the effect of two stage sampling on ordinary least squares methods abstract we look at the effect of intracluster correlation on standard procedures in linear regression the ordinary least squares estimator of the coefficient vector performs well in most cases but the usual estimator of cov and procedures based on this such as confidence intervals and hypothesis tests can be seriously misleading the size of the effect however tends to be smaller than the corresponding effect on the variance of an estimated mean in two stage sampling provided that the cluster sample sizes are approximately equal

ince s limits for confluent and double confluent heun equations we find pairs of solutions to a differential equation which is obtained as a special limit of a generalized spheroidal wave equation this is also known as confluent heun equation one solution in each pair is given by a series of hypergeometric functions and converges for any finite value of the independent variable z while the other is given by a series of modified bessel functions and converges for z z0 where z0 denotes a regular singularity for short the preceding limit is called ince s limit after ince who have used the same procedure to get the mathieu equations from the whittaker hill ones we find as well that when z0 tends to zero the ince limit of the generalized spheroidal wave equation turns out to be the ince limit of a double confluent heun equation for which solutions are provided finally we show that the schrodinger equation for inverse fourth and sixth power potentials reduces to peculiar cases of the double confluent heun equation and its ince s limit respectively

convergence and applications of some solutions of the confluent heun equation we study the convergence of a group of solutions in series of confluent hypergeometric functions for the confluent heun equation these solutions are expansions in two sided infinite series summation from minus to plus infinity which are interpreted as a modified version of expansions proposed by leaver 1986 we show that the two sided solutions yield two nonequivalent groups of one sided series solutions summation from zero to plus infinity in the second place we find that one sided solutions of one of these groups can be used to solve an equation which describes a time dependent two level system of quantum optics for this problem in addition to finite series solutions we obtain infinite series wavefunctions which are convergent and bounded for any value of the time t and vanish when t goes to infinity

comment on dirac equation in the background of the nutku helicoid metric j math phys 48 092301 2007 the dirac equation written on the boundary of the nutku helicoid space consists of a system of ordinary differential equations we tried to analyze this system and we found that it has a higher singularity than those of the heun equations which give the solutions of the dirac equation in the bulk we also lose an independent integral of motion on the boundary this facts explain why we could not find the solution of the system on the boundary in terms of known functions we make the stability analysis of the helicoid and catenoid cases and end up with an appendix which gives a new example wherein one encounters a form of the heun equation

bifurcation in lp r for a semilinear equation keywords weak solution second order differential equation bifurcations reference ana article 1986 001doi 10 1016 0022 0396 86 90077 x record created on 2008 12 10 modified on 2016 08 08

linear operators with invariant polynomial space and graded algebra the irreducible finite dimensional representations of the graded algebras osp j 2 j 1 2 3 are expressed in terms of differential operators some quantum deformations of these algebras are shown to admit similar kinds of representations these are formulated in terms of finite difference operators the results are discussed in the framework of the quasi exactly solvable equations

a new class of quasi exactly solvable potentials with a position dependent mass a new class of quasi exactly solvable potentials with a variable mass in the schrodinger equation is presented we have derived a general expression for the potentials including natanzon confluent potentials the general solution of the schrodinger equation is determined and the eigenstates are expressed in terms of the orthogonal polynomials

integral equations of fields on a rotating black hole it is known that the radial equation of the massless fields with spin around kerr black holes cannot be solved by special functions recently the analytic solution was obtained by use of the expansion in terms of the special functions and various astrophysical application have been discussed it was pointed out that the coefficients of the expansion by the confluent hypergeometric functions are identical to those of the expansion by the hypergeometric functions we explain the reason of this fact by using the integral equations of the radial equation it is shown that the kernel of the equation can be written by the product of confluent hypergeometric functions the integral equation transforms the expansion in terms of the confluent hypergeometric functions to that of the hypergeometric functions and vice versa which explains the reason why the expansion coefficients are universal

on the families of orthogonal polynomials associated to the razavy potential we show that there are two different families of weakly orthogonal polynomials associated to the quasi exactly solvable razavy potential v x cos 2x m 2 0 m one of these families encompasses the four sets of orthogonal polynomials recently found by khare and mandal while the other one is new these results are extended to the related periodic potential u x cos 2x m 2 for which we also construct two different families of weakly orthogonal polynomials we prove that either of these two families yields the ground state when m is odd and the lowest lying gaps in the energy spectrum of the latter periodic potential up to and including the m 1 th gap and having the same parity as m 1 moreover we show that the algebraic eigenfunctions obtained in this way are the well known finite solutions of the whittaker hill or hill s three term periodic differential equation thus the foregoing results provide a lie algebraic justification of the fact that the whittaker hill equation unlike for instance mathieu s equation admits finite solutions

confluent heun functions and separation of variables in spheroidal coordinates the helmholtz equation and schrodinger s equation for h2 are separable in respectively oblate and prolate spheroidal coordinates they share the same form of the angular equation in both cases the radial and angular equations have solutions in terms of confluent heun functions we show that the zeros of the wronskian of a pair of solutions to the angular equation give the allowed values of the separation of variables parameter since the heun functions and their derivatives are implemented in maple this provides a new method of calculating the physical values of the separation of variables parameter without programming we also derive the asymptotic forms of the radial solutions of the helmholtz equation and obtain integral relations between the radial and angular solutions

