adaptive estimation of regression models via moment restrictions abstract this paper considers adaptive estimation of regression models by means of generalized method of moments estimators two models are considered one with an i i d disturbance that is independent of the regressors and one with a conditionally symmetric but possibly heteroskedastic disturbance for both cases the paper develops linearized estimators that are asymptotically efficient if the number and variety of moment conditions are allowed to grow at an appropriate rate with the sample size in the general symmetric case no other adaptive estimator has yet been proposed also results of a small monte carlo study indicate that in the independence case the small sample performance of the generalized method of moments estimator can be quite good vis a vis other estimators previously proposed

superintegrable systems on the two dimensional sphere s2 and the hyperbolic plane h2 the existence of superintegrable systems with n 2 degrees of freedom possessing three independent globally defined constants of motion which are quadratic in the velocities is studied on the two dimensional sphere s2 and on the hyperbolic plane h2 the approach used is based on enforcing the conditions for the existence of two independent integrals further than the energy this is done in a way which allows us to discuss at once the cases of the sphere s2 and the hyperbolical plane h2 by considering the curvature as a parameter different superintegrable potentials are obtained as the solutions of certain systems of two dependent second order partial differential equations the euclidean results are directly recovered for 0 and the superintegrable potentials on either the standard unit sphere radius r 1 or the unit lobachewski plane radius r 1 appear as the particular values of the dependent superintegrable potentials for the values 1 and 1 some new superintegrable potentials are f

a stable normalized least mean fourth algorithm with improved transient and tracking behaviors the stability problems of the least mean fourth lmf algorithm put a limitation on its tracking capability the paper investigates the possibility of solving this problem via stabilization of the algorithm the analysis is done for a markov plant it is found that the available stable normalized lmf nlmf algorithm has a tracking limitation for high signal to noise ratio the paper presents a new stable nlmf algorithm that is free of this limitation mean square stability of the algorithm is proved expressions are derived for the minimum steady state mean square deviation msd and the corresponding convergence time the new algorithm outperforms the available stable nlmf algorithm in both the transient and steady states the new algorithm is also compared with the nlms algorithm when the adaptation parameter of each algorithm is set to the value that minimizes its steady state msd for large initial msd the algorithm outperforms the nlms algorithm even for gaussian noise for small initial msd the algorithm outperforms the nlms algorithm for sub gaussian noise while the situation is opposite for gaussian noise analytical results are supported by simulations

root n consistent semiparametric regression on considere une variable aleatoire x z dans 7b r p 7b r q on construit un estimateur generalisant l estimateur des moindres carres ordinaires en inserant des estimateurs non parametriques de la regression dans la projection orthogonale non lineaire sur z

nondegenerate 2d complex euclidean superintegrable systems and algebraic varieties a classical or quantum superintegrable system is an integrable n dimensional hamiltonian system with potential that admits 2n 1 functionally independent constants of the motion polynomial in the momenta the maximum possible if the constants are all quadratic the system is second order superintegrable the kepler coulomb system is the best known example such systems have remarkable properties multi integrability and multi separability an algebra of higher order symmetries whose representation theory yields spectral information about the schrodinger operator deep connections with special functions and with qes systems for complex riemannian spaces with n 2 the structure and classification of second order superintegrable systems is complete here however we present a new and conceptually simpler approach to the classification for complex euclidean 2 space in which the possible superintegrable systems with nondegenerate potentials correspond to points on an algebraic variety specifically we determine a variety in six variables subject to two cubic and one quartic polynomial constraints each point on the variety corresponds to a superintegrable system the euclidean group e 2 acts on the variety such that two points determine the same superintegrable system if and only if they lie on the same leaf of the foliation

semiparametric estimation of censored selection models with a nonparametric selection mechanism abstract in this paper estimation of the coefficients in a single index selectivity bias model is considered under the assumption that the selection correction function depends on the conditional mean of some observable selection variable the estimation method follows a familiar two step strategy the first step uses a nonparametric regression estimator for the selection variable while the second step uses a weighted instrumental variables estimator for the coefficients in the equation of interest the paper gives conditions under which the proposed estimator is root n consistent and asymptotically normal the proposed method is applied to data on labor supply

nonparametric estimates of regression quantiles and their local bahadur representation let x y be a random vector such that x is d dimensional y is real valued and y theta x varepsilon where x and varepsilon are independent and the alpha th quantile of varepsilon is 0 alpha is fixed such that 0 0 and set r p m 2p d where m is a nonnegative integer smaller than p let t theta denote a derivative of theta of order m it is proved that there exists a pointwise estimate hat t n of t theta based on a set of i i d observations x 1 y 1 cdots s n y n that achieves the optimal nonparametric rate of convergence n r under appropriate regularity conditions further a local bahadur type representation is shown to hold for the estimate hat t n and this is used to obtain some useful asymptotic results

