information theoretic lengths of jacobi polynomials the information theoretic lengths of the jacobi polynomials p n x which are information theoretic measures renyi shannon and fisher of their associated rakhmanov probability density are investigated they quantify the spreading of the polynomials along the orthogonality interval 1 1 in a complementary but different way as the root mean square or standard deviation because contrary to this measure they do not refer to any specific point of the interval the explicit expressions of the fisher length are given the renyi lengths are found by the use of the combinatorial multivariable bell polynomials in terms of the polynomial degree n and the parameters the shannon length which cannot be exactly calculated because of its logarithmic functional form is bounded from below by using sharp upper bounds to general densities on 1 1 given in terms of various expectation values moreover its asymptotics is also pointed out finally several computational issues relative to these three quantities are carefully analyzed

partially monotone tensor spline estimation of the joint distribution function with bivariate current status data the analysis of the joint cumulative distribution function cdf with bivariate event time data is a challenging problem both theoretically and numerically this paper develops a tensor spline based sieve maximum likelihood estimation method to estimate the joint cdf with bivariate current status data the i splines are used to approximate the joint cdf in order to simplify the numerical computation of a constrained maximum likelihood estimation problem the generalized gradient projection algorithm is used to compute the constrained optimization problem based on the properties of b spline basis functions it is shown that the proposed tensor spline based nonparametric sieve maximum likelihood estimator is consistent with a rate of convergence potentially better than n 1 3 under some mild regularity conditions the simulation studies with moderate sample sizes are carried out to demonstrate that the finite sample performance of the proposed estimator is generally satisfactory

a donsker varadhan type of invariance principle letx 1 x 2 h be i i d random variables in the domain of attraction of a stable lawg and denote sn x1 xn ln a n 1 l n omega a n 1 sum limits j 0 n 1 chi a s j mathord left vphantom s j a n right kern nulldelimiterspace a n satisfiesa n 1 s n g large deviation probability estimates of donsker varadhan type are obtained forl n and these are then used to study the behavior of small values of s n a n these latter results are analogues of strassen s results which described the behavior of large values of s n a n when the limit law was gaussian the limiting constants are seen to depend only on the limit lawg and not on the distribution ofx 1 the techniques used are those developed by donsker and varadhan in their theory of large deviations

path integral treatment for a screened potential the propagator relative to an effective screened potential which generalizes the hulthen potential and the kepler problem is calculated via the path integration over the noncompact group su 1 1 the energy spectrum and the wave functions of bound and scattering states are deduced

dirac equation with hulthen potential an algebraic approach the energy levels of the dirac equation with scalar and vector hulthen type potentials are obtained by means of algebraic perturbation calculations which are based upon the dynamical group structure so 2 1 numerical results are given for the particular case when the strengths of the vector and scalar potentials are equal

simple variational approaches to eigenvalues in quantum theory the authors develop a very simple variational procedure for obtaining quite accurate analytical expressions for the eigenvalues of quantum mechanical models it consists of finding the minimum value of a properly built functional form for the energy in the phase space the method enables one to make use of available information about the analytic structure of the eigenenergies results are shown for the linear confirming potential model

locally efficient estimation with bivariate right censored data estimation of the survival curve for independently right censored bivariate failure time data is a problem that has been studied extensively over the past 20 years in this article we propose a new class of estimators for the bivariate survivor function based on locally efficient le estimation the le estimator takes bivariate estimators fn and gn of the distributions of the time variables t1 t2 and the censoring variables c1 c2 and maps them to the resulting estimator le if fn and gn are appropriate consistent estimators of f and g then le will be nonparametrically efficient thus the term locally efficient however if either fn or gn but not both is not a consistent estimator of f or g then le will still be consistent and asymptotically normally distributed we propose a locally efficient estimator that uses a consistent nonparametric estimator for g and allows the user to supply lower dimensional semiparametric or parametric working model for f because the estimator that we cho

