efficient algorithms for computing sommerfeld integral tails sommerfeld integrals sis are ubiquitous in the analysis of problems involving antennas and scatterers embedded in planar multilayered media it is well known that the oscillating and slowly decaying nature of their integrands makes the numerical evaluation of the si real axis tail segment a very time consuming and computationally expensive task therefore si tails have to be specially treated in this paper we compare two recently developed techniques for their efficient numerical evaluation first a partition extrapolation method in which the integration then summation procedure is combined with a new version of the weighted averages wa extrapolation technique is summarized the previous variants of wa technique are also discussed then a review of double exponential de quadrature formulas for direct integration of the si tails is presented the efficient way of implementing the algorithms their pros and cons as well as comparisons of their performance are discussed in detail

nonparametric estimation for a class of levy processes we consider estimation for a class of levy processes modelled as a sum of a drift a symmetric stable process and a compound poisson process we propose a nonparametric approach to estimating unknown parameters of our model including the drift the scale and index parameters in the stable law the mean of the poisson process and the underlying jump size distribution we show that regression and nonparametric deconvolution methods based on the empirical characteristic function can be used for inference interesting connections are shown to exist between properties of our estimators and of those found in conventional deconvolution

local well posedness and quantitative ill posedness for the ostrovsky equation in this article we consider the initial value problem ivp for the ostrovsky equation tu x3u x 1u u xu 0 x r t r u x 0 u0 x with initial data in sobolev spaces hs r we prove that if s 34 this ivp is locally well posed in hs r and if s 34 the ivp is not quantitatively well posed in hs r

estimating dimension from small samples abstract details of a new algorithm to estimate the dimension of a measure from point samples are described the new algorithm emphasizes that dimension need not be the same at all scales that is there need not be a scaling region the algorithm is verified with gaussian measures and is found to be reliable and have less demanding data requirements than conventional estimators using certain convergence properties of gaussian measures it is shown that good estimates can be made from very small samples

a statistical rainfall runoff mixture model with heavy tailed components 1 we present a conditional density model of river runoff given covariate information which includes precipitation at four surrounding stations the proposed model is nonparametric in the central part of the distribution and relies on extreme value theory parametric assumptions for the upper tail of the distribution from the trained conditional density model we can compute quantiles of various levels the median can serve to simulate river runoff quantiles of level 5 and 95 can be used to form a 90 confidence interval and finally extreme quantiles can estimate the probability of large runoff the conditional density model is based on a mixture of hybrid paretos the hybrid pareto is built by stitching a truncated gaussian with a generalized pareto distribution the mixture is made conditional by considering its parameters as functions of covariates a neural network is used to implement those functions a penalty term on the tail indexes is added to the conditional log likelihood to guide the maximum likelihood estimator toward solutions that are preferred this alleviates the difficulties encountered with the maximum likelihood estimator of the tail index on small training sets we evaluate the proposed model on rainfall runoff data from the orgeval basin in france the effect of the tail penalty is further illustrated on synthetic data

stability of solitary waves and weak rotation limit for the ostrovsky equation abstract ostrovsky equation describes the propagation of long internal and surface waves in shallow water in the presence of rotation in this model dispersion is taken into account while dissipation is neglected existence and nonexistence of localized solitary waves is classified according to the sign of the dispersion parameter which can be either positive or negative it is proved that for the case of positive dispersion the set of solitary waves is stable with respect to perturbations the issue of passing to the limit as the rotation parameter tends to zero for solutions of the cauchy problem is investigated on a bounded time interval

fast computation of sommerfeld integral tails via direct integration based on double exponential type quadrature formulas a direct integration algorithm based on double exponential type quadrature rules is presented for the efficient computation of the sommerfeld integral tails arising in the evaluation of multilayered green s functions the proposed scheme maintains the error controllable nature of the so called partition extrapolation methods often used to tackle this problem whereas it requires substantially reduced computational time moreover the proposed method is very easy to implement since the associated weights and abscissas can be precomputed the overall behavior of the proposed method both in terms of accuracy and efficiency is demonstrated through a series of representative numerical experiments where compared with one of the most proven methods available in the literature

