three point boundary value problems for second order discrete equations we formulate existence results for solutions to discrete equations which approximate three point boundary value problems for second order ordinary differential equations c 2003 elsevier science ltd all rights reserved

rank based tests for randomness against first order serial dependence abstract optimal rank based procedures were derived in hallin ingenbleek and puri 1985 1987 and hallin and puri 1988 for some fundamental testing problems arising in time series analysis the optimality properties of these procedures are of an asymptotic nature however whereas much of the attractiveness of rank based methods lies in their small sample applicability and robustness features accordingly the objective of this article is twofold a a study of the finite sample behavior of the asymptotically optimal tests for randomness against first order autoregressive moving average dependence proposed in hallin et al 1985 both under the null hypothesis tables of critical values and under alternatives of serial dependence evaluation of the power function and b an heuristic investigation of the robustness properties of the proposed procedures with emphasis on the identification problem in the presence of outliers we begin sec 2 with a brief description of the rank based measu

rank based optimal tests of the adequacy of an elliptic varma model we are deriving optimal rank based tests for the adequacy of a vector autoregressive moving average varma model with elliptically contoured innovation density these tests are based on the ranks of pseudo mahalanobis distances and on narmed residuals computed from tyler s ann statist 15 1987 234 251 scatter matrix they generalize the univariate signed rank procedures proposed by hallin and puri j multivariate anal 39 1991 1 29 two types of optimality properties are considered both in the local and asymptotic sense a la le cam a fixed score procedures local asymptotic minimaxity at selected radial densities and b estimated score procedures local asymptotic minimaxity uniform over a class f of radial densities contrary to their classical counterparts based on cross covariance matrices these tests remain valid under arbitrary elliptically symmetric innovation densities including those with infinite variance and heavy tails we show that the ares of our fixed score procedures with respect to traditional gaussian methods are the same as for the tests of randomness proposed in hallin and paindaveine bernoulli 8 2002b 787 815 the multivariate serial extensions of the classical chernoff savage and hodges lehmann results obtained there thus also hold here in particular the van der waerden versions of our tests are uniformly more powerful than those based on cross covariances as for our estimated score procedures they are fully adaptive hence uniformly optimal over the class of innovation densities satisfying the required technical assomptions

optimal tests for multivariate location based on interdirections and pseudo mahalanobis ranks we propose a family of tests based on randies 1989 concept of inter directions and the ranks of pseudo mahalanobis distances computed with respect to a multivariate m estimator of scatter due to tyler 1987 for the multivariate one sample problem under elliptical symmetry these tests which generalize the univariate signed rank tests are affine invariant depending on the score function considered van der waerden laplace they allow for locally asymptotically maximin tests at selected densities multivariate normal multivariate double exponential local powers and asymptotic relative efficiencies are derived with respect to hotelling s test randies 1989 multivariate sign test peters and randies 1990 wilcoxon type test and with respect to the oja median tests we moreover extend to the multivariate setting two famous univariate results the traditional chemoff savage 1958 property showing that hotelling s traditional procedure is uniformly dominated in the pitman sense by the van der waerden version of our tests and the celebrated hodges lehmann 1956 864 result providing for any fixed space dimension k the lower bound for the asymptotic relative efficiency of wilcoxon type tests with respect to hotelling s these asymptotic results are confirmed by a monte carlo investigation and application to a real data set

solving singular second order three point boundary value problems using reproducing kernel hilbert space method this paper investigates the numerical solutions of singular second order three point boundary value problems using reproducing kernel hilbert space method it is a relatively new analytical technique the solution obtained by using the method takes the form of a convergent series with easily computable components however the reproducing kernel hilbert space method cannot be used directly to solve a singular second order three point boundary value problem so we convert it into an equivalent integro differential equation which can be solved using reproducing kernel hilbert space method four numerical examples are given to demonstrate the efficiency of the present method the numerical results demonstrate that the method is quite accurate and efficient for singular second order three point boundary value problems