on superintegrable symmetry breaking potentials in n dimensional euclidean space we give a graphical prescription for obtaining and characterizing all separable coordinates for which the schr dinger equation admits separable solutions for one of the superintegrable potentials here xn 1 is a distinguished cartesian variable the algebra of second order symmetries of the resulting schr dinger equation is given and for the first potential the closure relations of the corresponding quadratic algebra these potentials are particularly interesting because they occur in all dimensions n 1 the separation of variables problem is highly nontrivial for them and many other potentials are limiting cases

quadratic algebra as a hidden symmetry of the hartmann potential it is shown that operators commuting with the hamiltonian of the hartmann potential form the quadratic hahn algebra qh 3 the structure of this algebra and its finite dimensional representations are described an analysis of these representations is applied to obtain all the relevant physical results energy spectrum degree of degeneration and overlap functions

the equitable racah algebra from three mathfrak su 1 1 algebras the racah algebra a quadratic algebra with two independent generators is central in the analysis of superintegrable models and encodes the properties of the racah polynomials it is the algebraic structure behind the racah problem as it is realized by the intermediate casimir operators arising in the addition of three irreducible representations it has been shown that this racah algebra can also be obtained from quadratic elements in the enveloping algebra of the correspondence between these two realizations is here explained and made explicit

symmetrically trimmed least squares estimation for tobit models this paper proposes alternatives to maximum likelihood estimation of the censored and truncated regression models known to economists as tobit models the proposed estimators are based upon symmetric censoring or truncation of the upper tail of the distribution of the dependent variable unlike methods based on the assumption of identically distributed gaussian errors the estimators are semiparametric in the sense that they are consistent and asymptotically normal for a wide class of symmetric error distributions with heteroskedasticity of unknown form the paper gives the regularity conditions and proofs of these large sample properties demonstrates how to construct consistent estimators of the asymptotic covariance matrices and presents the results of a simulation study for the censored case extensions and limitations of the approach are also considered

classical and quantum superintegrability with applications a superintegrable system is roughly speaking a system that allows more integrals of motion than degrees of freedom this review is devoted to finite dimensional classical and quantum superintegrable systems with scalar potentials and integrals of motion that are polynomials in the momenta we present a classification of second order superintegrable systems in two dimensional riemannian and pseudo riemannian spaces it is based on the study of the quadratic algebras of the integrals of motion and on the equivalence of different systems under coupling constant metamorphosis the determining equations for the existence of integrals of motion of arbitrary order in real euclidean space e2 are presented and partially solved for the case of third order integrals a systematic exposition is given of systems in two and higher dimensional space that allow integrals of arbitrary order the algebras of integrals of motions are not necessarily quadratic but close polynomially or rationally the relation between superintegrability and the classification of orthogonal polynomials is analyzed

superintegrability in a two dimensional space of nonconstant curvature a hamiltonian with two degrees of freedom is said to be superintegrable if it admits three functionally independent integrals of the motion this property has been extensively studied in the case of two dimensional spaces of constant possibly zero curvature when all the independent integrals are either quadratic or linear in the canonical momenta in this article the first steps are taken to solve the problem of superintegrability of this type on an arbitrary curved manifold in two dimensions this is done by examining in detail one of the spaces of revolution found by g koenigs we determine that there are essentially three distinct potentials which when added to the free hamiltonian of this space have this type of superintegrability separation of variables for the associated hamilton jacobi and schrodinger equations is discussed the classical and quantum quadratic algebras associated with each of these potentials are determined

mean square stability analysis of a normalized least mean fourth algorithm for a markov plant recently it has been shown that the stability of the least mean fourth lmf algorithm depends on the nonstationarity of the plant the present paper investigates the possibility of overcoming this problem by normalization of the weight vector update term a rigorous mean square stability analysis is provided for a recent normalized lmf algorithm which is normalized by a term that is second order in the estimation error and fourth order in the regressor the analysis is done for a markov plant with a stationary white input with even probability density function and a stationary zero mean white noise it is proved that the mean square deviation msd of the algorithm is bounded for all finite values of the input variance noise variance initial msd and mean square plant parameter increment analytical results are supported by simulations

completeness of multiseparable superintegrability on the complex 2 sphere the possibility that schrodinger s equation with a given potential can separate in more than one coordinate system is intimately connected with the notion of superintegrability here we demonstrate for nondegenerate potentials how to establish a complete list of such potentials that are superintegrable on the complex 2 sphere using essentially algebraic means we classify all such potentials that admit a pair of second order constants of motion here nondegenerate means that the potentials depend on four independent parameters the method of proof generalizes to other spaces and dimensions we show for the 2 sphere that all these superintegrable systems possess the remarkable property that they correspond to quadratic algebras and we work out the detailed structure relations and their quantum analogues