tails of lorenz curves the lorenz dominance criterion is the centre piece of inequality analysis yet the appeal of this criterion which requires considering lorenz curves in their entirety is undermined by the practical problem that many sample lorenz curves intersect in the tails the commonly used inferential methods based on central limit theorem arguments do not apply to the tails since these contain too few observations by contrast we propose a test procedure for distributions whose tails lie in the domain of attraction of the frechet distribution which fully takes into account the tail behaviour of lorenz curves our experiments and empirical examples demonstrate the good performance of the proposed test in many cases we are able to infer that despite sample tail crossings the population lorenz curves do in fact exhibit lorenz dominance

semiparametric exponential families for heavy tailed data we propose a semiparametric method for fitting the tail of a heavy tailed population given a relatively small sample from that population and a larger sample from a related background population we model the tail of the small sample as an exponential tilt of the better observed large sample tail using a robust sufficient statistic motivated by extreme value theory in particular our method induces an estimator of the small population mean and we give theoretical and empirical evidence that this estimator outperforms methods that do not use the background sample we demonstrate substantial efficiency gains over competing methods in simulation and on data from a large controlled experiment conducted by facebook

some asymptotic theory for the bootstrap efron s bootstrap method of distribution approximation is shown to be asymptotically valid in a large number of situations including t statistics the empirical and quantile processes and von mises functionals some counter examples are also given to show that the approximation does not always succeed

a monte carlo method for estimating the correlation exponent we propose a monte carlo method for estimating the correlation exponent of a stationary ergodic sequence the estimator can be considered as a bootstrap version of the classical hill estimator a simulation study shows that the method yields reasonable estimates

a note on the ostrovsky equation in weighted sobolev spaces abstract in this work we consider the initial value problem ivp associated to the ostrovsky equations u t x 3 u x 1 u u x u 0 x r t r u x 0 u 0 x we study the well posedness of the ivp in the weighted sobolev spaces z s s 2 f h s r x 1 f l 2 r l 2 x s d x with 3 4 s 1

power law distributions in empirical data power law distributions occur in many situations of scientific interest and have significant consequences for our understanding of natural and man made phenomena unfortunately the detection and characterization of power laws is complicated by the large fluctuations that occur in the tail of the distribution the part of the distribution representing large but rare events and by the difficulty of identifying the range over which power law behavior holds commonly used methods for analyzing power law data such as least squares fitting can produce substantially inaccurate estimates of parameters for power law distributions and even in cases where such methods return accurate answers they are still unsatisfactory because they give no indication of whether the data obey a power law at all here we present a principled statistical framework for discerning and quantifying power law behavior in empirical data our approach combines maximum likelihood fitting methods with goodness of fit tests based on the kolmogorov smirnov ks statistic and likelihood ratios we evaluate the effectiveness of the approach with tests on synthetic data and give critical comparisons to previous approaches we also apply the proposed methods to twenty four real world data sets from a range of different disciplines each of which has been conjectured to follow a power law distribution in some cases we find these conjectures to be consistent with the data while in others the power law is ruled out

global well posedness of the kp i initial value problem in the energy space we prove that the kp i initial value problem begin cases partial tu partial x 3u partial x 1 partial y 2u partial x u 2 2 0 text on mathbb r 2 x y times mathbb r t u 0 phi end cases is globally well posed in the energy space mathbf e 1 mathbb r 2 big phi mathbb r 2 to mathbb r phi mathbf e 1 mathbb r 2 approx phi l 2 partial x phi l 2 big partial x 1 partial y phi big l 2 infty big

the asymptotic distribution of sums of extreme values from a regularly varying distribution on donne une description complete de la distribution asymptotique de la somme des k n valeurs extremes superieures x n 1 kd n n x n 2 kd n n x n n

bias corrected geometric type estimators of the tail index the estimation of the tail index is a central topic in extreme value analysis we consider a geometric type estimator for the tail index and study its asymptotic properties we propose here two asymptotic equivalent bias corrected geometric type estimators and establish the corresponding asymptotic behaviour we also apply the suggested estimators to construct asymptotic confidence intervals for this tail parameter some simulations in order to illustrate the finite sample behaviour of the proposed estimators are provided