the d dimensional softcore coulomb potential and the generalized confluent heun equation an analysis of the generalized confluent heun equation 2r2 1r y 2r2 1r 0 y e1r e0 y 0 in d dimensional space where i i ei are real parameters is presented with the aid of these general results the quasi exact solvability of the schrodinger eigenproblem generated by the softcore coulomb potential v r e2z r b b 0 is explicitly resolved necessary and sufficient conditions for polynomial solvability are given a three term recurrence relation is provided to generate the coefficients of polynomial solutions explicitly we prove that these polynomial solutions are sources of finite sequences of orthogonal polynomials properties such as the recurrence relations christoffel darboux formulas and moments of the weight function are discussed we also reveal a factorization property of these polynomials which permits the construction of other interesting related sequences of orthogonal polynomials an analysis of the generalized confluent heun equation 2r2 1r y 2r2 1r 0 y e1r e0 y 0 in d dimensional space where i i ei are real parameters is presented with the aid of these general results the quasi exact solvability of the schrodinger eigenproblem generated by the softcore coulomb potential v r e2z r b b 0 is explicitly resolved necessary and sufficient conditions for polynomial solvability are given a three term recurrence relation is provided to generate the coefficients of polynomial solutions explicitly we prove that these polynomial solutions are sources of finite sequences of orthogonal polynomials properties such as the recurrence relations christoffel darboux formulas and moments of the weight function are discussed we also reveal a factorization property of these polynomials which permits the construction of other interesting related sequences of orthogonal polynomials

higher order approximations for frequency domain time series regression second order expansions and mean squared error approximations are given for efficient frequency domain regression estimators while bandwidth choices do not figure in first order asymptotics for these estimators they do influence second order terms and it is shown how suitable choices will enhance second order efficiency data based bandwidth selection rules are given for practical implementation of these procedures two commonly used and asymptotically equivalent spectral regression estimators are studied and shown to differ in their second order asymptotic behavior some monte carlo evidence is reported

integral relations for solutions of the confluent heun equation firstly we construct kernels for integral relations among solutions of the confluent heun equation che additional kernels are systematically generated by applying substitutions of variables secondly we establish integral relations between known solutions of the che that are power series and solutions that are series of special functions thirdly by using one of the integral relations as an integral transformation we obtain a new series solution of the ordinary spheroidal wave equation a particular che from this solution we construct new series solutions of the general che and show that these are suitable for solving the radial part of the two center problem in quantum mechanics

a note on ordinary least squares methods for two stage sampling abstract a class of linear models for two stage sampling problems is considered ordinary least squares estimates are optimal for these models for some parameters the usual least squares estimates of standard errors are shown to be appropriate

generalized least squares f test in regression analysis with two stage cluster samples abstract when data are obtained by two stage cluster sampling serious problems can arise if conventional methods that ignore the intracluster correlations are used wu holt and holmes showed that the standard f statistic in regression analysis leads to inflated type i error rate or size due to correlated errors in the regression model appropriate for clustered data they proposed a simple correction to the standard f statistic that takes into account common intracluster correlation p and showed through simulation that the corrected f test performs much better than the standard f test in controlling the size for a scalar hypothesis it also performed almost as well as an iterative generalized least squares igls f statistic for large values of and better than the igls for small in controlling the size this article considers a two step generalized least squares gls f statistic by first estimating using the well known method of fitting constants due to henderson and then substituting the

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

exact finite sample relative efficiency of suboptimally weighted least squares estimators in models with ordered heteroscedasticity this paper is concerned with regression models having heteroscedastic errors whose variances are not precisely known but can be ordered by size the object is to use this limited information to devise regression coefficient estimators which are more efficient in finite samples than the ordinary least squares ols estimator three common relative efficiency measures determinant trace and extreme eigenvalue are considered a class of weighted least squares wls estimators is proved to be efficient relative to the ols estimator on the determinant generalized variance measure only a partial result can be obtained for the trace measure a counterexample shows that no similar results are possible for the eigenvalue measure under the weak conditions of this paper the efficiency value of additional prior information on variance is also investigated

generalized confluent hypergeometric solutions of the heun confluent equation abstract we show that the heun confluent equation admits infinitely many solutions in terms of the confluent generalized hypergeometric functions for each of these solutions a characteristic exponent of a regular singularity of the heun confluent equation is a non zero integer and the accessory parameter obeys a polynomial equation each of the solutions can be written as a linear combination with constant coefficients of a finite number of either the kummer confluent hypergeometric functions or the bessel functions

quasi exactly solvable quartic potential a new two parameter family of quasi exactly solvable quartic polynomial potentials is introduced heretofore it was believed that the lowest degree one dimensional quasi exactly solvable polynomial potential is sextic this belief is based on the assumption that the hamiltonian must be hermitian however it has recently been discovered that there are huge classes of non hermitian symmetric hamiltonians whose spectra are real discrete and bounded below replacing hermiticity by the weaker condition of symmetry allows for new kinds of quasi exactly solvable theories the spectra of this family of quartic potentials discussed here are also real discrete and bounded below and the quasi exact portion of the spectra consists of the lowest j eigenvalues these eigenvalues are the roots of a jth degree polynomial

bootstrap bartlett adjustment in seemingly unrelated regression abstract seemingly unrelated regression sur is a method introduced by zellner 1962 for estimating several regression equations simultaneously a common and important problem in econometrics this article considers hypothesis testing in sur using the log likelihood ratio llr test although the asymptotic distribution of this statistic is well known substantial departures occur in samples of the size commonly employed by economists significance levels given by the use of this asymptotic distribution are too low sometimes by an order of magnitude leading to the rejection of too many true null hypotheses a common approach to this problem is to use a bartlett 1937 adjustment to the llr statistic in which the test statistic is multiplied by a factor derived from second order asymptotics before it is referred to the asymptotic 2 q distribution this article proposes that the bartlett adjustment factor be computed using efron s bootstrap 1979 1982 in the cases examined this leads to more accu