semiparametric and nonparametric estimation of sample selection models under symmetry this paper considers the semiparametric estimation of binary choice sample selection models under a joint symmetry assumption our approaches overcome various drawbacks associated with existing estimators in particular our method provides root n consistent estimators for both the intercept and slope parameters of the outcome equation in a heteroscedastic framework without the usual cross equation exclusion restriction or parametric specification for the error distribution and or the form of heteroscedasticity our two step estimators are shown to be consistent and asymptotically normal a monte carlo simulation study indicates the usefulness of our approaches

pairwise difference estimators of censored and truncated regression models abstract this paper proposes a class of estimators of the semiparametric censored regression model under the assumption that the error terms are i i d and independent of the covariates the estimators exploit the fact that for a pair of observations a particular transformation of the censored variables depending upon the parameter vector and both covariate vectors will be identically distributed therefore the difference in the transformed dependent variables will be symmetrically distributed around zero when the transformation is evaluated at the true parameter vector an analogous class of estimators for the truncated regression model includes the bhattacharya chernoff and yang 1983 estimator as a special case the estimators are defined as minimizers of u processes so the large sample theory for classical m estimators is extended to this case conditions are given to ensure root n consistency and asymptotic normality of the estimators and a small scale simulation study for the censored regression estimator suggest that it will be well behaved in finite samples

a smoothed maximum score estimator for the binary response model this paper describes a semiparametric estimator for binary response models in which there may be arbitrary heteroskedasticity of unknown form the estimator is obtained by maximizing a smoothed version of the objective function of c manski s maximum score estimator the smoothing procedure is similar to that used in kernel nonparametric density estimation the resulting estimator s rate of convergence in probability is the fastest possible under the assumptions that are made the centered normalized estimator is asymptotically normally distributed methods are given for consistently estimating the parameters of the limiting distribution and for selecting the bandwidth required by the smoothing procedure copyright 1992 by the econometric society

exact solvability of superintegrable benenti systems we establish quantum and classical exact solvability for two large classes of maximally superintegrable benenti systems in n dimensions with arbitrarily large n namely we solve the hamilton jacobi and schrodinger equations for the systems in question the results obtained are illustrated for a model with the cubic potential

exact and quasiexact solvability of second order superintegrable quantum systems i euclidean space preliminaries we show that second order superintegrable systems in two dimensional and three dimensional euclidean space generate both exactly solvable es and quasiexactly solvable qes problems in quantum mechanics via separation of variables and demonstrate the increased insight into the structure of such problems provided by superintegrability a principal advantage of our analysis using nondegenerate superintegrable systems is that they are multiseparable most past separation of variables treatments of qes problems via partial differential equations have only incorporated separability not multiseparability also we propose another definition of es and qes the quantum mechanical problem is called es if the solution of schrodinger equation can be expressed in terms of hypergeometric functions fnm and is qes if the schrodinger equation admits polynomial solutions with coefficients necessarily satisfying a three term or higher order of recurrence relations in three dimensions we give an example of a system tha

nonparametric censored and truncated regression we propose new estimators of the latent regression function in nonparametric cen sored and truncated regression models our estimators are computationally convenient consisting only of two nonparametric regressions and a univariate integral we establish consistency and asymptotic normality for an implementation based on local linear ker nel estimators an extension permits estimation in the presence of a general form of heteroscedasticity both the function m and the distribution f of the error e are unknown the errors are not assumed to be symmetric this paper provides a simple consistent estimator of m x k for some constant k under an additional tail condition we provide a consistent estimator of m x when m is the conditional mean function of the uncensored population also we show that f can be estimated given m x the proposed estimator is extended to deal with the truncated regression model where yi is only observed when it is not censored we also describe extensions to deal with a general form of heteroskedasticity in which the distribution of e could depend in unknown ways on a subset of elements of x for any continuously distributed element xk of x let mk x am x axk we also provide direct estimators of the derivatives mk x in both the censored and truncated regression models these derivatives are interpretable as the marginal effect of a change in x on the underlying uncensored population they can also be used to test or estimate parametric or semiparametric specifications of m x for example mk x is constant if m x is linear in xk and mk x depends only on xk if m x is additive in a function of xk

group theory of the smorodinsky winternitz system the three degrees of freedom smorodinsky winternitz system is a degenerate or super integrable hamiltonian that possesses five functionally independent globally defined and single valued integrals of the motion in both classical and quantum mechanics this is explained in terms of a forced degeneracy imposed as a consequence of the invariance of the hamiltonian under a group of symmetry transformations isomorphic to the three dimensional unitary unimodular group su 3 in turn this degeneracy group is embedded in a larger group of transformations that maps all the bound energy levels among each other the so called dynamical group all the bound state eigenfunctions act as basis functions for a single irreducible representation of the dynamical group so in common with the hydrogen atom and the harmonic oscillator the quantum mechanics of the smorodinsky winternitz system may be completely solved within the framework of group theory alone