group theoretic approach for a dirac particle in coulomb like potentials the energy levels of the dirac equation with a kohn sham ks potential are obtained using algebraic perturbation theory based on the dynamical group structure so 2 1 without making any non relativistic approximation it has been shown that this formalism reproduces the exact analytical result for the eigenvalues of the dirac equation with both vector and scalar coulomb potentials the lowest order results obtained from the analytical formulae are found to be in excellent agreement with exact numerical results

spin and pseudospin symmetry along with orbital dependency of the dirac hulthen problem abstract the role of the hulthen potential on the spin and pseudospin symmetry solutions is investigated systematically by solving the dirac equation with attractive scalar s r and repulsive vector v r potentials the spin and pseudospin symmetry along with orbital dependency pseudospin orbit and spin orbit dependent couplings of the dirac equation are included to the solution by introducing the hulthen square approximation this effective approach is based on forming the spin and pseudo centrifugal kinetic energy term from the square of the hulthen potential the analytical solutions of the dirac equation for the hulthen potential with the spin orbit and pseudospin orbit dependent couplings are obtained by using the nikiforov uvarov nu method the energy eigenvalue equations and wave functions for various degenerate states are presented for several spin orbital pseudospin orbital and radial quantum numbers under the condition of the spin and pseudospin symmetry

projections of planar mandelbrot random measures abstract let be a planar mandelbrot measure and its orthogonal projection on one of the principal axes we study the thermodynamic and geometric properties of we first show that is exact dimensional with dim min dim dim where is the bernoulli product measure obtained as the expectation of we also prove that is absolutely continuous with respect to if and only if dim dim our results provides a new proof of dekking grimmett falconer formula for the hausdorff and box dimension of the topological support of as well as a new variational interpretation we obtain the free energy function of on a wide subinterval 0 q c of r for q 0 1 it is given by a variational formula which sometimes yields phase transitions of order larger than 1 for q 1 it is given by min which can exhibit first order phase transitions this is in contrast with the analyticity of over 0 q c also we prove the validity of the multifractal formalism for at each q c 0

the shifted 1 n expansion and the energy eigenvalues of the hulthen potential for l not 0 the shifted 1 n expansion has been used to obtain the energy values of the hulthen potential for non zero values of the orbital angular momentum the analytic expression for the energies en l yield fairly accurate results for a wide range of values of n l and the screening parameter lambda the energy values obtained by this method have been compared and found to be in excellent agreement with pade approximation calculations

a universal chung type law of the iterated logarithm let x 1 x 2 be a sequence of independent and identically distributed random variables we find sequences of norming and centering constants n and n such that a universal chung type law of the iterated logarithm holds namely lim inf n max 1 k n s k k n n almost surely where s k denotes the sum of the first k of x 1 x 2 k 1 if the underlying distribution function is in the feller class we show that this lim inf is strictly positive with probability 1

quantum theory of anharmonic oscillators a variational and systematic general approximation method this is an investigation of the energy levels and wavefunctions of an anharmonic oscillator characterised by the potential 1 2 omega 2q2 lambda q4 as a lowest order approximation an extremely simple formula for energy levels ei 0 i 1 2 1 4 3 alphai alphai is derived i being the quantum number of the energy level which covers any lambda i alphai is the real positive root of a cubic equation yi alphai3 alphai2 1 0 with yi 6lambda 2i2 2i 1 2i 1 this formula reproduces the exact energy levels within an error of about 1 for any lambda i the worst case is 2 for i 0 lambda to infinity systematically higher orders of this perturbation theory are developed which contains the usual perturbation theory for the limiting case of small lambda but this perturbation theory is valid for any i lambda the second order perturbation theory reduces the errors of our lowest order results by a factor of about 1 5 in general various ranges large intermediate small of i lambda are investigated and compared with the exact values obtained by the montroll group 1975 1978 for i 0 1 even the fourth order perturbation calculation can be elaborated explicitly which reduces the error to about 0 01 for any lambda for small lambda it gives correct numerical coefficients up to lambda4 terms as it should