random group effects and the precision of regression estimates abstract when explanatory variable data in a regression model are drawn from a population with grouped structure the regression errors are often correlated within groups error component and random coefficient regression models are considered as models of the intraclass correlation this paper analyzes several empirical examples to investigate the applicability of random effects models and the consequences of inappropriately using ordinary least squares ols estimation in the presence of random group effects the principal findings are that the assumption of independent errors is usually incorrect and the unadjusted ols standard errors often have a substantial downward bias suggesting a considerable danger of spurious regression

quasi exactly solvable extensions of the lame equation quasi exactly solvable equations are spectral equations which possess a finite number of algebraic eigenvectors for a few of these equations such as the lame equation the number of algebraic solutions is larger than predicted by the general theory a class of quasi exactly solvable equations in one variable is considered and the conditions under which a rich algebraization occurs is discussed families of equations which by themselves are not quasi exactly solvable but can be transformed into systems of coupled quasi exactly solvable equations are discussed these results suggest a scheme for the classification of quasi exactly solvable systems in one variable

analytical results for a modified trigonometric razavy potential we have presented analytical results for the modified trigonometric razavy potential with an additional relative phase between their components it is shown that the analytical solutions can be given in terms of the heun double confluent functions based on the properties of the heun double confluent functions the transfer matrix has been constructed to compute energy bands

rank based estimation and associated inferences for linear models with cluster correlated errors r estimators based on the joint ranks jr of all the residuals have been developed over the last 20 years for fitting linear models with independently distributed errors in this article we extend these estimators to estimating the fixed effects in a linear model with cluster correlated continuous error distributions for general score functions we discuss the asymptotic theory of the estimators and standard errors of the estimators for the related mixed model with a single random effect we discuss robust estimators of the variance components these are used to obtain studentized residuals for the jr fit a real example is discussed which illustrates the efficiency of the jr analysis over the traditional analysis and the efficiency of a prudent choice of a score function simulation studies over situations similar to the example confirm the validity and efficiency of the analysis

exact tests and confidence sets in linear regressions with autocorrelated errors this article proposes a general method to build exact tests and confidence sets in linear regressions with first order autoregressive gaussian disturbances because of a nuisance parameter problem we argue that generalized bounds tests and conservative confidence sets provide natural inference procedures in such a context given an exact confidence set for the autocorrelation coefficient we describe how to obtain a similar simultaneous confidence set for the autocorrelation coefficient and any subvector of regression coefficient conservative confidence sets for the regression coefficients are then deduced by a projection method for any hypothesis that specifies jointly the value of the autocorrelation coefficient and any set of linear restrictions on the regression coefficients we get exact similar tests for tesing linear hypotheses about the regression coefficients only we suggest bounds type procedures exact confidence sets for the autocorrelation coefficient are built by inverting autocorrelation tests the method is illustrated with two examples copyright 1990 by the econometric society

orthogonal polynomials from hermitian matrices a unified theory of orthogonal polynomials of a discrete variable is presented through the eigenvalue problem of hermitian matrices of finite or infinite dimensions it can be considered as a matrix version of exactly solvable schrodinger equations the hermitian matrices factorizable hamiltonians are real symmetric tridiagonal jacobi matrices corresponding to second order difference equations by solving the eigenvalue problem in two different ways the duality relation of the eigenpolynomials and their dual polynomials is explicitly established through the techniques of exact heisenberg operator solution and shape invariance various quantities the two types of eigenvalues the eigenvalues and the sinusoidal coordinates the coefficients of the three term recurrence the normalization measures and the normalisation constants etc are determined explicitly

a novel quasi exactly solvable model with total transmission modes in this communication we present a novel quasi exactly solvable model with symmetric inverted potentials which are unbounded from below the quasi exactly solvable states are shown to be total transmission or reflectionless modes from these modes even and odd wavefunctions can be constructed which are normalizable and flux zero under the procedure of self adjoint extension a discrete spectrum of bound states can be obtained for these inverted potentials and the solvable part of the spectrum is the quasi exactly solvable states we have discovered

new quasi exactly solvable hamiltonians in two dimensions quasi exactly solvable schrodinger operators have the remarkable property that a part of their spectrum can be computed by algebraic methods such operators lie in the enveloping algebra of a finite dimensional lie algebra of first order differential operators the hidden symmetry algebra in this paper we develop some general techniques for constructing quasi exactly solvable operators our methods are applied to provide a wide variety of new explicit two dimensional examples on both flat and curved spaces of quasi exactly solvable hamiltonians corresponding to both semisimple and more general classes of lie algebras

type b 3 fold supersymmetry and non polynomial invariant subspaces we obtain the most general type b 3 fold supersymmetry by solving directly the intertwining relation we then show that it is a necessary and sufficient condition for a second order linear differential operator to have three linearly independent local analytic solutions we find that there are eight linearly independent non trivial linear differential operators of this kind as a by product we find new quasi solvable second order operators preserving a monomial or polynomial subspace one in type b two in type c and four in type x2 all of which have been missed in the existing literature in addition we show that type a type b and type c 3 fold supersymmetries are connected continuously via one parameter a few new quasi solvable models are also presented

quasi exactly solvable systems and orthogonal polynomials this paper shows that there is a correspondence between quasi exactly solvable models in quantum mechanics and sets of orthogonal polynomials pn the quantum mechanical wave function is the generating function for the pn e which are polynomials in the energy e the condition of quasi exact solvability is reflected in the vanishing of the norm of all polynomials whose index n exceeds a critical value j the zeros of the critical polynomial pj e are the quasi exact energy eigenvalues of the system