local likelihood tracking of fault lines and boundaries summary we suggest locally parametric methods for estimating curves such as boundaries of density supports or fault lines in response surfaces in a variety of spatial problems the methods are based on spatial approximations to the local likelihood that the curve passes through a given point in the plane as a function of that point the local likelihood might be a regular likelihood computed locally with kernel weights e g in the case of support boundary estimation or a local version of a likelihood ratio statistic e g in fault line estimation in either case the local likelihood surface represents a function which is relatively large near the target curve and relatively small elsewhere therefore the curve may be estimated as a ridge line of the surface we require only a numerical algorithm for tracking the projection of a ridge into the plane this approach offers several potential advantages over alternative methods first the local log likelihood surface can be graphed and the degree of ridginess assessed visually to determine how the level of local smoothing should be varied in different spatial locations in order to emphasize the ridge and hence the curve adequately secondly the local likelihood surface does not need to be computed in anything like its entirety once we have a reasonable approximation to a point on the curve we may track it by numerically walking along the ridge line thirdly the method is appropriate without change for many different types of spatial explanatory vanables gridded stochastic or otherwise three examples are explored in detail fault lines in response surfaces and in intensity or density surfaces and boundaries of supports of probability densities

tail prepivoting for the hill estimator it is well known that prepivoting reduces level error of confidence sets we adapt this method to the context of the tail index estimation introducing a procedure that we call tail prepivoting we apply this procedure to the hill estimator and establish its consistency

hitting time statistics and extreme value theory we consider discrete time dynamical systems and show the link between hitting time statistics the distribution of the first time points land in asymptotically small sets and extreme value theory distribution properties of the partial maximum of stochastic processes this relation allows to study hitting time statistics with tools from extreme value theory and vice versa we apply these results to non uniformly hyperbolic systems and prove that a multimodal map with an absolutely continuous invariant measure must satisfy the classical extreme value laws with no extra condition on the speed of mixing for example we also give applications of our theory to higher dimensional examples for which we also obtain classical extreme value laws and exponential hitting time statistics for balls we extend these ideas to the subsequent returns to asymptotically small sets linking the poisson statistics of both processes

a quadrature formula with zeros of bessel functions as nodes a quadrature formula for entire functions of exponential type wherein the nodes are the zeros of the bessel function of the first kind was recently obtained by c frappier and p olivier here the condition imposed on the function is relaxed some applications of the formula are also given

symmetry and uniqueness of the solitary wave solution for the ostrovsky equation we consider herein the ostrovsky equation which arises in modeling the propagation of the surface and internal solitary waves in shallow water or the capillary waves in a plasma with the effects of rotation using the modified sliding method we prove that the solitary wave moving to the left to the ostrovsky equation is symmetric about the origin and unique up to translations we also establish the regularity and decay properties of solitary waves and obtain some results of the nonexistence of solitary wave solutions depending on the wave speed weak rotation and dispersive parameter

sharp well posedness and ill posedness results for a quadratic non linear schrodinger equation abstract we establish that the quadratic non linear schrodinger equation iu t u xx u 2 where u r r c is locally well posed in h s r when s 1 and ill posed when s 1 previous work in c kenig g ponce l vega quadratic forms for the 1 d semilinear schrodinger equation trans amer math soc 346 1996 3323 3353 had established local well posedness for s 3 4 the local well posedness is achieved by an iteration using a modification of the standard x s b spaces the ill posedness uses an abstract and general argument relying on the high to low frequency cascade present in the non linearity and a computation of the first non linear iterate

the cauchy problem for the ostrovsky equation with negative dispersion at the critical regularity abstract in this paper we investigate the cauchy problem for the ostrovsky equation x u t x 3 u 1 2 x u 2 u 0 in the sobolev space h 3 4 r here 0 0 corresponds to the positive negative dispersion of the media respectively p isaza and j mejia 2006 13 2009 15 k tsugawa 2009 26 proved that the problem is locally well posed in h s r when s 3 4 and ill posed when s 3 4 by using some modified bourgain spaces we prove that the problem is locally well posed in h 3 4 r with 0 and 0 the new ingredient that we introduce in this paper is lemma 2 1 lemma 2 2 lemma 2 3 lemma 2 4 lemma 2 5 lemma 2 6