solutions of the spatially dependent mass dirac equation with the spin and pseudospin symmetry for the coulomb like potential we study the effect of spatially dependent mass function over the solution of the dirac equation with the coulomb potential in the 3 1 dimensions for any arbitrary spin orbit k state in the framework of the spin and pseudospin symmetry concept the analytic bound state energy eigenvalues and the corresponding upper and lower two component spinors of the two dirac particles are obtained by means of the nikiforov uvarov method in closed form this physical choice of the mass function leads to an exact analytical solution for the pseudospin part of the dirac equation the special cases k 1 l l 0 i e s wave the constant mass and the non relativistic limits are briefly investigated

exact hausdorff measure on the boundary of a galton watson tree a necessary and sufficient condition for the almost sure existence of an absolutely continuous with respect to the branching measure exact hausdorff measure on the boundary of a galton watson tree is obtained in the case where the absolutely continuous exact hausdorff measure does not exist almost surely a criterion which classifies gauge functions 0 according to whether 0 hausdorff measure of the boundary minus a certain exceptional set is zero or infinity is given important examples are discussed in four additional theorems in particular hawkes s conjecture in 1981 is solved problems of determining the exact local dimension of the branching measure at a typical point of the boundary are also solved

any l state solutions of the hulthen potential in arbitrary dimensions the energy spectra of the hulthen effective potential in d dimensions are obtained within the new quantization rule approach for any l states the interdimensional degeneracies among states are also presented in the case of d 3 the agreement between our results and those obtained by other methods is excellent

the exact hausdorff dimension of a branching set we obtain a critical function for which the hausdorff measure of a branching set generated by a simple galton watson process is positive and finite

some universal results on the behavior of increments of partial sums we establish very general one sided results on the lim sup behavior of increments of suitably normalized partial sums of i i d random variables our main results apply to arbitrary nondegenerate positive random variables which need not have any finite moments as a corollary we can show that such results also hold for not necessarily positive random variables whose negative parts have finite moment generating functions

absolute continuity of some semi selfdecomposable distributions and self similar measures p s numbers are introduced in relation to absolute continuity of some infinite bernoulli convolutions absolute continuity and continuous singularity of some semi selfdecomposable distributions are studied as marginal distributions of subordinators it is shown that these properties are widely different according as their spans are p v numbers or the reciprocals of p s numbers a simple example of a subordinator whose distribution is continuous singular for small time and absolutely continuous for large time is given absolute continuity of convolutions of multidimensional homogeneous self similar measures are also discussed

a test of non identifying restrictions and confidence regions for partially identified parameters abstract we propose an easily implementable test of the validity of a set of theoretical restrictions on the relationship between economic variables which do not necessarily identify the data generating process the restrictions can be derived from any model of interactions allowing censoring and multiple equilibria when the restrictions are parameterized the test can be inverted to yield confidence regions for partially identified parameters thereby complementing other proposals primarily chernozhukov et al chernozhukov v hong h tamer e 2007 estimation and confidence regions for parameter sets in econometric models econometrica 75 1243 1285

hausdorff and packing spectra large deviations and free energy for branching random walks in mathbb r d consider an mathbb r d valued branching random walk brw on a supercritical galton watson tree without any assumption on the distribution of this brw we compute almost surely and simultaneously the hausdorff and packing dimensions of the level sets e k of infinite branches in the boundary of the tree endowed with its standard metric along which the averages of the brw have a given closed connected set of limit points k this goes beyond multifractal analysis which only considers those level sets when k ranges in the set of singletons alpha alpha in mathbb r d we also give a 0 law for the hausdorff and packing measures of the level sets e and compute the free energy of the associated logarithmically correlated random energy model in full generality moreover our results complete the previous works on multifractal analysis by including the levels which do not belong to the range of the gradient of the free energy this covers in particular a situation that was until now badly understood namely the case where a first order phase transition occurs as a consequence of our study we can also describe the whole singularity spectrum of mandelbrot measures as well as the associated free energy function or lq spectrum when a first order phase transition occurs

lower tail probability estimates for subordinators and nondecreasing random walks f xn ex 00 oo and a is the infimum of the support of the distribution of x1 for a xn lk we obtain the asymptotic behavior of log p sn nxn as n oo under the additional assumption of stochastic compactness a stronger result is obtained which gives the asymptotic behavior of p sn nxn itself analogues of these results are given for subordinators when t o oo or t o 0