second order superintegrable systems in conformally flat spaces ii the classical two dimensional stackel transform this paper is one of a series that lays the groundwork for a structure and classification theory of second order superintegrable systems both classical and quantum in conformally flat spaces here we study the stackel transform or coupling constant metamorphosis as an invertible mapping between classical superintegrable systems on different spaces through the use of this tool we derive and classify for the first time all two dimensional 2d superintegrable systems the underlying spaces are exactly those derived by koenigs in his remarkable paper giving all 2d manifolds with zero potential that admit at least three second order symmetries our derivation is very simple and quite distinct we also show that every superintegrable system is the stackel transform of a superintegrable system on a constant curvature space

a semiparametric maximum likelihood estimator this paper presents a procedure for analyzing a model in which the parameter vector has two parts a finite dimensional component 8 and a nonparametric component a the procedure does not require parametric modeling of but assumes that the true density of the data satisfies an index restriction the idea is to construct a parametric model passing through the true model and to estimate by setting the score for the parametric model to zero the score is estimated nonparametrically and the estimator is shown to be n consistent and asymptotically normal the estimator is then shown to attain the semiparametric efficiency bound characterized in begun et al 1983 for multivariate nonlinear regression simultaneous equations partially specified regression index regression censored regression switching regression and disequilibrium models in which the error densities are unknown

quadratic poisson algebras of two dimensional classical superintegrable systems and quadratic associative algebras of quantum superintegrable systems the integrals of motion of the classical two dimensional superintegrable systems with quadratic integrals of motion close in a restrained quadratic poisson algebra whose the general form is investigated each classical superintegrable problem has a quantum counterpart a quantum superintegrable system the quadratic poisson algebra is deformed into a quantum associative algebra the finite dimensional representations of this algebra are calculated by using a deformed parafermion oscillator technique it is shown that the finite dimensional representations of the quadratic algebra are determined by the energy eigenvalues of the superintegrable system the calculation of energy eigenvalues is reduced to the solution of algebraic equations which are universal that is for all two dimensional superintegrable systems with quadratic integrals of motion

endogenous selection or treatment model estimation in a sample selection or treatment effects model common unobservables may affect both the outcome and the probability of selection in unknown ways this paper shows that the distribution function of potential outcomes conditional on covariates can be identified given an observed variable v that affects the treatment or selection probability in certain ways and is conditionally independent of the error terms in a model of potential outcomes selection model estimators based on this identification are provided which take the form of simple weighted averages gmm or two stage least squares these estimators permit endogenous and mismeasured regressors empirical applications are provided to estimation of a firm investment model and a schooling effects on wages model

completeness of superintegrability in two dimensional constant curvature spaces we classify the hamiltonians h px2 py2 v x y of all classical superintegrable systems in two dimensional complex euclidean space with two additional second order constants of the motion we similarly classify the superintegrable hamiltonians h j12 j22 j32 v x y z on the complex two sphere where x2 y2 z2 1 this is achieved in all generality using properties of the complex euclidean group and the complex orthogonal group

nonparametric identification and estimation of a censored location scale regression model in this article we consider identification and estimation of a censored nonparametric location scale model we first show that in the case where the location function is strictly less than the fixed censoring point for all values in the support of the explanatory variables the location function is not identified anywhere in contrast when the location function is greater or equal to the censoring point with positive probability the location function is identified on the entire support including the region where the location function is below the censoring point in the latter case we propose a simple estimation procedure based on combining conditional quantile estimators for various higher quantiles the new estimator is shown to converge at the optimal nonparametric rate with a limiting normal distribution a small scale simulation study indicates that the proposed estimation procedure performs well in finite samples we also present an empirical illustration on unemployment insurance duration usin

stabilization of high order stochastic gradient adaptive filtering algorithms the paper is concerned with stabilizing the family of adaptive filtering algorithms based on minimizing the 2lth moment of the estimation error with l being an integer greater than 1 stabilization is attained via a proposed normalization of the algorithm mean square stability of the normalized algorithm is proved for a markov plant for all l 1 transient and steady state performances of the algorithm are analyzed for a time invariant plant expressions are derived for the steady state misadjustment and convergence time tradeoff between the transient and steady state performances is evaluated dependence of this tradeoff on the value of l is studied this tradeoff is compared with the tradeoff of the nlms algorithm the proposed algorithm is dramatically superior to the nlms algorithm for sub gaussian noise the superiority increases with l initial mean square deviation and signal to noise ratio analytical results are supported by simulations

second order superintegrable systems in conformally flat spaces i two dimensional classical structure theory this paper is part of a series that lays the groundwork for a structure and classification theory of second order superintegrable systems both classical and quantum in real or complex conformally flat spaces here we consider classical superintegrable systems with nondegenerate potentials in three dimensions we show that there exists a standard structure for such systems based on the algebra of 3 3 symmetric matrices and that the quadratic algebra always closes at order 6 we show that the spaces of truly second third fourth and sixth order constants of the motion are of dimension 6 4 21 and 56 respectively and we construct explicit bases for the fourth and sixth order constants in terms of products of the second order constants

