empirical likelihood based confidence interval for the mean with a heavy tailed distribution empirical likelihood based confidence intervals for a mean were introduced by owen biometrika 75 1988 237 249 where at least a finite second moment is required this excludes some important distributions for example those in the domain of attraction of a stable law with index between 1 and 2 in this article we use a method similar to qin and wong scand j statist 23 1996 209 219 to derive an empirical likelihood based confidence interval for the mean when the underlying distribution has heavy tails our method can easily be extended to obtain a confidence interval for any order of moment of a heavy tailed distribution

integrability of the 2d lotka volterra system via polynomial first integrals and polynomial inverse integrating factors we present new first integrals of the two dimensional lotka volterra systems which have a polynomial inverse integrating factor moreover we characterize all the polynomial first integrals of the two dimensional lotka volterra systems

analysis and solution of a nonlinear second order differential equation through rescaling and through a dynamical point of view the solutions of the equation y yy y3 0 where is a free parameter are investigated for 1 9 the equation is linearizable through an eight parameter symmetry group and is completely integrable for 1 9 only two symmetries subsist but through a dynamical description the analytical asymptotic solutions and their behavior are given according to the value of and according to the initial conditions

gel estimation for heavy tailed garch models with robust empirical likelihood inference abstract we construct a generalized empirical likelihood estimator for a garch 1 1 model with a possibly heavy tailed error the estimator imbeds tail trimmed estimating equations allowing for over identifying conditions asymptotic normality efficiency and empirical likelihood based confidence regions for very heavy tailed random volatility data we show the implied probabilities from the tail trimmed continuously updated estimator elevate weight for usable large values assign large but not maximum weight to extreme observations and give the lowest weight to non leverage points we derive a higher order expansion for gel with imbedded tail trimming gelitt which reveals higher order bias and efficiency properties available when the garch error has a finite second moment higher order asymptotics for gel without tail trimming requires the error to have moments of substantially higher order we use first order asymptotics and higher order bias to justify the choice of the number of trimmed observations in any given sample we also present robust versions of generalized empirical likelihood ratio wald and lagrange multiplier tests and an efficient and heavy tail robust moment estimator with an application to expected shortfall estimation finally we present a broad simulation study for gel and gelitt and demonstrate profile weighted expected shortfall for the russian ruble us dollar exchange rate we show that tail trimmed cue gmm dominates other estimators in terms of bias mse and approximate normality

empirical likelihood ratio confidence intervals for a single functional summary the empirical distribution function based on a sample is well known to be the maximum likelihood estimate of the distribution from which the sample was taken in this paper the likelihood function for distributions is used to define a likelihood ratio function for distributions it is shown that this empirical likelihood ratio function can be used to construct confidence intervals for the sample mean for a class of m estimates that includes quantiles and for differentiable statistical functionals the results are nonparametric extensions of wilks s 1938 theorem for parametric likelihood ratios the intervals are illustrated on some real data and compared in a simulation to some bootstrap confidence intervals and to intervals based on student s t statistic a hybrid method that uses the bootstrap to determine critical values of the likelihood ratio is introduced

symmetry and singularity properties of second order ordinary differential equations of lie s type iii abstract we extend the work of abraham shrauner b abraham shrauner hidden symmetries and linearization of the modified painleve ince equation j math phys 34 1993 4809 4816 on the linearization of the modified painleve ince equation to a wider class of nonlinear second order ordinary differential equations invariant under the symmetries of time translation and self similarity in the process we demonstrate a remarkable connection with the parameters obtained in the singularity analysis of this class of equations

polynomial first integrals of quadratic vector fields abstract we classify all quadratic polynomial differential systems having a polynomial first integral and provide explicit normal forms for such systems and for their first integrals

autonomous self similar ordinary differential equations and the painleve connection we demonstrate an intimate connection between nonlinear higher order ordinary differential equations possessing the two symmetries of autonomy and self similarity and the leading order behaviour and resonances determined in the application of the painleve test similar behaviour is seen for systems of first order differential equations several examples illustrate the theory in an integrable case of the abc system the singularity analysis reveals a positive and a negative resonance and the method of leading order behaviour leads naturally to a laurent expansion containing both

polynomial first integrals via the poincare series in appl math 28 2001 17 a method to compute the poincare liapunov constants for an arbitrary analytic differential system which has a linear center at the origin in function of the coefficients of the system was given this method also computes the coefficients of the poincare series in function of the same coefficients in this work we use this method to determine polynomial differential systems which have a polynomial first integral