a fourth order compact finite difference method for nonlinear higher order multi point boundary value problems a fourth order compact finite difference method is proposed for a class of nonlinear 2nth order multi point boundary value problems the multi point boundary condition under consideration includes various commonly discussed boundary conditions such as the three or four point boundary condition n 2 point boundary condition and 2 n m point boundary condition the existence and uniqueness of the finite difference solution are investigated by the method of upper and lower solutions without any monotone requirement on the nonlinear term the convergence and the fourth order accuracy of the method are proved an efficient monotone iterative algorithm is developed for solving the resulting nonlinear finite difference systems various sufficient conditions for the construction of upper and lower solutions are obtained some applications and numerical results are given to demonstrate the high efficiency and advantages of this new approach

a simpler affine invariant multivariate distribution free sign test abstract a simpler multivariate sign test is proposed that uses the transformation retransformation approach of chakraborty chaudhuri and oja together with a directional transformation due to tyler this produces a multivariate sign test that is practical to apply to data of any dimension makes minimal assumptions about the underlying distribution and has a small sample distribution free property over a broad class of population models it is shown to perform very well in comparison to hotelling s t 2 and other multivariate sign tests for heavy tailed and skewed distributions

optimal rank based procedures for time series analysis testing an arma model against other arma models the problem of testing a given arma model in which the density of the generating white noise is unspecified against other arma models is considered a distribution free asymptotically most powerful test based on a generalized linear serial rank statistic is provided against contiguous arma alternatives with specified coefficients in the case when the arma model in the alternative has unspecified coefficients the asymptotic sufficiency in the sense of le cam of a finite dimensional vector of rank statistics is established this asymptotic sufficiency is used to derive an asymptotically maximin most powerful test based on a generalized quadratic serial rank statistic the asymptotically maximin optimal test statistic can be interpreted as a rank based weighted version of the classical box pierce portmanteau statistic to which it reduces in some particular problems asymptotically and under gaussian assumptions

the solution of multipoint boundary value problems by the optimal homotopy asymptotic method in this work we solve multipoint boundary value problems using the optimal homotopy asymptotic method oham the proposed method is tested upon several problems from the literature and the results are compared with the available exact solution this method provides easy tools to control the convergence region of approximating solution series where ever necessary

monge kantorovich depth quantiles ranks and signs we propose new concepts of statistical depth multivariate quantiles ranks and signs based on canonical transportation maps between a distributionof interest on ird and a reference distribution on the d dimensionalunit ball the new depth concept called monge kantorovich depth specializesto halfspace depth in the case of elliptical distributions but for more generaldistributions differs from the latter in the ability for its contours to account fornon convex features of the distribution of interest we propose empirical counterpartsto the population versions of those monge kantorovich depth contours quantiles ranks and signs and show their consistency by establishing a uniformconvergence property for transport maps which is of independent interest

a multivariate signed rank test for the one sample location problem abstract an affine invariant signed rank test is proposed for the one sample multivariate location problem the test suggested is a modification of randles s multivariate sign test based on interdirections which extends blumen s bivariate procedure to the multidimensional setting comparisons are made between the proposed statistic and several competitors via pitman asymptotic relative efficiencies and monte carlo results the signed rank statistic appears to be robust it performs better than its competitors when the distribution is light tailed and virtually as well as hotelling s t 2 under multivariate normality for heavy tailed distributions the signed rank statistic performs better than hotelling s t 2 but not as well as randles s statistic

optimal rank based tests for homogeneity of scatter we propose a class of locally and asymptotically optimal tests based on multivariate ranks and signs for the homogeneity of scatter matrices in m elliptical populations contrary to the existing parametric procedures these tests remain valid without any moment assumptions and thus are perfectly robust against heavy tailed distributions validity robustness nevertheless they reach semiparametric efficiency bounds at correctly specified elliptical densities and maintain high powers under all efficiency robustness in particular their normal score version outperforms traditional gaussian likelihood ratio tests and their pseudo gaussian robustifications under a very broad range of non gaussian densities including for instance all multivariate student and power exponential distributions