a parametric solution for simple stress strength model of failure with an application abstract estimation of p r s is considered for the simple stress strength model of failure using the pareto and power distributions together with their combined form a useful parametric solution is obtained and is illustrated numerically it is shown that these models are also applicable when only the tails of distributions for r and s are considered an application to the failure study concerning the fractures is also included

on some simple estimates of an exponent of regular variation summary we present a class of simple estimates of an exponent of regular variation unlike those proposed recently by de haan and resnick 1980 ours converge at an algebraic rather than a logarithmic rate and unlike those considered by teugels 1980 their form does not depend on the unknown exponent

multilinear weighted convolution of l 2 functions and applications to nonlinear dispersive equations the x s b spaces as used by beals bourgain kenig ponce vega klainerman machedon and others are fundamental tools to study the low regularity behavior of nonlinear dispersive equations it is of particular interest to obtain bilinear or multilinear estimates involving these spaces by plancherel s theorem and duality these estimates reduce to estimating a weighted convolution integral in terms of the l 2 norms of the component functions in this paper we systematically study weighted convolution estimates on l 2 as a consequence we obtain sharp bilinear estimates for the kdv wave and schrodinger x s b spaces

global cauchy problem for the ostrovsky equation abstract in this article we consider the initial value problem for the ostrovsky equation t u x 3 u x 1 u u x u 0 x r t r u x 0 u 0 x with initial data in sobolev spaces h s r using bourgain spaces we prove that the problem is globally well posed for s 3 10 for both signs in the equation

local and global well posedness for the ostrovsky equation abstract we consider the initial value problem for 0 1 t u x 3 u x 1 u uu x 0 x t r where u is a real valued function and are real numbers such that 0 and x 1 f i 1 f this equation differs from korteweg de vries equation in a nonlocal term nevertheless we obtained local well posedness in x s f h s r x 1 f l 2 r s 3 4 using techniques developed in c e kenig g ponce l vega well posedness of the initial value problem for the korteweg de vries equation j amer math soc 4 1991 323 347 for the case 0 we also obtain a global result in x 1 using appropriate conservation laws

unique continuation principle for the ostrovsky equation with negative dispersion abstract in this article we prove that if the difference of two solutions of the ostrovsky equation with negative dispersion t u x 3 u x u u x u 0 has certain exponential decay for x 0 at two different times then both solutions are equal

prepivoting to reduce level error of confidence sets summary approximate confidence sets for a parameter 0 may be obtained by referring a function of 0 and of the sample to an estimated quantile of that function s sampling distribution we call this function the root of the confidence set either asymptotic theory or bootstrap methods can be used to estimate the desired quantile when the root is not a pivot in the sense of classical statistics the actual level of the approximate confidence set may differ substantially from the intended level prepivoting is the transformation of a confidence set root by its estimated bootstrap cumulative distribution function prepivoting can be iterated bootstrap confidence sets generated from a root prepivoted one or more times have smaller error in level than do confidence sets based on the original root the first prepivoting is nearly equivalent to studentizing when that operation is appropriate further iterations of prepivoting make higher order corrections automatically

on the support of solutions to the ostrovsky equation with negative dispersion abstract in this article we prove that sufficiently smooth solutions of the ostrovsky equation with negative dispersion t u x 3 u x 1 u u x u 0 that have compact support for two different times are identically zero

a diagnostic for selecting the threshold in extreme value analysis summary a new approach is suggested for choosing the threshold when fitting the hill estimator of a tail exponent to extreme value data our method is based on an easily computed diagnostic which in turn is founded directly on the hill estimator itself symmetrized to remove the effect of the tail exponent but designed to emphasize biases in estimates of that exponent the attractions of the method are its accuracy its simplicity and the generality with which it applies this generality implies that the technique has somewhat different goals from more conventional approaches which are designed to accommodate the minor component of a postulated two component pareto mixture our approach does not rely on the second component being pareto distributed nevertheless in the conventional setting it performs competitively with recently proposed methods and in more general cases it achieves optimal rates of convergence a by product of our development is a very simple and practicable exponential approximation to the distribution of the hill estimator under departures from the pareto distribution

spherical bessel functions and explicit quadrature formula an evaluation of the derivative of spherical bessel functions of order n 1 2 at its zeros is obtained consequently an explicit quadrature formula for entire functions of exponential type is given