on the singularity analysis of ordinary differential equations invariant under time translation and rescaling the pt is applied to the general second and third order ordinary differential equations invariant under the two symmetries associated with time translation and rescaling in order to investigate their solvability and global integrability the effect of the two symmetries on the compatibility conditions is determined and we show that generally these conditions are automatically a consequence of the resonance condition use is made of truncated laurent series both in ascending and descending powers as an example the case of the generalized chazy equation is presented

the asymptotic distribution of trimmed sums on donne une condition necessaire et suffisante pour l existence de constantes de normalisation et de centrage

darbouxian first integrals and invariants for real quadratic systems having an invariant conic we apply the darboux theory to study the integrability of real quadratic differential systems having an invariant conic the fact that two intersecting straight lines or two parallel straight lines are particular cases of conics allows us to study simultaneously the integrability of quadratic systems having at least two invariant straight lines real or complex

on the polynomial first integrals of the ital a ital b ital c lotka endash volterra system using elementary differential algebraic techniques we prove that the 3d lotka endash volterra dynamical system has no other nontrivial polynomial first integrals than the previously known ones copyright ital 1996 american institute of physics

asymptotic behavior of fractional laplacians in the half space in this article we first give the asymptotic behaviors of fractional laplacians in the half space of r n next we discuss the weighted boundary limits of modified fractional laplacians the asymptotic behaviors hold outside an explicitly defined exceptional set whose size can be controlled and set to be as small as required the estimation of the asymptotic behavior is the generalization of the argument reported in qiao and deng 2010

exponential factors and darbouxian first integrals of the lorenz system in this article we characterize all the exponential factors and the darbouxian first integrals for the lorenz system s y x rx y xz z bz xy in the proofs we use the weight homogeneous polynomials and the method of characteristic curves for solving linear partial differential equations

hidden symmetries and linearization of the modified painleve ince equation the linearization and hidden symmetries of the modified painleve ince equation y yy y3 0 where and are constants are presented the linearization of this equation by a nonlocal transformation yields a damped stable or growing unstable harmonic oscillator equation for 0 hidden symmetries are analyzed by transforming the modified painleve ince equation to a third order ordinary differential equation ode which in general is invariant under a three parameter group by a riccati transformation a type i hidden symmetry is found of a second order ode found from the third order ode where a symmetry is lost in the reduction of order by the non normal subgroup invariants a type ii hidden symmetry occurs in the third order ode because the symmetries of a second order ode reduced from the third order ode by another set of normal subgroup invariants are increased

properties of second order ordinary differential equations invariant under time translation and self similar transformation the general form of a second order ordinary differential equation invariant under time translation and self similarity is obtained together with its solution in parametric form next two representatives of two different families of such equations are studied in terms of rescaled phase space variables the first example has a simple physical interpretation and its limit cycle properties and period are easily derived similar results are found for the second example but the physical interpretation is less obvious the invariant equations are used as a basis to study the asymptotic behavior of related noninvariant equations thereby underlining the critical value nature of the parameters for which a self similar solution sss exists

darboux integrability for 3d lotka volterra systems we describe the improved darboux theory of integrability for polynomial ordinary differential equations in three dimensions using this theory and computer algebra we study the existence of first integrals for the three dimensional lotka volterra systems only working up to degree two with the invariant algebraic surfaces and the exponential factors we find the major part of the known first integrals for such systems and in addition we find three new classes of integrability the method used is of general interest and can be applied to any polynomial ordinary differential equations in arbitrary dimension

on the polynomial first integrals of the a b c lotka volterra system using elementary differential algebraic techniques we prove that the 3 d lotka volterra dynamical system has no other nontrivial polynomial first integrals than the previously known ones

linearization of fourth order ordinary differential equations by point transformations the solution of the problem on linearization of fourth order equations by means of point transformations is presented here we show that all fourth order equations that are linearizable by point transformations are contained in the class of equations which is linear in the third order derivative we provide the linearization test and describe the procedure for obtaining the linearizing transformations as well as the linearized equation for ordinary differential equations of order greater than 4 we obtain necessary conditions which separate all linearizable equations into two classes

first integrals for the modified emden equation q α t q qn 0 it is shown that the modified emden equation q t q qn 0 possesses first integrals for functions t other than kt 1 the function t is obtained explicitly in the case n 3 and parametrically for other n 2 the case n 2 is seen to be particularly difficult to solve