a distribution free m estimator of multivariate scatter the existence and uniqueness of a limiting form of a huber type m estimator of multivariate scatter is established under certain conditions on the observed sample these conditions hold with probability one when sampling randomly from a continuous multivariate distribution the existence of the estimator is proven by showing that it is the limiting point of a specific algorithm hence the proof is constructive for continuous populations the estimator of multivariate scatter is shown to be strongly consistent and asymptotically normal an important property of the estimator is that its asymptotic distribution is distribution free with respect to the class of continuous elliptically distributed populations this distribution free property also holds for the finite sample size distribution when the location parameter is known in addition the estimator is the most robust estimator of the scatter matrix of an elliptical distribution in the sense of minimizing the maximum asymptotic variance

on multivariate runs tests for randomness this paper proposes several extensions of the concept of runs to the multivariate setup and studies the resulting tests of multivariate randomness against serial dependence two types of multivariate runs are defined i an elliptical extension of the spherical runs proposed by marden 1999 and ii an original concept of matrix valued runs the resulting runs tests themselves exist in various versions one of which is a function of the number of data based hyperplanes separating pairs of observations only all proposed multivariate runs tests are affine invariant and highly robust in particular they allow for heteroscedasticity and do not require any moment assumption their limiting distributions are derived under the null hypothesis and under sequences of local vector arma alternatives asymptotic relative efficiencies with respect to gaussian portmanteau tests are computed and show that while marden type runs tests suffer severe consistency problems tests based on matrix valued runs perform u

analytic perturbation of the taylor spectrum let tf z tm z z g c ck be analytic families of bounded operators in a complex banach space x such that for each z g the operators ti z and tj z i j l n commute main result if k z denotes the taylor spectrum of the tuple tf z tm z then the set valued function k g 2c is analytic analyticity of such set valued functions is defined here by a simultaneous local maximum property of fc tuples of complex polynomials on the graph of k introduction background and results the following result due to the author is the starting point of this paper theorem ift z z g g c c is an analytic family of bounded operators on a banach space then the set valued function k z sp t z is analytic in the following sense the set 1 u z w zgg w k z

a numerical algorithm for nonlinear multi point boundary value problems in this paper an algorithm is presented for solving second order nonlinear multi point boundary value problems bvps the method is based on an iterative technique and the reproducing kernel method rkm two numerical examples are provided to show the reliability and efficiency of the present method

tests for conditional ellipticity in multivariate garch models tests are proposed for the assumption that the conditional distribution of a multivariate garch process is elliptic these tests are of kolmogorov smirnov and cramer von mises type and make use of the common geometry underlying the characteristic function of any spherically symmetric distribution the asymptotic null distribution of the test statistics as well as the consistency of the tests is investigated under general conditions it is shown that both the finite sample and the asymptotic null distribution depend on the unknown distribution of the euclidean norm of the innovations therefore a conditional monte carlo procedure is used to actually carry out the tests the validity of this resampling scheme is formally justified results on the behavior of the new tests in finite samples are included along with comparisons with other tests

a distribution free multivariate sign test based on interdirections abstract distribution free tests are investigated for the one sample multivariate location problem counts called interdirections which measure the angular distance between two observation vectors relative to the positions of the other observations are introduced these counts are invariant under nonsingular linear transformations and have a small sample distribution free property over a broad class of population models called distributions with elliptical directions which includes all elliptically symmetric populations and many skewed populations a sign test based on interdirections is described including as special cases the two sided univariate sign test and blumen s bivariate sign test the statistic is shown to have a limiting 2 p null distribution and because it is based on interdirections it is also seen to be invariant and to have a small sample distribution free property pitman asymptotic relative efficiencies and a monte carlo study show the test to perform well compared with hotell

a quasilinearization method for second order four point boundary value problems abstract in this paper we obtain the existence and uniqueness results for general second order four point boundary value problems by applying the method of upper and lower solutions together with leray schauder degree also we obtain monotone iteration schemes which converge quadratically to the unique solution of some specific second order four point boundary value problems by using the method of quasilinearization