unified treatment and classification of superintegrable systems with integrals quadratic in momenta on a two dimensional manifold in this paper we prove that the two dimensional superintegrable systems with quadratic integrals of motion on a manifold can be classified by using the poisson algebra of the integrals of motion there are six general fundamental classes of superintegrable systems analytic formulas for the involved integrals are calculated in all the cases all the known superintegrable systems are classified as special cases of these six general classes

completely integrable relativistic hamiltonian systems and separation of variables in hermitian hyperbolic spaces the hamilton jacobi and laplace beltrami equations on the hermitian hyperbolic space hh 2 are shown to allow the separation of variables in precisely 12 classes of coordinate systems the isometry group of this two complex dimensional riemannian space su 2 1 has four mutually nonconjugate maximal abelian subgroups these subgroups are used to construct the separable coordinates explicitly all of these subgroups are two dimensional and this leads to the fact that in each separable coordinate system two of the four variables are ignorable ones the symmetry reduction of the free hh 2 hamiltonian by a maximal abelian subgroup of su 2 1 reduces this hamiltonian to one defined on an o 2 1 hyperboloid and involving a nontrivial singular potential separation of variables on hh 2 and more generally on hh n thus provides a new method of generating nontrivial completely integrable relativistic hamiltonian systems

completeness of multiseparable superintegrability in e2 c the possibility that schrodinger s equation with a given potential can separate in more than one coordinate system is intimately connected with the notion of superintegrability examples of this type of system are well known in this paper we demonstrate how to establish a complete list of such potentials using essentially algebraic means our approach is to classify all nondegenerate potentials that admit a pair of second order constants of motion here nondegenerate means that the potentials depend on four independent parameters this is carried out for two dimensional complex euclidean space though the method generalizes to other spaces and dimensions we show that all these superintegrable systems correspond to quadratic algebras and we work out the detailed structure relations and their quantum analogues

global stabilization of the least mean fourth algorithm the least mean fourth algorithm has several stability problems its stability depends on the variance and distribution type of the adaptive filter input the noise variance and the initialization of the filter weights the present correspondence provides a global solution to all these stability problems this is achieved by normalizing the weight vector update term by a term that is fourth order in the regressor and second order in the estimation error the former property stabilizes the algorithm against the variance and distribution type of the filter input while the latter stabilizes the algorithm against the noise variance and the weight initialization the obtained algorithm is stable for all values of the step size between 0 and 2 the stability of the algorithm is supported by simulations

maximal superintegrability on n dimensional curved spaces a unified algebraic construction of the classical smorodinsky winternitz systems on the nd sphere euclidean and hyperbolic spaces through the lie groups so n 1 iso n and so n 1 is presented firstly general expressions for the hamiltonian and its integrals of motion are given in a linear ambient space n 1 and secondly they are expressed in terms of two geodesic coordinate systems on the nd spaces themselves with an explicit dependence on the curvature as a parameter on the sphere the potential is interpreted as a superposition of n 1 oscillators furthermore each lie algebra generator provides an integral of motion and a set of 2n 1 functionally independent ones are explicitly given in this way the maximal superintegrability of the nd euclidean smorodinsky winternitz system is shown for any value of the curvature

invariant classification of second order conformally flat superintegrable systems in this paper we continue the work of kalnins et al in classifying all second order conformally superintegrable laplace type systems over conformally flat spaces using tools from algebraic geometry and classical invariant theory the results obtained show through stackel equivalence that the list of known nondegenerate superintegrable systems over three dimensional conformally flat spaces is complete in particular a seven dimensional manifold is determined such that each point corresponds to a conformal class of superintegrable systems this manifold is foliated by the nonlinear action of the conformal group in three dimensions two systems lie in the same conformal class if and only if they lie in the same leaf of the foliation this foliation is explicitly described using algebraic varieties formed from representations of the conformal group the proof of these results rely heavily on grobner basis calculations using the computer algebra software packages maple and singular

maximal superintegrability of benenti systems for a class of hamiltonian systems naturally arising in the modern theory of separation of variables we establish their maximal superintegrability by explicitly constructing the additional integrals of motion

superintegrable systems in darboux spaces almost all research on superintegrable potentials concerns spaces of constant curvature in this paper we find by exhaustive calculation all superintegrable potentials in the four darboux spaces of revolution that have at least two integrals of motion quadratic in the momenta in addition to the hamiltonian these are two dimensional spaces of nonconstant curvature it turns out that all of these potentials are equivalent to superintegrable potentials in complex euclidean 2 space or on the complex 2 sphere via coupling constant metamorphosis or equivalently via stackel multiplier transformations we present a table of the results

stochastic analysis of a stable normalized least mean fourth algorithm for adaptive noise canceling with a white gaussian reference the least mean fourth lmf algorithm has several stability problems its stability depends on the variance and distribution type of the adaptive filter input the noise variance and the initialization of the filter weights a global solution to these stability problems was presented recently for a normalized lmf nlmf algorithm here a stochastic analysis of the mean square deviation msd of the globally stable nlmf algorithm is provided the analysis is done in the context of adaptive noise canceling with a white gaussian reference input and gaussian binary and uniform desired signals the analytical model is shown to accurately predict the results of monte carlo simulations comparisons of the nlmf and nlms algorithms are then made for various parameter selections it is then shown under what conditions the nlmf algorithm is superior to nlms algorithm for adaptive noise canceling