model averaging in semiparametric estimation of treatment effects choosing the covariates and functional form of the propensity score is an important choice for estimating treatment effects this paper proposes a data driven way of averaging the estimators over candidate specifications to resolve the specification uncertainty in the propensity score weighting estimation of the att the proposed procedures minimize the estimated mse of the att estimator in a local asymptotic framework we formulate model averaging as a statistical decision problem in a limit experiment and derive an averaging scheme that is bayes optimal with respect to a given prior the averaging estimator outperforms selection estimators and the estimators in any of the candidate models in terms of bayes asymptotic mse our monte carlo studies illustrate the size of the mse gains we apply the averaging procedure to evaluate the effect of a labor market program

optimal aggregation of classifiers in statistical learning classification can be considered as nonparametric estimation of sets where the risk is defined by means of a specific distance between sets associated with misclassification error it is shown that the rates of convergence of classifiers depend on two parameters the complexity of the class of candidate sets and the margin parameter the dependence is explicitly given indicating that optimal fast rates approaching o n 1 can be attained where n is the sample size and that the proposed classifiers have the property of robustness to the margin the main result of the paper concerns optimal aggregation of classifiers we suggest a classifier that automatically adapts both to the complexity and to the margin and attains the optimal fast rates up to a logarithmic factor

on profile likelihood abstract we show that semiparametric profile likelihoods where the nuisance parameter has been profiled out behave like ordinary likelihoods in that they have a quadratic expansion in this expansion the score function and the fisher information are replaced by the efficient score function and efficient fisher information the expansion may be used among others to prove the asymptotic normality of the maximum likelihood estimator to derive the asymptotic chi squared distribution of the log likelihood ratio statistic and to prove the consistency of the observed information as an estimator of the inverse of the asymptotic variance

sample function behavior of increasing processes of class l we consider increasing processes x t t 0 of classl that is increasing self similar processes with inswpendent increments leth t be an increasing positive function on 0 withh 0 0 andh by virtue of the zero one laws there existsc resp c 0 such that lim inf resp lim sup x t h t c resp c a s both ast tends to 0 and ast tends to we decide a necessary and sufficient condition for the existence ofh t withc orc 1 and explicitly constructh t in caseh t exists withc orc 1 moreover we give a criterion to classify functionsh t withc orc 0 andh t withc orc in caseh t does not exist withc orc 1

locally efficient estimation of a multivariate survival function in longitudinal studies we consider estimation of the joint distribution of multivariate survival times t t1 tk which are subject to right censoring by a common censoring variablec two estimators are proposed an initial inverse probability of censoring weighted ipcw estimator and a one step estimator both estimators incorporate information on available time independent and time dependent prognostic factor covariate data the ipcw estimator is consistent and asymptotically normal can under coarsening at random car and a correct specification of a model for the hazard of censoring given the past covariate and failure data the one step estimator is a locally efficient doubly robust estimator that is i it is can under the assumption of car and either but not necessarily both correct specification of a model for the hazard of censoring given the past or correct specification of a model for the conditional distribution of t given past failure and covariate information and ii it is efficient when both these

strong laws for the k th order statistic when k c log2n we give several bounds and characterizations for the lower almost sure classes of uk n where k kn denotes a nondecreasing integer sequence such that 1 kn 0 log2n as n and where u1 n un n denote the order statistics of the first n observations from a sequence of independent and uniformly distributed random variables on 0 1

