The Empirical Beta Copula

Given a sample from a multivariate distribution $F$, the uniform random variates generated independently and rearranged in the order specified by the componentwise ranks of the original sample look like a sample from the copula of $F$. This idea can be regarded as a variant on Baker's [J. Multivariate Anal. 99 (2008) 2312--2327] copula construction and leads to the definition of the empirical beta copula. The latter turns out to be a particular case of the empirical Bernstein copula, the degrees of all Bernstein polynomials being equal to the sample size. Necessary and sufficient conditions are given for a Bernstein polynomial to be a copula. These imply that the empirical beta copula is a genuine copula. Furthermore, the empirical process based on the empirical Bernstein copula is shown to be asymptotically the same as the ordinary empirical copula process under assumptions which are significantly weaker than those given in Janssen, Swanepoel and Veraverbeke [J. Stat. Plan. Infer. 142 (2012) 1189--1197]. A Monte Carlo simulation study shows that the empirical beta copula outperforms the empirical copula and the empirical checkerboard copula in terms of both bias and variance. Compared with the empirical Bernstein copula with the smoothing rate suggested by Janssen et al., its finite-sample performance is still significantly better in several cases, especially in terms of bias.Comment: 23 pages, 3 figure

The potential (iz)^m generates real eigenvalues only, under symmetric rapid decay conditions

We consider the eigenvalue problems -u"(z) +/- (iz)^m u(z) = lambda u(z), m >= 3, under every rapid decay boundary condition that is symmetric with respect to the imaginary axis in the complex z-plane. We prove that the eigenvalues lambda are all positive real.Comment: 23 pages and 1 figur

Further results on the reverse-order law

AbstractAn explicit expression is obtained for a pair of generalized inverses (B−,A−) such that B−A−=(AB)+MN, and a class of pairs (B−,A− of this property is shown. A necessary and sufficient condition for (AB)− to have the expression B−A− is also given

Processo de Elaboração do Termo de Uso das Farinheiras Comunitárias de Açungui e Potinga, Guaraqueçaba - PR

Trabalho apresentado no 31º SEURS - Seminário de Extensão Universitária da Região Sul, realizado em Florianópolis, SC, no período de 04 a 07 de agosto de 2013 - Universidade Federal de Santa Catarina.O Litoral do Paraná está inserido no mais preservado espaço contínuo de Mata Atlântica do Brasil, rico em sociobiodiversidade e pluralidade cultural. A produção de mandioca atua como atividade estratégica para o desenvolvimento local, visto que contribui para a segurança alimentar e é um potencial de geração de renda para as famílias rurais. Neste contexto, o Programa de Extensão Farinheiras, da Universidade Federal do Paraná – Setor Litoral desenvolve desde o ano de 2008, ações de ensino, pesquisa e extensão relacionadas à gestão e organização de farinheiras comunitárias, localizadas nas comunidades de Açungui e Potinga, no município de Guaraqueçaba. Ao longo do Programa e através do diálogo constante com os agricultores familiares, houve a necessidade de elaborar o termo de uso dos equipamentos das farinheiras comunitárias, foi utilizado o espaço das reuniões das Associações para a realização da ação. A atividade teve a duração de três meses e a metodologia aplicada foi a participativa e a técnica do grupo focal. A finalização do termo de uso das farinheiras permitiu ao Programa, juntamente com os agricultores, estabelecer a organização e autogestão da unidade produtiva

Asymptotic Conditional Distribution of Exceedance Counts: Fragility Index with Different Margins

Let $\bm X=(X_1,...,X_d)$ be a random vector, whose components are not necessarily independent nor are they required to have identical distribution functions $F_1,...,F_d$. Denote by $N_s$ the number of exceedances among $X_1,...,X_d$ above a high threshold $s$. The fragility index, defined by $FI=\lim_{s\nearrow}E(N_s\mid N_s>0)$ if this limit exists, measures the asymptotic stability of the stochastic system $\bm X$ as the threshold increases. The system is called stable if $FI=1$ and fragile otherwise. In this paper we show that the asymptotic conditional distribution of exceedance counts (ACDEC) $p_k=\lim_{s\nearrow}P(N_s=k\mid N_s>0)$, $1\le k\le d$, exists, if the copula of $\bm X$ is in the domain of attraction of a multivariate extreme value distribution, and if $\lim_{s\nearrow}(1-F_i(s))/(1-F_\kappa(s))=\gamma_i\in[0,\infty)$ exists for $1\le i\le d$ and some $\kappa\in{1,...,d}$. This enables the computation of the FI corresponding to $\bm X$ and of the extended FI as well as of the asymptotic distribution of the exceedance cluster length also in that case, where the components of $\bm X$ are not identically distributed

On eigenvalues of the Schr\"odinger operator with a complex-valued polynomial potential

In this paper, we generalize a recent result of A. Eremenko and A. Gabrielov on irreducibility of the spectral discriminant for the Schr\"odinger equation with quartic potentials. We consider the eigenvalue problem with a complex-valued polynomial potential of arbitrary degree d and show that the spectral determinant of this problem is connected and irreducible. In other words, every eigenvalue can be reached from any other by analytic continuation. We also prove connectedness of the parameter spaces of the potentials that admit eigenfunctions satisfying k>2 boundary conditions, except for the case d is even and k=d/2. In the latter case, connected components of the parameter space are distinguished by the number of zeros of the eigenfunctions.Comment: 23 page

Quantum toboggans with two branch points

Wave functions describing quantum toboggans with two branch points (QT2) are defined along complex contours of coordinates which spiral around these branch points. The classification of QT2 is found in terms of certain ``winding descriptors" $\varrho$. A mapping $x^{(\varrho)}(s) \to y^{(0)}(s)$ is then presented which rectifies the contours for a subset of the simplest $\varrho=\varrho_0$.Comment: 19pp, 3 figs, presented also as a part of the lecture for QTS-5 conference in Valladolid, Spain, during July 22-28, 2007 (see its webpage http://tristan.fam.cie.uva.es/qts5

Quasi-exactly solvable quartic: real algebraic spectral locus

We describe the real quasi-exactly solvable spectral locus of the PT-symmetric quartic using the Nevanlinna parametrization.Comment: 17 pages, 11 figure

Second-order linear differential equations with two irregular singular points of rank three: the characteristic exponent

For a second-order linear differential equation with two irregular singular points of rank three, multiple Laplace-type contour integral solutions are considered. An explicit formula in terms of the Stokes multipliers is derived for the characteristic exponent of the multiplicative solutions. The Stokes multipliers are represented by converging series with terms for which limit formulas as well as more detailed asymptotic expansions are available. Here certain new, recursively known coefficients enter, which are closely related to but different from the coefficients of the formal solutions at one of the irregular singular points of the differential equation. The coefficients of the formal solutions then appear as finite sums over subsets of the new coefficients. As a by-product, the leading exponential terms of the asymptotic behaviour of the late coefficients of the formal solutions are given, and this is a concrete example of the structural results obtained by Immink in a more general setting. The formulas displayed in this paper are not of merely theoretical interest, but they also are complete in the sense that they could be (and have been) implemented for computing accurate numerical values of the characteristic exponent, although the computational load is not small and increases with the rank of the singular point under consideration.Comment: 33 page