiced, NHST has been characterized as a null ritual that is overused and too often misapplied and misinterpreted. NHST is in fact a patchwork of two fundamentally different classical statistical testing models, often blended with some wishful quasi-Bayesian interpretations. This is undoubtedly a major reason why NHST is very often misunderstood. But NHST also has intrinsic logical problems and the epistemic range of the information provided by such tests is much more limited than most researchers recognize. In this article we introduce to the scientometric community the theoretical origins of NHST, which is mostly absent from standard statistical textbooks, and we discuss some of the most prevalent problems relating to the practice of NHST and trace these problems back to the mixup of the two different theoretical origins. Finally, we illustrate some of the misunderstandings with examples from the scientometric literature and bring forward some modest recommendations for a more sound practice in quantitative data analysis.
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Non-Orthogonal Tensor Diagonalization
Tensor diagonalization means transforming a given tensor to an exactly or nearly diagonal form through multiplying the tensor by non-orthogonal invertible matrices along selected dimensions of the tensor. It is generalization of approximate joint diagonalization (AJD) of a set of matrices. In particular, we derive (1) a new algorithm for symmetric AJD, which is called two-sided symmetric diagonalization of order-three tensor, (2) a similar algorithm for non-symmetric AJD, also called general two-sided diagonalization of an order-3 tensor, and (3) an algorithm for three-sided diagonalization of order-3 or order-4 tensors. The latter two algorithms may serve for canonical polyadic (CP) tensor decomposition, and they can outperform other CP tensor decomposition methods in terms of computational speed under the restriction that the tensor rank does not exceed the tensor multilinear rank. Finally, we propose (4) similar algorithms for tensor block diagonalization, which is related to the tensor block-term decomposition.
math.NA↗
Donald Arthur Preece: A life in statistics, mathematics and music
Biography and publications list for Donald Arthur Preece, who died on 6 January 2014, who made many contributions in statistics (experimental design) and in combinatorics.
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A method for comparing chess openings
A quantitative method is described for comparing chess openings. Test openings and baseline openings are run through chess engines under controlled conditions and compared to evaluate the effectiveness of the test openings. The results are intuitively appealing and in some cases they agree with expert opinion. The specific contribution of this work is the development of an objective measure that may be used for the evaluation and refutation of chess openings, a process that had been left to thought experiments and subjective conjectures and thereby to a large variety of opinion and a great deal of debate.
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Spectral Correlation Hub Screening of Multivariate Time Series
This chapter discusses correlation analysis of stationary multivariate Gaussian time series in the spectral or Fourier domain. The goal is to identify the hub time series, i.e., those that are highly correlated with a specified number of other time series. We show that Fourier components of the time series at different frequencies are asymptotically statistically independent. This property permits independent correlation analysis at each frequency, alleviating the computational and statistical challenges of high-dimensional time series. To detect correlation hubs at each frequency, an existing correlation screening method is extended to the complex numbers to accommodate complex-valued Fourier components. We characterize the number of hub discoveries at specified correlation and degree thresholds in the regime of increasing dimension and fixed sample size. The theory specifies appropriate thresholds to apply to sample correlation matrices to detect hubs and also allows statistical significance to be attributed to hub discoveries. Numerical results illustrate the accuracy of the theory and the usefulness of the proposed spectral framework.
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Principal bundles over statistical manifolds
In this paper, we introduce the concept of principal bundles on statistical manifolds. After necessary preliminaries on information geometry and principal bundles on manifolds, we study the $α$-structure of frame bundles over statistical manifolds with respect to $α$-connections, by giving geometric structures. The manifold of one-dimensional normal distributions appears in the end as an application and a concrete example.
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Claude Bouchu, intendant de Bourgogne au 17ème siècle, a-t-il inventé le mot "statistique"
The objective of this paper is to examine the assertion that the word "statistics" would have been used for the first time in the 17th century, in a report written by Claude Bouchu, administrator of Bourgogne. A historical and bibliographical analysis is carried out to judge the credibility of this thesis. The physical inspection of the report then makes it possible to bring a final answer.
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Extremes of Order Statistics of Stationary Processes
Let $\{X_i(t),t\ge0\}, 1\le i\le n$ be independent copies of a stationary process $\{X(t), t\ge0\}$. For given positive constants $u,T$, define the set of $r$th conjunctions $ C_{r,T,u}:= \{t\in [0,T]: X_{r:n}(t) > u\}$ with $X_{r:n}(t)$ the $r$th largest order statistics of $X_1(t), \ldots , X_n(t), t\ge 0$. In numerous applications such as brain mapping and digital communication systems, of interest is the approximation of the probability that the set of conjunctions $C_{r,T,u}$ is not empty. Imposing the Albin's conditions on $X$, in this paper we obtain an exact asymptotic expansion of this probability as $u$ tends to infinity. Further, we establish the tail asymptotics of the supremum of a generalized skew-Gaussian process and a Gumbel limit theorem for the minimum order statistics of stationary Gaussian processes. As a by-product we derive a version of Li and Shao's normal comparison lemma for the minimum and the maximum of Gaussian random vectors.
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Validating Sample Average Approximation Solutions with Negatively Dependent Batches
Sample-average approximations (SAA) are a practical means of finding approximate solutions of stochastic programming problems involving an extremely large (or infinite) number of scenarios. SAA can also be used to find estimates of a lower bound on the optimal objective value of the true problem which, when coupled with an upper bound, provides confidence intervals for the true optimal objective value and valuable information about the quality of the approximate solutions. Specifically, the lower bound can be estimated by solving multiple SAA problems (each obtained using a particular sampling method) and averaging the obtained objective values. State-of-the-art methods for lower-bound estimation generate batches of scenarios for the SAA problems independently. In this paper, we describe sampling methods that produce negatively dependent batches, thus reducing the variance of the sample-averaged lower bound estimator and increasing its usefulness in defining a confidence interval for the optimal objective value. We provide conditions under which the new sampling methods can reduce the variance of the lower bound estimator, and present computational results to verify that our scheme can reduce the variance significantly, by comparison with the traditional Latin hypercube approach.
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