a mean square stability analysis of the least mean fourth adaptive algorithm this paper presents a new convergence analysis of the least mean fourth lmf adaptive algorithm in the mean square sense the analysis improves previous results in that it is valid for non gaussian noise distributions and explicitly shows the dependence of algorithm stability on the initial conditions of the weights analytical expressions are derived presenting the relationship between the step size the initial weight error vector and mean square stability the analysis assumes a white zero mean gaussian reference signal and an independent identically distributed i i d measurement noise with any even probability density function pdf it has been shown by nascimento and bermudez probability of divergence for the least mean fourth lmf algorithm ieee transactions on signal processing vol 54 no 4 pp 1376 1385 apr 2006 that the lmf algorithm is not mean square stable for reference signals whose pdfs have infinite support however the probability of divergence as a function of the step size value tends to rise abruptly only when it moves past a given threshold our analysis provides a simple and yet precise estimate of the region of quick rise in the probability of divergence hence the present analysis is useful for predicting algorithm instability in most practical applications

the least mean fourth lmf adaptive algorithm and its family new steepest descent algorithms for adaptive filtering and have been devised which allow error minimization in the mean fourth and mean sixth etc sense during adaptation the weights undergo exponential relaxation toward their optimal solutions time constants have been derived and surprisingly they turn out to be proportional to the time constants that would have been obtained if the steepest descent least mean square lms algorithm of widrow and hoff had been used the new gradient algorithms are insignificantly more complicated to program and to compute than the lms algorithm their general form is w j 1 w j 2 mu k epsilon j 2k 1 x j where w j is the present weight vector w j 1 is the next weight vector epsilon j is the present error x j is the present input vector mu is a constant controlling stability and rate of convergence and 2k is the exponent of the error being minimized conditions have been derived for weight vector convergence of the mean and of the variance for the new gradient algorithms the behavior of the least mean fourth lmf algorithm is of special interest in comparing this algorithm to the lms algorithm when both are set to have exactly the same time constants for the weight relaxation process the lmf algorithm under some circumstances will have a substantially lower weight noise than the lms algorithm it is possible therefore that a minimum mean fourth error algorithm can do a better job of least squares estimation than a mean square error algorithm this intriguing concept has implications for all forms of adaptive algorithms whether they are based on steepest descent or otherwise

nondegenerate three dimensional complex euclidean superintegrable systems and algebraic varieties a classical or quantum second order superintegrable system is an integrable n dimensional hamiltonian system with potential that admits 2n 1 functionally independent second order constants of the motion polynomial in the momenta the maximum possible such systems have remarkable properties multi integrability and multiseparability an algebra of higher order symmetries whose representation theory yields spectral information about the schrodinger operator deep connections with special functions and with quasiexactly solvable systems here we announce a complete classification of nondegenerate i e four parameter potentials for complex euclidean 3 space we characterize the possible superintegrable systems as points on an algebraic variety in ten variables subject to six quadratic polynomial constraints the euclidean group acts on the variety such that two points determine the same superintegrable system if and only if they lie on the same leaf of the foliation there are exactly ten nondegenerat

wilson polynomials and the generic superintegrable system on the 2 sphere for complex two dimensional riemannian spaces every classical or quantum second order superintegrable system can be obtained from a single generic 3 parameter potential on the complex 2 sphere by delicate limit operations and through stackel transforms between manifolds here we derive families of finite and infinite dimensional irreducible representations of the corresponding quadratic quantum algebra for the 2 sphere and point out their role in explaining the degeneracy of the energy eigenspaces corresponding to bound state and continuous spectra of quantum and wave equation analogs of this system the algebra is exactly the one that describes the wilson and racah polynomials in their full generality

integrable potentials on spaces with curvature from quantum groups a family of classical integrable systems defined on a deformation of the two dimensional sphere hyperbolic and anti de sitter spaces is constructed through hamiltonians defined on the non standard quantum deformation of a sl 2 poisson coalgebra all these spaces have a non constant curvature that depends on the deformation parameter z as particular cases the analogues of the harmonic oscillator and kepler coulomb potentials on such spaces are proposed another deformed hamiltonian is also shown to provide superintegrable systems on the usual sphere hyperbolic and anti de sitter spaces with a constant curvature that exactly coincides with z according to each specific space the resulting potential is interpreted as the superposition of a central harmonic oscillator with either two more oscillators or centrifugal barriers the non deformed limit z 0 of all these hamiltonians can then be regarded as the zero curvature limit contraction which leads to the corresponding super integrable systems on the flat euclidean and minkowskian spaces