nonlinear pseudo supersymmetry in the framework of fold supersymmetry we recall the importance of recognizing the different mathematical nature of various concepts relating to symmetric quantum theories after clarifying the relation between supersymmetry and pseudo supersymmetry we prove generically that nonlinear pseudo supersymmetry recently proposed by sinha and roy is just a special case of fold supersymmetry in particular we show that all the models constructed by these authors have type a two fold supersymmetry furthermore we prove that an arbitrary one body quantum hamiltonian which admits two local solutions in closed form belongs to type a two fold supersymmetry irrespective of whether or not it is hermitian symmetric pseudo hermitian and so on

examples of heun and mathieu functions as solutions of wave equations in curved spaces we give examples of where the heun function exists as solutions of wave equations encountered in general relativity as a new example we find that while the dirac equation written in the background of nutku helicoid metric yields mathieu functions as its solutions in four spacetime dimensions the trivial generalization to five dimensions results in the double confluent heun function we reduce this solution to the mathieu function with some transformations

new quasi exactly solvable double well potentials a new three parameter family of quasi exactly solvable double well potentials is introduced we show that the solutions of this family of double well potentials are expressed in terms of the heun confluent functions under a certain parameter condition some of the bound state wavefunctions and associated energies can be found exactly in explicit form in particular we develop an analytical method to derive the conditions for the energy eigenvalues of the bound states it is also shown that our analytical results can be applied to construct exact solutions of the nonlinear schrodinger equation with double well potentials and spatially localized nonlinearities

higher order approximations for wald statistics in time series regressions with integrated processes asymptotic expansions are developed for wald test statistics in time series regressions with integrated processes these expansions provide an opportunity to reduce size distortion in testing by suitable bandwidth selection and automated rules for doing so are calculated a band spectral regression and the associated wald test are also considered both the first order and second order properties of the estimator are studied

longitudinal data analysis using generalized linear models summary this paper proposes an extension of generalized linear models to the analysis of longitudinal data we introduce a class of estimating equations that give consistent estimates of the regression parameters and of their variance under mild assumptions about the time dependence the estimating equations are derived without specifying the joint distribution of a subject s observations yet they reduce to the score equations for multivariate gaussian outcomes asymptotic theory is presented for the general class of estimators specific cases in which we assume independence m dependence and exchangeable correlation structures from each subject are discussed efficiency of the proposed estimators in two simple situations is considered the approach is closely related to quasi likelih ood some key ironh estimating equation generalized linear model longitudinal data quasi likelihood repeated measures

representation reduction and solution space contraction in quasi exactly solvable systems in quasi exactly solvable problems partial analytic solutions energy spectrum and associated wavefunctions are obtained if some potential parameters are assigned specific values we introduce a new class in which exact solutions are obtained at a given energy for a special set of values of the potential parameters to obtain a larger solution space one varies the energy over a discrete set the spectrum by simply changing the value of a given integer a unified treatment that includes the standard as well as the new class of quasi exactly solvable problems is presented and a few examples are given the solution space is spanned by discrete square integrable basis functions in which the matrix representation of the hamiltonian is tridiagonal consequently the wave equation gives a three term recursion relation for the expansion coefficients of the wavefunction imposing quasi exact solvability constraints results in a complete reduction of the representation to the direct sum of a finite and an infinite component the finite is real and exactly solvable whereas the infinite is complex and associated with zero norm states consequently the whole physical space contracts to a finite dimensional subspace with normalizable states

fully nonparametric hypotheses for factorial designs i multivariate repeated measures designs abstract we introduce nonparametric versions for many of the hypotheses tested in analysis of variance and repeated measures models such as the hypotheses of no main effects no interaction effects and no factor effects these natural extensions of the nonparametric hypothesis of equality of the k distributions in the k sample problem have appealing practical interpretations we concentrate on multivariate repeated measures designs and obtain simple rank statistics for testing these hypotheses these statistics are the rank transform rt versions of the classical statistics for testing hypotheses in repeated measures designs we emphasize that even though recent research has demonstrated the inappropriateness of the rt method for many parametric hypotheses the rt procedure is always valid for testing our nonparametric hypotheses we show that the rank statistics converge in distribution to central chi squared distributions under their respective nonparametric null hypotheses the noncentrality paramet

regression diagnostics for rank based methods abstract residual plots and diagnostic techniques have become important tools in examining the least squares fit of a linear model in this article we explore the properties of the residuals from a rank based fit of the model we present diagnostic techniques that detect outlying cases and cases that have an influential effect on the rank based fit we show that the residuals from this fit can be used to detect curvature not accounted for by the fitted model furthermore our diagnostic techniques inherit the excellent efficiency properties of the rank based fit over a wide class of error distributions including asymmetric distributions we illustrate these techniques with several examples

some properties of double morse potentials in the framework of various exact and approximate methods the properties of the quasi exactly solvable double morse potential are discussed the relation between the hidden algebraic structure of the model and its geometrical properties is examined the potential is used as a testing model for a semiclassical approximation which proves to work well both in the symmetric and asymmetric cases the relation between semiclassical energy level splitting and perturbation theory is underlined leading to conclusions on the interpretation of the semiclassical results

existence and bifurcation theorems for nonlinear elliptic eigenvalue problems on unbounded domains abstract we consider nonlinear elliptic eigenvalue problems on unbounded domains g r n using an extended ljusternik schnirelman theory we prove the existence of infinitely many eigenfunctions on every sphere in l 2 g moreover we establish that the infimum of the spectrum of the linearized problem l is always a bifurcation point in addition there is an infinity of branches emanating at from the trivial line of solutions if belongs to the essential spectrum of l