eca high dimensional elliptical component analysis in non gaussian distributions abstractwe present a robust alternative to principal component analysis pca called elliptical component analysis eca for analyzing high dimensional elliptically distributed data eca estimates the eigenspace of the covariance matrix of the elliptical data to cope with heavy tailed elliptical distributions a multivariate rank statistic is exploited at the model level we consider two settings either that the leading eigenvectors of the covariance matrix are nonsparse or that they are sparse methodologically we propose eca procedures for both nonsparse and sparse settings theoretically we provide both nonasymptotic and asymptotic analyses quantifying the theoretical performances of eca in the nonsparse setting we show that eca s performance is highly related to the effective rank of the covariance matrix in the sparse setting the results are twofold i we show that the sparse eca estimator based on a combinatoric program attains the optimal rate of convergence ii based on some recent d

the uniqueness of the complete norm topology in complete normed nonassociative algebras abstract a theorem on uniqueness of the complete norm topology for complete normed nonassociative algebras is proved this theorem contains the well known one by johnson for associative banach algebras and the recent analogous result by aupetit for banach jordan algebras

convergence properties of the nelder mead simplex method in low dimensions the nelder mead simplex algorithm first published in 1965 is an enormously popular direct search method for multidimensional unconstrained minimization despite its widespread use essentially no theoretical results have been proved explicitly for the nelder mead algorithm this paper presents convergence properties of the nelder mead algorithm applied to strictly convex functions in dimensions 1 and 2 we prove convergence to a minimizer for dimension 1 and various limited convergence results for dimension 2 a counterexample of mckinnon gives a family of strictly convex functions in two dimensions and a set of initial conditions for which the nelder mead algorithm converges to a nonminimizer it is not yet known whether the nelder mead method can be proved to converge to a minimizer for a more specialized class of convex functions in two dimensions

a semi analytical approach to solve integro differential equations in this work we present a semi analytical method for solving integro differential equations under multi point or mixed boundary conditions the proposed method solves linear and nonlinear fredholmvolterra integro differential equations a convergence analysis of the proposed method is directly examined numerical examples are worked out to demonstrate the main results moreover proper graphs are provided to confirm the efficiency and the accuracy of the proposed scheme we show that with a few number of obtained approximating terms we achieve a high accuracy level of the obtained results however increasing the number of approximating terms yields a significant decrease of the error of the approximation the proposed method is very useful reliable and flexible for solving different kinds of integro differential equations

a numerical method for singularly perturbed turning point problems with an interior layer the objective of this paper is to present a numerical method for solving singularly perturbed turning point problems exhibiting an interior layer the method is based on the asymptotic expansion technique and the reproducing kernel method rkm the original problem is reduced to interior layer and regular domain problems the regular domain problems are solved by using the asymptotic expansion method the interior layer problem is treated by the method of stretching variable and the rkm four numerical examples are provided to illustrate the effectiveness of the present method the results of numerical examples show that the present method can provide very accurate approximate solutions

a semi analytical algorithm to solve systems of integro differential equations under mixed boundary conditions in this work we present a semi analytical method to solve systems of integro differential equations under mixed boundary conditions the proposed method handles linear and nonlinear systems of fredholmvolterra integro differential equations in a reliable manner we present a convergence analysis for the proposed method to emphasize its reliability numerical examples are examined to show the efficiency and the accuracy of the scheme we show that a few number of approximating terms gives results of high accuracy level and by increasing the number of these terms the error of the approximated solution decreases rapidly the proposed method is powerful and reliable to solve other kinds of systems of integro differential equations further it has a small cpu time and its computational time is less than the other methods a new method to solve systems of ides under mixed boundary conditions is presented this algorithm can solve different kinds of systems of integro differential equations convergence analysis for the method is proved and accuracy of the method is high accuracy of results is truly fine even if the number of approximating terms is small cpu time and computational time of the new method is less than the other methods