tail index estimation for heavy tailed models accommodation of bias in weighted log excesses we are interested in the derivation of the distributional properties of a weighted es estimatorof a positive tail index one of the main objectives of such an estilog excesses estimator of a positive tail index one of the main objectives of such an estimator is the accommodation of the dominant component of asymptotic bias together with the maintenance of the asymptotic variance of the maximum likelihood estimator of under a strict pareto model we consider the external estimation not only of a second order shape parameter p but also of a second order scale parameter this will enable us to reduce the asymptotic variance of the final estimators under consideration compared with second order reduced bias estimators that are already available in the literature the second order reduced bias estimators that are considered are also studied for finite samples through monte carlo techniques as well as applied to real data in the field of finance

on the support of solutions to the ostrovsky equation with positive dispersion abstract in this paper we prove that sufficiently smooth solutions of the ostrovsky equation with positive dispersion t u x 3 u x 1 u u x u 0 that have exponential decay for x 0 and such that their supports for two different times are contained in an interval b are identically zero

estimating tails of probability distributions we study the asymptotic properties of estimators of the tail of a distribution based on the excesses over a threshold a key idea is the use of pickands generalised pareto distribution and its fitting in most cases by the method of maximum likelihood the results cover all three limiting types of extreme value theory we propose a new estimator for an index of regular variation and show that it often performs better than hill s estimator we give new results for estimating the endpoint of a distribution extending earlier work by hall and by smith and weissman finally we give detailed results for the domain of attraction of exp e x and show that in most cases our proposed estimator is more efficient than two others one based on the exponential distribution and the other due to davis and resnick we also touch briefly on the problem of large deviations from a statistical point of view the results make extensive use of existing work on rates of convergence

on the estimation of extreme tail probabilities applications of extreme value theory to problems of statistical inference typically involve estimating tail probabilities well beyond the range of the data without the benefit of a concise mathematical model for the sampling distribution the available model is generally only an asymptotic one that is an approximation to probabilities of extreme deviation is supposed which is assumed to become increasingly accurate as one moves further from the range of the data but whose concise accuracy is unknown quantification of the level of accuracy is essential for optimal estimation of tail probabilities in the present paper we suggest a practical device based on a nonstandard application of the bootstrap for determining empirically the accuracy of the approximation and thereby constructing appropriate estimators

quadrature formulae using zeros of bessel functions as nodes a gaussian type quadrature formula where the nodes are the zeros of bessel functions of the first kind of order r 1 was recently proved for entire functions of exponential type here we relax the restriction on as well as on the function some applications are also given

a quadrature formula involving zeros of bessel functions an exact quadrature formula for entire functions of exponential type is obtained the nodes of the formula are zeros of the bessel function of the first kind of order a it generalizes and refines a known quadrature formula related to the sampling theorem the uniqueness of the nodes is studied

the global attractor of the damped forced ostrovsky equation abstract the existence of the global attractor of the damped forced ostrovsky equation in l 2 r is proved for the forces in l 2 r moreover the global attractor of the equation in l 2 r is actually a compact set in h 3 r

cauchy problem for the ostrovsky equation in spaces of low regularity abstract in this article we consider the initial value problem for the ostrovsky equation t u x 3 u x 1 u u x u 0 x r t r u x 0 u 0 x with initial data in sobolev spaces h s r using bourgain spaces we prove that the problem is locally well posed for s 1 2 if the sign of the third term of the equation is and for s 3 4 if the sign of this term is

statistics of extremes by oracle estimation we use the fitted pareto law to construct an accompanying approximation of the excess distribution function a selection rule of the location of the excess distribution function is proposed based on a stagewise lack of fit testing procedure our main result is an oracle type inequality for the kullback leibler loss

well posedness and weak rotation limit for the ostrovsky equation we consider the cauchy problem of the ostrovsky equation we first prove the time local well posedness in the anisotropic sobolev space hs a with s a 2 3 4 and 0 a 1 by the fourier restriction norm method this result include the time local well posedness in hs with s 3 4 for both positive and negative dissipation namely for both 0 and 0 we next consider the weak rotation limit we prove that the solution of the ostrovsky equation converges to the solution of the kdv equation when the rotation parameter goes to 0 and the initial data of the kdv equation is in l2 to show this result we prove a bilinear estimate which is uniform with respect to