the eigenvalue equation on the eguchi hanson space we consider the eigenvalue equation for the laplace beltrami operator acting on scalar functions on the noncompact eguchi hanson space the corresponding differential equation is reducible to a confluent heun equation with ince symbol 0 2 12 we construct approximations for the eigenfunctions and their asymptotic scattering phases with the help of the liouville green approximation wkb furthermore for specific discrete eigenvalues obtained by a continued t fraction we construct the solution by the frobenius method and determine its scattering phase by a monodromy computation

failure of the hill determinant method for the sextic anharmonic oscillator an extended analysis of eigenvalues and wavefunctions resulting from the hill determinant method is carried out for the one dimensional sextic anharmonic oscillator it is shown that in spite of a seemingly natural ansatz of the method irrelevant wavefunctions may appear leading to incorrect eigenvalues the domain of applicability of the method is limited and its practical use demands a priori investigation before concrete calculations of spectral characteristics effective variational calculations based on adequate trial functions are performed and comparison of the results of both approaches is presented

transformations of heun s equation and its integral relations using the transformation theory for the heun equation we find substitutions of variables which preserve the form of the equation for the kernels of integral relations among solutions of the heun equation these transformations lead to new kernels for the heun equation given by single hypergeometric functions lambe ward type kernels and by products of two hypergeometric functions erd lyi type such kernels by a limiting process also afford new kernels for the confluent heun equation

hypothesis testing with complex survey data the use of classical quadratic test statistics with particular reference to regression problems abstract sample surveys often have complex sample designs with multistage cluster sampling stratification and differential selection probabilities this article is concerned with testing the null hypothesis h 0 where the p dimensional parameter g and is a q dimensional vector of means the asymptotic framework that consists of a sequence of increasing finite populations is used to define as the limit of finite population means as part of the inference we use replicated estimates of variances that take into account the complex sample design the wald statistic can be used to test h 0 but inference for based on the wald statistic can have low power thus an alternative to using a wald test is pursued in this article first define a classical quadratic test statistic that would be used if one had a simple random sample of the population second treating this quadratic form as a population parameter use design based methods to estimate it from the observed survey data last use

estimation of a linear regression model with stationary arma p q errors abstract it is well known that consistent estimation of a linear regression model with a stationary gaussian arma process in the errors can be carried out by maximum likelihood or alternatively by two stage procedures involving estimation of the nuisance parameters followed by feasible generalized least squares for the model parameters we show that the estimators coincide up to o p t 3 2 and derive the variance to o t 2 which up to terms of this order is the same for both estimators considering the form of the error covariance matrix for an arma p q process allows us to examine a computationally convenient algorithm for estimation of the parameters of the regression model finally we provide a monte carlo comparison of the small sample properties of ols and two versions of the proposed estimator

normalizability of one dimensional quasi exactly solvable schrodinger operators we completely determine necessary and sufficient conditions for the normalizability of the wave functions giving the algebraic part of the spectrum of a quasi exactly solvable schrodinger operator on the line methods from classical invariant theory are employed to provide a complete list of canonical forms for normalizable quasi exactly solvable hamiltonians and explicit normalizability conditions in general coordinate systems

the effect of estimating weights in weighted least squares abstract in weighted least squares it is typical that the weights are unknown and must be estimated most packages provide standard errors assuming that the weights are known this is fine for sufficiently large sample sizes but what about for small to moderate sample sizes the investigation of this article into the effect of estimating weights proceeds under the assumption typical in practice that one has a parametric model for the variance function in this context generalized least squares consists of a an initial estimate of the regression parameter b a method for estimating the variance function and c the number of iterations in reweighted least squares by means of expansions for the covariance it is shown that each of a c can matter in problems of small to moderate size a few conclusions may be of practical interest first estimated standard errors assuming that the weights are known can be too small in practice the investigation indicates that a simple bootstrap operation resul

solutions to a spheroidal wave equation fackerell crossman s solutions to a generalized spheroidal wave equation are extended for the case in which there is no free parameter in that equation the results together with leaver s expansions in series of coulomb wave functions are used to find the time dependence of massive scalar test fields in nonflat friedmann robertson walker models of universes with dust

small sample comparisons of level and power for simple goodness of fit statistics under cluster sampling abstract monte carlo methods were used to examine the type i error performance of a number of goodness of fit statistics under cluster sampling the study has reduced the comparison to four procedures a x 2 s the rao scott satterthwaite adjusted x 2 b xj fay s jackknifed x 2 c fw a modified wald statistic referred to an f distribution and d fx 2 c an f based version of the rao scott adjusted x 2 depending only on the cell design effects de unlike the others the statistic fx 2 c performs well provided that the coefficient of variation a of the eigenvalues i of the design effects matrix is small in general both x 2 s and xj perform well even when a is not small fw behaves reasonably well for tests of uniform probability 0 1 k 1 k where k is the number of categories and 1 k 1 k 1 1 k 1 but it is sensitive to skewness in 0 the power performance of procedures a d was also examined attention was focused on 0 1 k 1 k

asymptotically efficient estimation of a nonlinear model of the heteroscedasticity and the calibration of measurement systems the measurement system calibration includes the estimation of the sensor error models in order to get an optimal estimation of the measured parameters the paper is devoted to the estimation of a nonlinear parametric model of the heteroscedasticity and its application to the calibration of measurement systems the heteroscedasticity occurs in regression when the measurement noise variance is nonconstant the maximum likelihood ml estimation of variance function parameters leads to a system of nonlinear equations the iterative solution of these nonlinear equations is based entirely on a successful choice of initial conditions which is intractable in practice to overcome this difficulty another linear quasi ml estimator is proposed it is strongly consistent asymptotically gaussian and only slightly less efficient than the cramer rao lower bound by using this estimator as an initial condition an asymptotically efficient estimation is obtained by using one step noniterative newton method the theoretical findings have been applied to the calibration of the egnos gps positioning algorithm in the sub urban environments