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

smoothed quantile regression for panel data abstract this paper studies fixed effects estimation of quantile regression models for panel data under an asymptotic framework where both the numbers of individuals and time periods grow at the same rate we show that the fixed effects estimator for the smoothed objective function has a limiting normal distribution with a bias in the mean and provide the analytic form of the asymptotic bias we propose a one step bias correction estimator based on the analytic bias formula obtained from the asymptotic analysis importantly our results cover the case that observations are dependent over time we illustrate the effects of the bias correction through simulations

exact packing dimension in random recursive constructions we explore the exact packing dimension of certain random recursive constructions in case of polynomial decay at 0 of the distribution function of random variable x associated with the construction we prove that it does not exist and in case of exponential decay it is t log logt where is the fractal dimension of the limit set and 1 is the rate of exponential decay

on the bootstrap of the maximum score estimator this paper shows that the bootstrap does not consistently estimate the asymptotic distribution of the maximum score estimator the theory developed also applies to other estimators within a cube root convergence class for some single parameter estimators in this class the results suggest a simple method for inference based upon the bootstrap

bound state solutions of the dirac equation for the trigonometric and hyperbolic scarf grosche potentials using the nikiforov uvarov method the relativistic dirac equation with equal trigonometric and hyperbolic scarf grosche scalar and vector potentials in 3 1 and 1 1 space time dimension are solved by using the nikiforov uvarov method the exact energy equations and the spinor wave functions are obtained for these sets of potentials the bound states energy are calculated numerically also with appropriate transformations solutions of the fourth potential v4 can be deduced from the solutions of the third potential v3

exact solution of the schrodinger and klein gordon equations for generalised hulthen potentials for the class of generalised hulthen s wave potentials vgh p r sigma i 1p giai exp gamma ir 1 sigma j 1p aj exp gamma jr p or 1 the exact solution is presented for both the schrodinger and klein gordon equations this simultaneous solubility represents a new non perturbative treatment of the kinematical relativistic corrections and is based on a non standard generalisation of the hypergeometric function

exact solution of the n dimensional generalized dirac coulomb equation an exact solution to the bound state problem for the n dimensional generalized dirac coulomb equation whose potential contains both the lorentz vector and lorentz scalar terms of the coulomb form is obtained

random walks and percolation on trees there is a way to define an average number of branches per vertex for an arbitrary infinite locally finite tree it equals the exponential of the hausdorff dimension of the boundary in an appropriate metric its importance for probabilistic processes on a tree is shown in several ways including random walk and percolation where it provides points of phase transition

semi selfdecomposable distributions and a new class of limit theorems self decomposable distributions are given as limits of normalized sums of independent random variables we define semi selfdecomposable distributions as limits of subsequences of normalized sums more generally we introduce a way of making a new class of limiting distributions derived from a class of distributions by taking the limits through subsequences of normalized sums and define the class of semi selfdecomposable distributions and a decreasing sequence of subclasses of it we give two kinds of necessary and sufficient conditions for distributions belonging to those classes one is in terms of the decomposability of random variables and another is in terms of levy measures

the class of limit laws for stochastically compact normed sums khintchine showed that every infinitely divisible law can be obtained as the limit of a subsequence of normed sums of independent identically distributed random variables here we restrict the summands to be in a class which makes the normed sums stochastically compact i e so that every subsequence has a further subsequence which converges to a nondegenerate limit a nice analytic condition for stochastic compactness was obtained by feller our result is an analogous characterization of the class of limit laws of subsequences of stochastically compact normed sums one consequence is that they have c infty densities

a recurrence formula for obtaining certain matrix elements in the base of eigenfunctions of the hamiltonian for a particular screened potential a recurrence formula for obtaining matrix elements of products of powers of exp nu r and 1 exp nu r is derived the matrix elements are calculated in the base of eigenfunctions of the hamiltonian for the effective potential lambda exp nu r 1 exp nu r mu exp nu r 1 exp nu r 2 where lambda nu and mu are positive constants this potential can be considered as a generalisation of a potential suggested by hylleraas and risberg 1941 and by hulthen 1942 in the limit of the parameter nu tending to zero the recurrence formula is transformed into a recurrence formula is transformed into a recurrence formula given by badawi et al 1973 for matrix elements of powers of r for the hydrogen atom

nested classes of c decomposable laws a random variable x is c decomposable if x d cx y c for all c in c where y c is a random variable independent of x and c is a closed multiplicative subsemigroup of 0 1 x is self decomposable if c 0 1 extending an idea of urbanik in the self decomposable case we define a decreasing sequence of subclasses of the class of c decomposable laws for any c we give a structural representation for laws in these classes and we show that laws in the limiting subclass are infinitely divisible we also construct noninfinitely divisible examples some of which are continuous singular

