Goodness-of-fit testing for left-truncated two-parameter weibull distributions with known truncation point

The left-truncated Weibull distribution is used in life-time analysis, it has many applications ranging from financial market analysis and insurance claims to the earthquake inter-arrival times. We present a comprehensive analysis of the left-truncated Weibull distribution when the shape, scale or both parameters are unknown and they are determined from the data using the maximum likelihood estimator. We demonstrate that if both the Weibull parameters are unknown then there are sets of sample configurations, with measure greater than zero, for which the maximum likelihood equations do not possess non trivial solutions. The modified critical values of the goodness-of-fit test from the Kolmogorov-Smirnov test statistic when the parameters are unknown are obtained from a quantile analysis. We find that the critical values depend on sample size and truncation level, but not on the actual Weibull parameters. Confirming this behavior, we present a complementary analysis using the Brownian bridge approach as an asymptotic limit of the Kolmogorov-Smirnov statistics and find that both approaches are in good agreement. A power testing is performed for our Kolmogorov-Smirnov goodness-of-fit test and the issues related to the left-truncated data are discussed. We conclude the paper by showing the importance of left-truncated Weibull distribution hypothesis testing on the duration times of failed marriages in the US, worldwide terrorist attacks, waiting times between stock market orders, and time intervals of radioactive decay.Ayşe Kızılersü, Markus Kreer, Anthony W. Thoma

Building the Full Fermion-Photon Vertex of QED by Imposing Multiplicative Renormalizability of the Schwinger-Dyson Equations for the Fermion and Photon Propagators

In principle, calculation of a full Green's function in any field theory requires knowledge of the infinite set of multi-point Green's functions, unless one can find some way of truncating the corresponding Schwinger-Dyson equations. For the fermion and boson propagators in QED this requires an {\it ansatz} for the full three point vertex. Here we illustrate how the properties of gauge invariance, gauge covariance and multiplicative renormalizability impose severe constraints on this fermion-boson interaction, allowing a consistent truncation of the propagator equations. We demonstrate how these conditions imply that the 3-point vertex {\bf in the propagator equations} is largely determined by the behaviour of the fermion propagator itself and not by knowledge of the many higher point functions. We give an explicit form for the fermion-photon vertex, which in the fermion and photon propagator fulfills these constraints to all orders in leading logarithms for massless QED, and accords with the weak coupling limit in perturbation theory at ${\cal O}(\alpha)$. This provides the first attempt to deduce non-perturbative Feynman rules for strong physics calculations of propagators in massless QED that ensures a more consistent truncation of the 2-point Schwinger-Dyson equations. The generalisation to next-to-leading order and masses will be described in a longer publication.Comment: 57 pages, 3 figure

Constructing the fermion-boson vertex in QED3

We derive perturbative constraints on the transverse part of the fermion-boson vertex in massive QED3 through its one loop evaluation in an arbitrary covariant gauge. Written in a particular form, these constraints naturally lead us to the first non-perturbative construction of the vertex, which is in complete agreement with its one loop expansion in all momentum regimes. Without affecting its one-loop perturbative properties, we also construct an effective vertex in such a way that the unknown functions defining it have no dependence on the angle between the incoming and outgoing fermion momenta. Such a vertex should be useful for the numerical study of dynamical chiral symmetry breaking, leading to more reliable results.Comment: 13 pages, 2 figure

Regularization-independent study of renormalized non-perturbative quenched QED

A recently proposed regularization-independent method is used for the first time to solve the renormalized fermion Schwinger-Dyson equation numerically in quenched QED$_4$. The Curtis-Pennington vertex is used to illustrate the technique and to facilitate comparison with previous calculations which used the alternative regularization schemes of modified ultraviolet cut-off and dimensional regularization. Our new results are in excellent numerical agreement with these, and so we can now conclude with confidence that there is no residual regularization dependence in these results. Moreover, from a computational point of view the regularization independent method has enormous advantages, since all integrals are absolutely convergent by construction, and so do not mix small and arbitrarily large momentum scales. We analytically predict power law behaviour in the asymptotic region, which is confirmed numerically with high precision. The successful demonstration of this efficient new technique opens the way for studies of unquenched QED to be undertaken in the near future.Comment: 20 pages,5 figure

Gauge Dependence of Mass and Condensate in Chirally Asymmetric Phase of Quenched QED3

We study three dimensional quenched Quantum Electrodynamics in the bare vertex approximation. We investigate the gauge dependence of the dynamically generated Euclidean mass of the fermion and the chiral condensate for a wide range of values of the covariant gauge parameter $\xi$. We find that (i) away from $\xi=0$, gauge dependence of the said quantities is considerably reduced without resorting to sophisticated vertex {\em ansatze}, (ii) wavefunction renormalization plays an important role in restoring gauge invariance and (iii) the Ward-Green-Takahashi identity seems to increase the gauge dependence when used in conjunction with some simplifying assumptions. In the Landau gauge, we also verify that our results are in agreement with those based upon dimensional regularization scheme within the numerical accuracy available.Comment: 14 pages, 11 figures, uses revte

Landau-Khalatnikov-Fradkin Transformations and the Fermion Propagator in Quantum Electrodynamics

We study the gauge covariance of the massive fermion propagator in three as well as four dimensional Quantum Electrodynamics (QED). Starting from its value at the lowest order in perturbation theory, we evaluate a non-perturbative expression for it by means of its Landau-Khalatnikov-Fradkin (LKF) transformation. We compare the perturbative expansion of our findings with the known one loop results and observe perfect agreement upto a gauge parameter independent term, a difference permitted by the structure of the LKF transformations.Comment: 9 pages, no figures, uses revte

Solution of coupled vertex and propagator Dyson-Schwinger equations in the scalar Munczek-Nemirovsky model

In a scalar $\phi^2\chi$ model, we exactly solve the vertex and $\phi$ propagator Dyson-Schwinger equations under the assumption of a spatially constant (Munczek-Nemirovsky) propagator for the $\chi$ field. Various truncation schemes are also considered.Comment: 7 pages,4 figures, minor changes, reference added for published versio

An improved hadronic model for pion electroproduction

Current measurements of the high energy behavior of the pion form factor are obtained from pion electroproduction data. These values are model dependent, utilizing the Vanderhaeghen, Guidal and Laget Regge (VGL) Model for their extraction. Recent work which examined the implementation of gauge invariance in that model suggested that it might lead to extracted pion form factors larger than the true values. Here we introduce a new model which preserves the successes of the VGL Model but implements gauge invariance in a new way. To demonstrate the validity of this new approach, we first use it to extract the pion form factor in a simple toy model. When compared with the previous extraction method, the improved model leads to a more reliable extraction. The success in this simple model leads us to reanalyze the electroproduction cross section data, where we obtain comparable values for the pion form factor to those obtained using the VGL procedure.Robert J.Perry, Ayşe Kızılersü, Anthony W.Thoma