variance function estimation abstract heteroscedastic regression models are used in fields including economics engineering and the biological and physical sciences often the heteroscedasticity is modeled as a function of the covariates or the regression and other structural parameters standard asymptotic theory implies that how one estimates the variance function in particular the structural parameters has no effect on the first order properties of the regression parameter estimates there is evidence however both in practice and higher order theory to suggest that how one estimates the variance function does matter further in some settings estimation of the variance function is of independent interest or plays an important role in estimation of other quantities in this article we study variance function estimation in a unified way focusing on common methods proposed in the statistical and other literature to make both general observations and compare different estimation schemes we show that there are significant di

confluent heun equations convergence of solutions in series of coulomb wavefunctions the leaver solutions in series of coulomb wavefunctions for the confluent heun equation are given by two sided infinite series that is by series where the summation index n runs from minus to plus infinity leaver 1986 j math phys 27 1238 first we show that in contrast to the d alembert test under certain conditions the raabe test ensures that the domains of convergence of these solutions include an additional singular point we also consider solutions for a limit of the confluent heun equation for both equations new solutions are generated by transformations of variables finally we discuss the time dependence of the klein gordon equation in two cosmological models and the spatial dependence of the schr dinger equation to a family of quasi exactly solvable potentials for a subfamily of these potentials we obtain infinite series solutions which converge and are bounded for all values of the independent variable in opposition to a common belief

generalized spheroidal wave equation and limiting cases we find sets of solutions to the generalized spheroidal wave equation gswe or equivalently to the confluent heun equation each set is constituted by three solutions one given by a series of ascending powers of the independent variable and the others by series of regular and irregular confluent hypergeometric functions for a fixed set the solutions converge over different regions of the complex plane but present series coefficients proportional to each other these solutions for the gswe afford solutions to a double confluent heun equation by a taking limit process due to leaver e w leaver j math phys 27 1238 1986 another procedure called whittaker ince limit b d figueiredo j math phys 46 113503 2005 provides solutions in series of powers and bessel functions for two other equations with a different type of singularity at infinity in addition new solutions are obtained for the whittaker hill and mathieu equations f m arscott proc r soc edinburg a67 265 1967 b

physical applications of second order linear differential equations that admit polynomial solutions in this paper conditions for the second order linear differential equation to have polynomial solutions are given several applications of these results to schrodinger s equation are discussed conditions under which the confluent biconfluent and general heun equation yields polynomial solutions are explicitly given some new classes of exactly solvable differential equations are also discussed the results of this work are expressed in such a way as to allow direct use without preliminary analysis

efficiency of iterative estimators in the regression model with ar 1 disturbances abstract the mean square error approximation method of nagar is applied to the iterated prais winsten and iterated maximum likelihood estimators of regression coefficients in the model with ar 1 disturbances their mean square errors are found to equal that of the two stage prais winsten estimator at the second order level of approximation

heun functions and quasi exactly solvable double well potentials we investigate a type of one dimensional quasi exactly solvable double well potential whose analytical solution can be constructed in terms of the heun functions it is shown that for certain special values of the potential parameters two energy eigenvalues and eigenstates of the lowest part of the energy spectrum can be found exactly in explicit form in addition the wronskian method has been applied to derive the conditions for the energy eigenvalues of the bound states our analytical results may find applications in the tunnelling problem for the double well potentials

global bifurcation analysis and uniqueness for a semilinear problem we study the bifurcation diagram and uniqueness of solutions of by using a rescaling technique and the implicit function theorem we establish the global bifurcation diagram uniqueness is proved by a separation argument to complete the bifurcation picture of the problem our study suggests that the bifurcation diagrams have different behaviour at 0 depending on whether g 0 or g l norm but quite similar behaviour in l p or w 2 1

generalized least squares inference in panel and multilevel models with serial correlation and fixed effects in this paper i consider generalized least squares gls estimation in fixed effects panel and multilevel models with autocorrelation the presence of fixed effects complicates implementation of gls as estimating the fixed effects will typically render standard estimators of the covariance parameters necessary for obtaining feasible gls estimates inconsistent i focus on the case where the disturbances follow an ar p process and offer a simple to implement bias correction for the ar coefficients the usefulness of gls and the derived bias correction for the parameters of the autoregressive process is illustrated through a simulation study which uses data from the current population survey

high breakdown rank regression abstract a weighted rank estimate is proposed that has 50 breakdown and is asymptotically normal at rate n based on this theory inferential procedures including asymptotic confidence and tests and diagnostic procedures such as studentized residuals are developed the influence function of the estimate is derived and shown to be continuous and bounded everywhere in x y space examples show that robustness against outlying high leverage clusters may approach that of the least median of squares while retaining more stability against inliers the estimator uses weights that correct for both factor and response spaces a monte carlo study shows that the estimate is more efficient than the generalized rank estimates which are generalized r estimates with weights that only correct for factor space when weights are constant the estimate reduces to the regular wilcoxon rank estimate

new quasi exactly solvable periodic potentials we present two new classes of quasi exactly solvable periodic potentials it is shown that their solutions are given in terms of a known special function when the parameters in the two periodic potentials satisfy a certain condition some of the bloch waves and associated energies can be found exactly and in closed form at either the center or the edge of the first brillouin zone two fundamental sets of solutions have been constructed to calculate all the bloch waves and energy bands