covariance and survivor function estimation using censored multivariate failure time data summary the covariance between counting process martingales is used to characterize the dependence between two failure time variates a representation of the bivariate survivor function is obtained in terms of the marginal survivor functions and this covariance function a closely related representation expresses the bivariate survivor function in terms of marginal survivor functions and a conditional covariance function leading to a new nonparametric survivor function estimator generalizations to higher dimensional failure time variates are also given simulation evaluations of the survivor function estimator are presented and generalizations to regression problems are outlined

the small time chung wichura law for levy processes with non vanishing brownian component we give a small time functional version of chung s other law of the iterated logarithm for levy processes with non vanishing brownian component this is an analogue of the other law of the iterated logarithm at large times for levy processes and random walks with finite variance as extended to a functional version by wichura as one of many possible applications we mention a functional law for a two sided passage time process

a simple approximation for eigenvalues in quantum theory an approximation method for the determination of energy levels in quantum theory is discussed which starts from a scaled set of eigenstates of a solvable model it is shown that the lowest approximation fulfils certain relations which also hold exactly the approximation is tested by comparison with numerically computed eigenvalues in several cases the errors turn out to be moderate in most of these and depend very little on typical coupling constants

solution of the relativistic dirac hulthen problem the one particle three dimensional dirac equation with spherical symmetry is solved for the hulthen potential the s wave relativistic energy spectrum and two component spinor wavefunctions are obtained analytically conforming to the standard feature of the relativistic problem the solution space splits into two distinct subspaces depending on the sign of a fundamental parameter in the problem unique and interesting properties of the energy spectrum are pointed out and illustrated graphically for several values of the physical parameters the square integrable two component wavefunctions are written in terms of the jacobi polynomials the nonrelativistic limit reproduces the well known nonrelativistic energy spectrum and results in schrodinger equation with a generalized three parameter hulthen potential which is the sum of the original hulthen potential and its square

using a monotone single index model to stabilize the propensity score in missing data problems and causal inference the augmented inverse weighting method is one of the most popular methods for estimating the mean of the response in causal inference and missing data problems an important component of this method is the propensity score popular parametric models for the propensity score include the logistic probit and complementary log log models a common feature of these models is that the propensity score is a monotonic function of a linear combination of the explanatory variables to avoid the need to choose a model we model the propensity score via a semiparametric single index model in which the score is an unknown monotonic nondecreasing function of the given single index under this new model the augmented inverse weighting estimator aiwe of the mean of the response is asymptotically linear semiparametrically efficient and more robust than existing estimators moreover we have made a surprising observation the inverse probability weighting and aiwes based on a correctly specified parametric model may have worse performance than their counterparts based on a nonparametric model a heuristic explanation of this phenomenon is provided a real data example is used to illustrate the proposed methods

variational perturbation theory for summing divergent tunnelling amplitudes we present a method for evaluating divergent series with factorially growing coefficients of equal sign the method is based on an analytic continuation of variational perturbation theory from the regime of alternating signs we demonstrate its power first by applying it to the exactly known partition function of the anharmonic oscillator in zero space time dimensions the simple integral then we consider the quantum mechanical case of one space time dimension and derive the imaginary part of the ground state energy of the anharmonic oscillator for all negative values of the coupling constant g including the non analytic tunnelling regime at small g as a highlight of the theory we extract from the divergent perturbation expansion the action of the critical bubble and the contribution of the higher loop fluctuations around the bubble some figures in this article are in colour only in the electronic version

kaplan meier estimate on the plane estimation of the bivariate survival function from censored data is considered the product integral representation of univariate survival functions is generalized to the bivariate case and used to determine identifiability of the survival function of the partially observed data a bivariate analogue of the kaplan meier estimate is introduced and its almost sure consistency is studied extensions to the general multivariate case are sketched

