Statistics

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.

Do there exist circular and spherical copulas in R d ? That is, do there exist circularly symmetric distributions on the unit disk in R 2 and spherically symmetric distributions on the unit ball in R d , d≥3 , whose one-dimensional marginal distributions are uniform? The answer is yes for d=2 and 3, where the circular and spherical copulas are unique and can be determined explicitly, but no for d≥4 . A one-parameter family of elliptical bivariate copulas is obtained from the unique circular copula in R 2 by oblique coordinate transformations. Copulas obtained by a non-linear transformation of a uniform distribution on the unit ball in R d are also described, and determined explicitly for d=2 .

In this paper, we study classes of graphs with three types of edges that capture the modified independence structure of a directed acyclic graph (DAG) after marginalisation over unobserved variables and conditioning on selection variables using the m -separation criterion. These include MC, summary, and ancestral graphs. As a modification of MC graphs, we define the class of ribbonless graphs (RGs) that permits the use of the m -separation criterion. RGs contain summary and ancestral graphs as subclasses, and each RG can be generated by a DAG after marginalisation and conditioning. We derive simple algorithms to generate RGs, from given DAGs or RGs, and also to generate summary and ancestral graphs in a simple way by further extension of the RG-generating algorithm. This enables us to develop a parallel theory on these three classes and to study the relationships between them as well as the use of each class.

Quantum computers use quantum resources to carry out computational tasks and may outperform classical computers in solving certain computational problems. Special-purpose quantum computers such as quantum annealers employ quantum adiabatic theorem to solve combinatorial optimization problems. In this paper, we compare classical annealings such as simulated annealing and quantum annealings that are done by the D-Wave machines both theoretically and numerically. We show that if the classical and quantum annealing are characterized by equivalent Ising models, then solving an optimization problem, i.e., finding the minimal energy of each Ising model, by the two annealing procedures, are mathematically identical. For quantum annealing, we also derive the probability lower-bound on successfully solving an optimization problem by measuring the system at the end of the annealing procedure. Moreover, we present the Markov chain Monte Carlo (MCMC) method to realize quantum annealing by classical computers and investigate its statistical properties. In the numerical section, we discuss the discrepancies between the MCMC based annealing approaches and the quantum annealing approach in solving optimization problems.

The monograph investigates the misapplication of conventional statistical techniques to fat tailed distributions and looks for remedies, when possible. Switching from thin tailed to fat tailed distributions requires more than "changing the color of the dress". Traditional asymptotics deal mainly with either n=1 or n=∞ , and the real world is in between, under of the "laws of the medium numbers" --which vary widely across specific distributions. Both the law of large numbers and the generalized central limit mechanisms operate in highly idiosyncratic ways outside the standard Gaussian or Levy-Stable basins of convergence. A few examples: + The sample mean is rarely in line with the population mean, with effect on "naive empiricism", but can be sometimes be estimated via parametric methods. + The "empirical distribution" is rarely empirical. + Parameter uncertainty has compounding effects on statistical metrics. + Dimension reduction (principal components) fails. + Inequality estimators (GINI or quantile contributions) are not additive and produce wrong results. + Many "biases" found in psychology become entirely rational under more sophisticated probability distributions + Most of the failures of financial economics, econometrics, and behavioral economics can be attributed to using the wrong distributions. This book, the first volume of the Technical Incerto, weaves a narrative around published journal articles.

Several authors, including the American Statistician (ASA), have noted the challenges facing statisticians when attacking large, complex, unstructured problems, as opposed to well-defined textbook problems. Clearly, the standard paradigm of selecting the one "correct" statistical method for such problems is not sufficient; a new paradigm is needed. Statistical engineering has been proposed as a discipline that can provide a viable paradigm to attack such problems, used in conjunction with sound statistical science. Of course, in order to develop as a true discipline, statistical engineering needs a well-developed theory, not just a formal definition and successful case studies. This article documents and disseminates the current state of the underlying theory of statistical engineering. Our purpose is to provide a vehicle for applied statisticians to further enhance the practice of statistics, and for academics so interested to continue development of the underlying theory of statistical engineering.

Statistics has moved beyond the frequentist-Bayesian controversies of the past. Where does this leave our ability to interpret results? I suggest that a philosophy compatible with statistical practice, labeled here statistical pragmatism, serves as a foundation for inference. Statistical pragmatism is inclusive and emphasizes the assumptions that connect statistical models with observed data. I argue that introductory courses often mischaracterize the process of statistical inference and I propose an alternative "big picture" depiction.

Generalized Chinese Remainder Theorem (CRT) is a well-known approach to solve ambiguity resolution related problems. In this paper, we study the robust CRT reconstruction for multiple numbers from a view of statistics. To the best of our knowledge, it is the first rigorous analysis on the underlying statistical model of CRT-based multiple parameter estimation. To address the problem, two novel approaches are established. One is to directly calculate a conditional maximum a posteriori probability (MAP) estimation of the residue clustering, and the other is based on a generalized wrapped Gaussian mixture model to iteratively search for MAP of both estimands and clustering. Residue error correcting codes are introduced to improve the robustness further. Experimental results show that the statistical schemes achieve much stronger robustness compared to state-of-the-art deterministic schemes, especially in heavy-noise scenarios.

With the aid of mathematical modelling (basic tool is the random walk with absorbing barriers) we derive subsequent formulas to study the effect of different versions of possible rules. For different rules the probability of winning a game, the probability of break point occurrence, the mathematical expectation of the number of rallies (points) and, the mathematical expectation of the number of break points in a game are expressed. We check these rules against ATP statistics for the Top-200 men players. In conclusion, we suggest a slight but essential modification for the rule of a tennis game, namely , second service ( in case of a first service fault) is to be allowed only at the first three points (rallies). This would partially preserve the traditions (server has an advantage in the modern game) and at the same time it would reduce the predictability of the game, significantly increasing in this way the excitement for the spectators.

In this paper, we consider the exponential functional \(A_{\infty}=\int_0^\infty e^{-\xi_s}ds\) of a L{é}vy process \(\xi_s\) and aim to estimate the characteristics of \(\xi_{s}\) from the distribution of \(A_{\infty}\). We present a new approach, which allows to statistically infer on the L{é}vy triplet of \(\xi_{t}\), and study the theoretical properties of the proposed estimators. The suggested algorithms are illustrated with numerical simulations.