transfer of tail information in censored regression models consider a heteroscedastic regression model y m x sigma x varepsilon where the functions m and sigma are smooth and varepsilon is independent of x the response variable y is subject to random censoring but it is assumed that there exists a region of the covariate x where the censoring of y is light under this condition it is shown that the assumed nonparametric regression model can be used to transfer tail information from regions of light censoring to regions of heavy censoring crucial for this transfer is the estimator of the distribution of varepsilon based on nonparametric regression residuals whose weak convergence is obtained the idea of transferrring tail information is applied to the estimation of the conditional distribution of y given x x with information on the upper tail borrowed from the region of light censoring and to the estimation of the bivariate distribution p x leq x y leq y with no regions of undefined mass the weak convergence of the two estimators is obtained by products of this investigation include the uniform consistency of the conditional kaplan meier estimator and its derivative the location and scale estimators and the estimators of their derivatives

superintegrability and associated polynomial solutions euclidean space and the sphere in two dimensions in this work we examine the basis functions for those classical and quantum mechanical systems in two dimensions which admit separation of variables in at least two coordinate systems we do this for the corresponding systems defined in euclidean space and on the two dimensional sphere we present all of these cases from a unified point of view in particular all of the special functions that arise via variable separation have their essential features expressed in terms of their zeros the principal new results are the details of the polynomial bases for each of the nonsubgroup bases not just the subgroup cartesian and polar coordinate cases and the details of the structure of the quadratic algebras we also study the polynomial eigenfunctions in elliptic coordinates of the n dimensional isotropic quantum oscillator

dependence of the stability of the least mean fourth algorithm on target weights non stationarity the paper investigates a new stability problem of the least mean fourth lmf algorithm which is the dependence of the algorithm stability on the time variation of the target weights of the adaptive filter the analysis is done in the context of tracking a markov plant with a stationary white gaussian input it is found that the algorithm diverges if the mean square increment of the plant parameter vector exceeds a threshold value that depends on the step size input variance and noise moments the paper also derives a closed form of the steady state mean square deviation without the usual assumption of a strong noise comparison of the tracking capabilities of the lmf and lms algorithms is provided the comparison is done in terms of the minimum mean square deviation attained by each algorithm over the stability range of its step size gaussian uniform and binary distributions of the noise are considered conditions that make one algorithm outperform the other are determined analytical results are supported by simulations

probability of divergence for the least mean fourth algorithm in this paper it is shown that the least mean fourth lmf adaptive algorithm is not mean square stable when the regressor input is not strictly bounded as happens for example if the input has a gaussian distribution for input distributions with infinite support even for the gaussian distribution the lmf always has a nonzero probability of divergence no matter how small the step size is chosen this result is proven for a slight modification of the gaussian distribution in a one tap filter and corroborated with several simulations in addition an upper bound is given for the probability of divergence of lmf as a function of the filter length input power step size and noise variance for the case of gaussian regressors the results reported in this paper provide tools for designers to better understand the behavior of the lmf algorithm and decide on the convenience or not of its use for a given application

quantum superintegrable systems with quadratic integrals on a two dimensional manifold there are two classes of quantum integrable systems on a manifold with quadratic integrals the liouville and the lie integrable systems as it happens in the classical case the quantum liouville quadratic integrable systems are defined on a liouville manifold and the schrodinger equation can be solved by separation of variables in one coordinate system the lie integrable systems are defined on a lie manifold and are not generally separable ones but can be solved therefore there are superintegrable systems with two quadratic integrals of motion not necessarily separable in two coordinate systems the quantum analogs of the two dimensional superintegrable systems with quadratic integrals of motion on a manifold are classified by using the quadratic associative algebra of the integrals of motion there are six general fundamental classes of quantum superintegrable systems corresponding to the classical ones analytic formulas for the involved integrals are calculated in all the cases all the known quan

universal integrals for superintegrable systems on n dimensional spaces of constant curvature an infinite family of classical superintegrable hamiltonians defined on the n dimensional spherical euclidean and hyperbolic spaces are shown to have a common set of 2n 3 functionally independent constants of the motion among them two different subsets of n integrals in involution including the hamiltonian can always be explicitly identified as particular cases we recover in a straightforward way most of the superintegrability properties of the smorodinsky winternitz and generalized kepler coulomb systems on spaces of constant curvature and we introduce as well new classes of quasi maximally superintegrable potentials on these spaces results presented here are a consequence of the poisson coalgebra symmetry of all the hamiltonians together with an appropriate use of the phase spaces associated with poincare and beltrami coordinates

efficiency bounds for semiparametric regression efficiency bounds for conditional moment restrictions with a nonparametric component are derived there is a given function of the data a random sample from a distribution f and a parameter the restriction is that a conditional expectation of this function is zero at some point in the parameter space the parameter has two parts a finite dimensional component and a general function evaluated at a subset of the conditioning variables an example is a regression function that is additive in parametric and nonparametric component as arises in sample selection models copyright 1992 by the econometric society