semiparametrically efficient rank based inference for shape ii optimal r estimation of shape a class of r estimators based on the concepts of multivariate signed ranks and the optimal rank based tests developed in hallin and paindaveine 2006a is proposed for the estimation of the shape matrix of an elliptical distribution these r estimators are root n consistent under any radial density g without any moment assumptions and semiparametrically efficient at some prespecified density f when based on normal scores they are uniformly more efficient thanthe traditional normal theory estimator based on empirical covariance matrices the asymptotic normality of which moreover requires finite moments of order four irrespective of the actual underlying elliptical density they rely on an original rank based version of le cam s one step methodology which avoids the unpleasant nonparametric estimation of cross information quantities that is generally required in the context of r estimation although they are not strictly equivariant they are shown to be equivariant in a weak asymptotic sense simulations confirm their feasability and excellent finite sample performances

a simple r estimation method for semiparametric duration models modeling nonnegative financial variables e g durations between trades traded volumes or asset volatilities is central to a number of studies across financial econometrics and despite the efforts still poses several statistical challenges among them the efficiency aspects of semiparametric estimation in this paper we concentrate on estimation problems in autoregressive conditional duration acd models with unspecified innovation densities exponential quasi likelihood estimators qmle are the usual practice in that context the efficiency of those qmles the only fisher consistent qmles unfortunately rapidly deteriorates away from the reference exponential density a phenomenon that has been emphasized earlier by drost and werker 2003 who propose various semiparametrically efficient procedures to palliate that phenomenon those procedures rely on a general semiparametric approach which typically requires kernel estimation of the underlying innovation density we propose rank based estimators r estimators as a substitute just as the qmle r estimators remain root n consistent irrespective of the underlying density and rely on the choice of a reference density under which they achieve semiparametric efficiency that density however needs not be the exponential one contrary to the semiparametric estimators proposed by drost and werker 2003 r estimators neither require tangent space calculations nor kernel based density estimation numerical results moreover indicate that r estimators based on exponential reference densities uniformly outperform the exponential qmle under such families of innovations as the weibull or burr densities a real data example about modeling the price range of the swiss stock market index concludes the paper

application of reproducing kernel method to third order three point boundary value problems abstract this paper investigates the analytical approximate solutions of third order three point boundary value problems using reproducing kernel method the solution obtained by using the method takes the form of a convergent series with easily computable components however the reproducing kernel method can not be used directly to solve third order three point boundary value problems since there is no method of obtaining reproducing kernel satisfying three point boundary conditions this paper presents a method for solving reproducing kernel satisfying three point boundary conditions so that reproducing kernel method can be used to solve third order three point boundary value problems results of numerical examples demonstrate that the method is quite accurate and efficient for singular second order three point boundary value problems

numerical simulation for the space fractional diffusion equations abstract this paper presents a novel semi analytical algorithm based on the chebyshev collocation method for the solution of space fractional diffusion equations the original fractional equation is transformed into a system of ordinary differential equations odes by the chebyshev collocation method a new semi analytical method is then used to approximate the solution of the resulting system to emphasize the reliability of the new scheme a convergence analysis is presented it is shown that highly accurate solutions can be achieved with relatively few approximating terms and absolute errors are rapidly decrease as the number of approximating terms is increased by presenting some numerical examples we show that the proposed method is a powerful and reliable algorithm for solving space fractional diffusion equations and can be extended to solve another space fractional partial differential equations comparison with other methods in the literature demonstrates that the proposed method is both efficient and accurate

successive iteration and positive solution for nonlinear second order three point boundary value problems in this paper we study the existence of positive solution for two classes of nonlinear second order three point boundary value problems using some monotone iterative schemes in both cases such schemes start off with known constant functions and therefore are useful to computation purpose

on linear maps preserving generalized invertibility and related properties let h be an infinite dimensional complex hilbert space b h be the algebra of all bounded linear operators on h we study surjective linear maps on b h preserving generalized invertibility we also investigate surjective linear maps preserving fredholm respectively semi fredholm operators our results improve those of mbekhta rodman and semrl