difference in difference meets generalized least squares higher order properties of hypotheses tests we investigate the estimation and inference in difference in difference econometric models used in the analysis of treatment effects when the innovations in such models display serial correlation commonly used ordinary least squares ols procedures are inefficient and may lead to tests with incorrect size implementation of feasible generalized least squares fgls procedures is often hindered by too few observations in the cross section to allow for unrestricted estimation of the weight matrix without leading to tests with similar size distortions as conventional ols based procedures we analyze the small sample properties of fgls based tests with a formal higher order edgeworth expansion that allows us to construct a size corrected version of the test we also address the question of optimal temporal aggregation as a method to reduce the dimension of the weight matrix we apply our procedure to data on regulation of mobile telephone service prices we find that a size corrected fgls based test outperforms tests based on ols

on the sensitivity of the usual t and f tests to covariance misspecification we consider the standard linear regression model with all standard assumptions except that the disturbances are not white noise but distributed n 0 p2x h where x 0 i n our interest lies in testing linear restrictions using the usual f statistic based on ols residuals we are not interested in nding out whether h 0 or not instead we want to nd out what the e ect is of possibly nonzero h on the f statistic itself we propose a sensitivity statistic for this purpose discuss its distribution and obtain a practical and easy to use decision rule to decide whether the f test is sensitive or not to covariance misspeci cation when h is close to zero some nite and asymptotic properties of u are studied as well as its behaviour in the special case of an ar 1 process near the unit root 2000 elsevier science s a all rights reserved jel classixcation c12 c22 c51 c52

nonparametric analysis of clustered multivariate data there is wide interest in extending univariate and multivariate nonparametric procedures to clustered and hierarchical data traditionally parametric mixed models have been used to account for the correlation structures among the dependent observational units in this work we extend multivariate nonparametric procedures for one sample and several sample location problems to clustered data settings the results are given for a general score function but with an emphasis on spatial sign and rank methods mixed models notation involving design matrices for fixed and random effects is used throughout the asymptotic variance formulas and limiting distributions of the test statistics under the null hypothesis and under a sequence of alternatives are derived as are the limiting distributions for the corresponding estimates the approach based on a general score function also shows for example how m estimates behave with clustered data efficiency studies demonstrate the practical advantages and disadvantag

a robust two stage multiple comparison procedure with application to a random drug screen summary a high volume random drug screen is used to detect drugs that are effective in reducing atherogenic low density lipoproteins a two stage analysis consisting of an overall f test along with a multiple comparison procedure to detect differences between treatments and a placebo seemed appropriate due to observations with heavy right tails and more than occasional outliers in the left tail this analysis based on least squares estimates had insufficient power as a solution to this problem we present a similar analysis based on the class of robust r estimates in general it was more powerful than the least squares analysis furthermore by a prudent choice of rank score functions this robust analysis can take advantage of the underlying skewed error structure its validity and power are verified in a monte carlo study atherosclerosis is a leading cause of death in the united states the search continues for an effective agent to treat this condition rather than searching for drugs that reduce serum total cholesterol the random drug screen discussed in this paper attempts to find agents effective in reducing atherogenic low density lipoproteins the screen uses the normocholesterolemic male sea susceptible to experimental atherosclerosis japanese quail as the animal model see chapman stafford and day 1976 for further details on the use of this animal for such studies prior to drug testing the birds were randomly allocated to 10 15 groups of 10 quails each they were housed individually in ten cage units and fed a commercial diet for 5 days the drugs to be tested were dissolved or dispersed in ethanol and mixed with the diet control groups received a diet mixed with ethanol alone after 1 week on the diets each bird was bled from the right jugular vein and serum samples were obtained the response of interest was the amount of beta cholesterol obtained from these birds from a statistical point of view each application of the screen results in a one way design a two stage procedure seems to be the appropriate analysis this would include an overall

on some solutions to generalized spheroidal wave equations and applications expansions in series of coulomb and hypergeometric functions for the solutions of the generalized spheroidal wave equations gswes are analysed and written together in pairs each pair consists of a solution in series of hypergeometric functions and another in series of coulomb wavefunctions and has the same recurrence relations for the series coefficients but the solutions inside it present different radii of convergence expansions without a phase parameter are derived by truncating the series with a phase parameter for the whittaker hill equation solutions are found by treating that equation as a particular case of gswe while for the confluent gswe solutions with and without a phase parameter are given as pairs of series of coulomb wavefunctions amongst the applications there are equations for the time dependence of dirac test fields in some nonflat friedmann robertson walker spacetimes the radial schrodinger equation for an electron in the field of two coulombian centres and the schrodinger equation for the razavy type quasi exactly solvable potentials for these problems it is possible to find wavefunctions in terms of infinite series regular and convergent over the entire range of the independent variable by matching expansions belonging to one or more of the above pairs the infinite series solutions for the razavy type potentials in addition to the polynomial ones suggest that the whole energy spectra may be determined without appealing to perturbation theory or semi classical methods of approximation

a limit of the confluent heun equation and the schrodinger equation for an inverted potential and for an electric dipole we re examine and extend a group of solutions in series of bessel functions for a limiting case of the confluent heun equation and then apply such solutions to the one dimensional schrodinger equation with an inverted quasiexactly solvable potential as well as to the angular equation for an electron in the field of a point electric dipole for the first problem we find finite and infinite series solutions which are convergent and bounded for any value of the independent variable for the angular equation we also find expansions in series of jacobi polynomials

