Matrix compatibility and correlation mixture representation of generalized Gini's gamma

Representations of measures of concordance in terms of Pearson' s correlation coefficient are studied. All transforms of random variables are characterized such that the correlation coefficient of the transformed random variables is a measure of concordance. Next, Gini' s gamma is generalized and it is shown that the resulting generalized Gini' s gamma can be represented as a mixture of measures of concordance that are Pearson' s correlation coefficients of transformed random variables. As an application of this correlation mixture representation of generalized Gini' s gamma, lower and upper bounds of the compatible set of generalized Gini' s gamma, which is the collection of all possible square matrices whose entries are pairwise bivariate generalized Gini' s gammas, are derived.Comment: 15 page

Estimation and Comparison of Correlation-based Measures of Concordance

We address the problem of estimating and comparing measures of concordance that arise as Pearson's linear correlation coefficient between two random variables transformed so that they follow the so-called concordance-inducing distribution. The class of such transformed rank correlations includes Spearman's rho, Blomqvist's beta and van der Waerden's coefficient as special cases. To answer which transformed rank correlation is best to use, we propose to compare them in terms of their best and worst asymptotic variances on a given set of copulas. A criterion derived from this approach is that concordance-inducing distributions with smaller variances of squared random variables are more preferable. In particular, we show that Blomqvist's beta is the optimal transformed rank correlation, and Spearman's rho outperforms van der Waerden's coefficient. Moreover, we find that Kendall's tau also attains the optimal asymptotic variances that Blomqvist's beta does, although Kendall's tau is not a transformed rank correlation.Comment: 30 pages, 1 figur

Tail concordance measures: A fair assessment of tail dependence

A new class of measures of bivariate tail dependence called tail concordance measures (TCMs) is proposed, which is defined as the limit of a measure of concordance of the underlying copula restricted to the tail region of interest. TCMs captures the extremal relationship between random variables not only along the diagonal but also along all angles weighted by a tail generating measure. Axioms of tail dependence measures are introduced, and TCMs are shown to characterize linear tail dependence measures. The infimum and supremum of TCMs over all generating measures are considered to investigate the issue of under- and overestimation of the degree of extreme co-movements. The infimum is shown to be attained by the classical tail dependence coefficient, and thus the classical notion always underestimates the degree of tail dependence. A formula for the supremum TCM is derived and shown to overestimate the degree of extreme co-movements. Estimators of the proposed measures are studied, and their performance is demonstrated in numerical experiments. For a fair assessment of tail dependence and stability of the estimation under small sample sizes, TCMs weighted over all angles are suggested, with tail Spearman's rho and tail Gini's gamma being interesting novel special cases of TCMs.Comment: 42 pages, 10 figures, 1 tabl

Avoiding zero probability events when computing Value at Risk contributions

This paper is concerned with the process of risk allocation for a generic multivariate model when the risk measure is chosen as the Value-at-Risk (VaR). We recast the traditional Euler contributions from an expectation conditional on an event of zero probability to a ratio involving conditional expectations whose conditioning events have stricktly positive probability. We derive an analytical form of the proposed representation of VaR contributions for various parametric models. Our numerical experiments show that the estimator using this novel representation outperforms the standard Monte Carlo estimator in terms of bias and variance. Moreover, unlike the existing estimators, the proposed estimator is free from hyperparameters

裾共単調性尺度による裾従属性の定量化

ISM Online Open House, 2021.6.18統計数理研究所オープンハウス（オンライン開催）、R3.6.18ポスター発

Anomalous momentum dependence of the multiband electronic structure of FeSe_1-xTe_x superconductors induced by atomic disorder

When periodicity of crystal is disturbed by atomic disorder, its electronic state becomes inhomogeneous and band dispersion is obscured. In case of Fe-based superconductors, disorder of chalcogen/pnictogen height causes disorder of Fe 3d level splitting. Here, we report an angle-resolved photoemission spectroscopy study on FeSe_1-xTe_x with the chalcogen height disorder, showing that the disorder affects the Fe 3d band dispersions in an orbital-selective way instead of simple obscuring effect. The reverse of the Fe 3d level splitting due to the chalcogen height difference causes the splitting of the hole band with Fe 3d x^2-y^2 character around the Gamma point.Comment: 5 pages, 4 figure

Model building by coset space dimensional reduction in ten-dimensions with direct product gauge symmetry

We investigate ten-dimensional gauge theories whose extra six-dimensional space is a compact coset space, $S/R$, and gauge group is a direct product of two Lie groups. We list up candidates of the gauge group and embeddings of $R$ into them. After dimensional reduction of the coset space,we find fermion and scalar representations of $G_{\mathrm{GUT}} \times U(1)$ with $G_{\mathrm{GUT}}=SU(5), SO(10)$ and $E_6$ which accomodate all of the standard model particles. We also discuss possibilities to generate distinct Yukawa couplings among the generations using representations with a different dimension for $G_{\mathrm{GUT}}=SO(10)$ and $E_6$ models.Comment: 14 pages; added local report number, added refferenc