a new iterative technique for solving nonlinear second order multi point boundary value problems abstract we present a semi analytical iterative method for solving nonlinear second order multi point boundary value problems to demonstrate the working of the method we consider a particular example of this class of problems in this example we demonstrate the accuracy and convergence of the method to the solution we demonstrate clearly that the method is accurate fast and has a reasonable order of convergence

center outward r estimation for semiparametric varma models we propose a new class of r estimators for semiparametric varma models in which the innovation density plays the role of the nuisance parameter our estimators are based on the novel concepts of mu

optimal rank based testing for principal components this paper provides parametric and rank based optimal tests for eigenvectors and eigenvalues of covariance or scatter matrices in elliptical families the parametric tests extend the gaussian likelihood ratio tests of anderson 1963 and their pseudo gaussian robustifications by davis 1977 a nd tyler 1981 1983 the rank based tests address a much broader class of problems where covariance matrices need not exist and principal components are associated with more general scatter matrices the proposed tests are shown to outperform daily practice both from the point of view of validity as from the point of view of efficiency this is achieved by utilizing the le cam theory of locally asymptotically normal experiments in the nonstandard context however of a curved parametrization the results we derive for curved experiments are of independent interest and likely to apply in other contexts 1 introduction this fairly detailed introduction aims at providing a comprehensive and nontechnical overview of the paper including its asymptotic theory aspects and a rough description of some of the rank based test statistics to be derived it is expected to be accessible to a broad readership it should be sufficiently informative for the reader not interested in the technical aspects of asymptotic theory to proceed to sections 5 gaussian and pseudo gaussian tests and 6 rank based tests where the proposed testing procedures are described and for the reader mainly interested in asymptotics to decide whether he she is interested in the treatment of a lan family with curved parametrization developed in sections 3 and 4

positive solutions of three point boundary value problems for systems of nonlinear second order ordinary differential equations abstract in this paper we study the three point boundary value problems for systems of nonlinear second order ordinary differential equations of the form u f t v t 0 1 v g t u t 0 1 u 0 v 0 0 u u 1 v v 1 under some conditions we show the existence and multiplicity of positive solutions of the above problem by applying the fixed point index theory in cones

on the efficiency of multivariate spatial sign and rank tests conditionally and asymptotically distribution free rotation but not scale invariant competitors of the well known hotelling s t 2 tests in this paper efficiency results are found in the one sample location case but they natu rally hold in the multivariate two sample multisample and general linear model case the lack of means of fully ordering or ranking observations in the multivariate setting seems at first sight to be an obstacle to generalizing the concepts of sign and rank and signed rank however these concepts can be defined also by using l criterion functions as follows consider k 1 variate observations y y utilizing the euclidean distance y s 1 n 2 2 1r2 y q qy the three objective functions 1 k t y s s y y i i i i i

the uniqueness of the complete norm topology in banach algebras and banach jordan algebras abstract in this paper we give a very simple subharmonic proof of an extension of the famous theorem of b e johnson on the equivalence of complete norms in semi simple banach algebras this proof avoids irreducible representations so that it can be adapted to the situation of banach jordan algebras in order to give a similar result

an efficient technique to find semi analytical solutions for higher order multi point boundary value problems abstract a new semi analytical algorithm is presented to solve general multi point boundary value problems this method can be applied on nth order linear nonlinear singular and nonsingular multi point boundary value problems mathematical base of the method is presented convergence of the method is proved also the algorithm is applied to solve multi point boundary value problems including nonlinear sixth order nonlinear singular second order five point boundary value problems and a singularly perturbed boundary value problem comparison results show that the new method works more accurate than the other methods

adaptive estimation in time series models in a framework particularly suited for many time series models we obtain a lan result under quite natural and economical conditions this enables us to construct adaptive estimators for part of the euclidean parameter in these semiparametric models special attention is directed to group models in time series with the important subclass of models with time varying location and scale our set up is confronted with the existing literature and as examples we reconsider linear regression and arma tar and arch models