two step estimation of heteroskedastic sample selection models abstract this paper considers two step estimation of a sample selection model in which there is heteroskedasticity of unknown form in the latent errors we propose an estimator which uses recent developments in nonparametric regression estimation involving series approximations the estimator is shown to be consistent and asymptotically normally distributed under reasonable conditions a small monte carlo experiment demonstrates the usefulness of the estimator and highlights the bias inherent in the usual heckman 1979 estimator when there is heteroskedasticity

superintegrability on the two dimensional hyperboloid ii this work is devoted to the investigation of the quantum mechanical systems on the two dimensional hyperboloid which admits separation of variables in at least two coordinate systems here we consider two potentials introduced in a paper of c p boyer e g kalnins and p winternitz j math phys 24 2022 1983 which have not yet been studied we give an example of an interbasis expansion and work out the structure of the quadratic algebra generated by the integrals of motion

efficiency bounds for distribution free estimators of the binary choice and the censored regression models we derive lower bounds on the asymptotic variances for regular distribution free estimators of the parameters of the binary choice model and the censored regression tobit model a distribution free or semiparametric estimator is one that does not require any assumption about the distribution of the stochastic error term in the model apart from regularity conditions for the binary choice model we obtain an explicit lower bound for the asymptotic variance for the slope parameters or more generally the parameters of a nonlinear regression function in the underlying latent variable model but we find that there is no regular semiparametric estimator of the constant term identified by requiring the error distribution to have zero median lower bounds are also obtained under the further assumption that the error distribution is symmetric and in this case there is a finite lower bound for the constant term too comparison of the bounds with those for the classical parametric problem shows the loss of information due to lack of a priori knowledge of the functional form of the error distribution we give the conditions for equality of the parametric and semiparametric lower bounds in which case adaptive estimation may be possible both with and without the assumption of a symmetric error distribution in general adaptive estimation is not possible but one special case where these conditions hold is when the regression function is linear and the explanatory variables have a multivariate normal distribution the tobit model considered here is the censored nonlinear regression model with a fixed censoring point we again give an explicit lower bound for the asymptotic variance for the regression parameters this time including a constant term if the error term has zero median comparison with the corresponding lower bound for the parametric case shows that adaptive estimation is in general not possible for this model

quantum superintegrability and exact solvability in n dimensions a family of maximally superintegrable systems containing the coulomb atom as a special case is constructed in n dimensional euclidean space two different sets of n commuting second order operators are found overlapping in the hamiltonian alone the system is separable in several coordinate systems and is shown to be exactly solvable it is solved in terms of classical orthogonal polynomials the hamiltonian and n further operators are shown to lie in the enveloping algebra of a hidden affine lie algebra

hidden symmetry separation of variables and interbasis expansions in the two dimensional hydrogen atom expansions for each fundamental basis of the hydrogen atom over two others are found and an additional integral of motion corresponding to an elliptic basis is determined rcpresentations of the elliptic basis as a superposition of polar and parabolic states are obtained certain interesting limiting cases are investigated

a novel family of adaptive filtering algorithms based on the logarithmic cost we introduce a novel family of adaptive filtering algorithms based on a relative logarithmic cost inspired by the competitive methods from the online learning literature the competitive or regret based approaches stabilize or improve the convergence performance of adaptive algorithms through relative cost functions the new family elegantly and gradually adjusts the conventional cost functions in its optimization based on the error amount we introduce important members of this family of algorithms such as the least mean logarithmic square lmls and least logarithmic absolute difference llad algorithms however our approach and analysis are generic such that they cover other well known cost functions as described in the paper the lmls algorithm achieves comparable convergence performance with the least mean fourth lmf algorithm and enhances the stability performance significantly the llad and least mean square lms algorithms demonstrate similar convergence performance in impulse free noise environments while the llad algorithm is robust against impulsive interferences and outperforms the sign algorithm sa we analyze the transient steady state and tracking performance of the introduced algorithms and demonstrate the match of the theoretical analyses and simulation results we show the enhanced stability performance of the lmls algorithm and analyze the robustness of the llad algorithm against impulsive interferences finally we demonstrate the performance of our algorithms in different scenarios through numerical examples

fine structure for 3d second order superintegrable systems three parameter potentials a classical or quantum superintegrable system of second order is an integrable n dimensional hamiltonian system with potential that admits 2n 1 functionally independent constants of the motion quadratic in the momenta the maximum possible for n 3 on conformally flat spaces with nondegenerate i e four parameter potentials the extreme case we have worked out the structure and classified most of the possible spaces and potentials here we extend the analysis to a more degenerate class of functionally linearly independent superintegrable systems the three parameter potential case we show that for true three parameter potentials the algebra of constants of the motion no longer closes at order 6 but still all such systems are stackel transforms of systems on complex euclidean space or the complex 3 sphere this is a significant step towards the complete structure analysis of all types of second order superintegrable systems