conditionally consistent estimators using only probabilities of selection in complex sample surveys abstract predictive regression estimators using functions of the probability of selection i as covariates are studied as an alternative to classical p weighting estimators for the finite population mean in complex sample surveys variance estimators are developed based on the relationship var y i i where y is the response and i is the probability of selection extensions to cluster sample designs are outlined and an example is given from the national educational longitudinal survey of 1988

solutions for confluent and double confluent heun equations this paper examines some solutions for confluent and double confluent heun equations in the first place we review two leaver s solutions in series of regular and irregular confluent hypergeometric functions for the confluent equation leaver e w j math phys 27 1238 1986 and introduce an additional expansion in series of irregular confluent hypergeometric functions then we find the conditions under which one of these solutions can be written as a linear combination of the others in the second place by means of limiting procedures we generate solutions for the double confluent equation as well as for special limits of both the confluent and double confluent equations finally we present some problems which are ruled by one or the other of these four equations

hidden algebras of the super calogero and sutherland models we propose to parametrize the configuration space of one dimensional quantum systems of n identical particles by the elementary symmetric polynomials of bosonic and fermionic coordinates it is shown that in this parametrization the hamiltonians of the an bcn bn cn and dn calogero and sutherland models as well as their supersymmetric generalizations can be expressed for arbitrary values of the coupling constants as quadratic polynomials in the generators of a borel subalgebra of the lie algebra gl n 1 or the lie superalgebra gl n 1 n for the supersymmetric case these algebras are realized by first order differential operators this fact establishes the exact solvability of the models according to the general definition given by turbiner and implies that the calogero and jack sutherland polynomials as well as their supersymmetric generalizations are related to finite dimensional irreducible representations of the lie algebra gl n 1 and the lie superalgebra gl n 1 n

quasi exactly solvable problems and sl 2 algebra recently discovered quasi exactly solvable problems of quantum mechanics are shown to be related to the existence of the finite dimensional representations of the groupsl 2 q whereq r c it is proven that the bilinear formh a j j b j stand for the generators allows one to generate a set of quasi exactly solvable problems of different types including those that are already known we get in particular problems in which the spectral riemannian surface containing an infinite number of sheets is split off one or two finite sheet pieces in the general case the transitionh h d2 dx2 v x is realized with the aim of the elliptic functions all known exactly solvable quantum problems with known spectrum and factorized riemannian surface can be obtained in this approach

global bifurcation for neumann problems without eigenvalues the spectrum of this linear problem consists of the half line 0 co and contains no eigenvalue corresponding to an eigenfunction in l because of this the nonlinear sturm liouville problem 1 1 is not one to which standard bifurcation theory applies nevertheless the question of bifurcation of nontrivial solutions from the line k a 0 a e ir of trivial solutions has been considered and we shall be concerned here with some further developments of the theory in a recent paper 5 kupper and riemer proved among other things that if y is a constant then nontrivial solutions of 1 1 bifurcate from g at 0 o in r x w2 2 0 a if and only if 0 u 4 their methods however depend crucially on the autonomous nature of the problem when r is constant in 7 and 81 stuart and in lo chiapinelli and stuart have considered the general question of the bifurcation of solutions of nonlinear operator equaions when the linearised problem about the trivial solution has no eigenvalues the variational approach of 7 8 has no difficulty in treating 82 0022 0396 82 040082 29 02 00 o

the effect of two stage sampling on the f statistic abstract the assumption of iid observations that underlies many statistical procedures is called into question when analyzing complex survey data the population structure particularly the existence of clusters in two stage samples that usually exhibit positive intracluster correlation invalidates the independence assumption kish and frankel 1974 investigated the impact of this fact on regression analysis by using the standard sample survey theory framework campbell 1977 and scott and holt 1982 used the linear model framework in general although ordinary least squares ols procedures are unbiased but not fully efficient for estimation of the regression coefficients serious difficulties can arise when using ols estimators for second order terms variances of the ols estimators for the regression coefficients can be larger sometimes much larger than the usual ols variance expression would indicate failure to consider this possibility leads to underestimation of variances with consequences fo

efficient estimation for rank based regression with clustered data rank based inference is widely used because of its robustness this article provides optimal rank based estimating functions in analysis of clustered data with random cluster effects the extensive simulation studies carried out to evaluate the performance of the proposed method demonstrate that it is robust to outliers and is highly efficient given the existence of strong cluster correlations the performance of the proposed method is satisfactory even when the correlation structure is misspecified or when heteroscedasticity in variance is present finally a real dataset is analyzed for illustration

traditional and rank based tests for ordered alternatives in a cluster correlated model methods for the analysis of one factor randomized groups designs with ordered treatments are well established but they do not apply in the case of more complex experiments this article describes ordered treatment methods based on maximum likelihood and robust estimation that apply to designs with clustered data including those with a vector of covariates the contrast coefficients proposed for the ordered treatment estimates yield higher power than those advocated by abelson and tukey the proposed robust estimation method is shown using theory and simulation to yield both high power and robustness to outliers extensions for nonmonotonic alternatives are easily obtained

reply to comment on the darboux transformation and algebraic deformations of shape invariant potentials we reply to the comment on our recent paper made by drs sinha and roy we agree that the backward darboux transformation method used in our paper is equivalent to the approach based on ces potentials but we stress that the emphasis and the results of our paper are different

a global branch of solutions to a semilinear equation on an unbounded interval keywords semilinear second order differential equation bifurcation asymptotic bifurcation neumann problem global branch reference ana article 1985 001view record in web of science record created on 2008 12 10 modified on 2016 08 08