on the smallest maximal increment of partial sums of i i d random variables we study the almost sure limiting behavior of the smallest maximal increment of partial sums of n independent identically distributed random variables for a variety of increment sizes k n where k n is a sequence of integers satisfying 1 le k n le n and going to infinity at various rates our aim is to obtain universal results on such behavior under little or no assumptions on the underlying distribution function

extended supersymmetry for the bound states of the generalized hulthen potential hierarchy using the associated hypergeometric differential equation we analytically solve the bound states corresponding to a hierarchy of the radial potential v0 e r 1 e r c e r 1 e r 2 as a generalization of the hulth n potential then an analytic solution corresponding to a special case for which the parameter c is expected to be in terms of l l 1 is also derived meanwhile without introducing a superpotential and in the framework of supersymmetric quantum mechanics it is shown that these bound states can be calculated by two different algebraic methods based on these two approaches it is noted that the bound states realize an extended supersymmetry structure

an approximate κ state solutions of the dirac equation for the generalized morse potential under spin and pseudospin symmetry by using an improved approximation scheme to deal with the centrifugal pseudo centrifugal term we solve the dirac equation for the generalized morse potential with arbitrary spin orbit quantum number in the presence of spin and pseudospin symmetry the analytic bound state energy eigenvalues and the associated upper and lower spinor components of two dirac particles are found by using the basic concepts of the nikiforov uvarov method we study the special cases when 1 l l 0 s wave the non relativistic limit and the limit when becomes zero kratzer potential model the present solutions are compared with those obtained by other methods

an algebraic approach to the exponential cosine screened coulomb potential an analytical formula is obtained for the energy eigenvalues of the exponential cosine screened coulomb potential by a method that is a combination of algebraic perturbation theory based on the lie algebra of the so 2 1 group and the scaling variational method the results are found to be in excellent agreement with pade approximation calculations

parametric rate inference for one sided differentiable parameters suppose one has a collection of parameters indexed by a possibly infinite dimensional set given data generated from some distribution the objective is to estimate the maximal parameter in this

shifted 1 n expansion and exact solutions for the potential v r z r gr λr2 the authors have found an infinite number of exact solutions for the hydrogenic atom in the external potential v r gr r2 not only for an s wave state but for higher waves as well from supersymmetric considerations the general schrodinger equation has been treated by the shifted 1 n expansion method the eigenvalues obtained from the shifted 1 n expansion are compared with those obtained by bessis et al 1987 and also with the supersymmetric exact values

the lower limit of a normalized random walk soit sn une marche aleatoire et n une suite de constantes telles que n n est non decroissante on donne un test integral universel qui determine la limite inferieure de sn n jusqu a une echelle constante pour lim sup 2n n

shifted 1 n expansion and scalar potential in the dirac equation the shifted 1 n expansion method has been applied to a linear scalar potential in the dirac equation to obtain the relativistic energy eigenvalues the results are compared with those obtained by rein 1977 and by gunion and li 1975

self similar processes with independent increments a stochastic process x t t 0 onr d is called wide sense self similar if for eachc 0 there are a positive numbera and a functionb t such that x ct and ax t b t have common finite dimensional distributions if x t is widesense self similar with independent increments stochastically continuous andx 0 const then for everyt the distribution ofx t is of classl conversely if is a distribution of classl then for everyh 0 there is a unique process x selfsimilar with exponenth with independent increments such thatx 1 has distribution consequences of this characterization are discussed the properties finitedimensional distributions behaviors for small time etc of the process x called the process of classl with exponenth induced by are compared with those of the levy process y t such thaty 1 has distribution results are generalized to operator self similar processes and distributions of classol a process x t onr d is called wide sense operator self similar if for eachc 0 there are a linear operatora c and a functionb c t such that x ct and a c x t b c t have common finite dimensional distributions it is proved that if x t is wide sense operator self similar and stochastically continuous then thea c can be chosen asa c c q with a linear operatorq with some special spectral properties this is an extension of a theorem of hudson and mason 4

bound states of the potential v r ze2 r β energy eigenvalues of various states from n 1 to n 4 have been calculated in the framework of the shifted 1 n expansion method results for s states are compared with those of mehta and patil 1978 for small values a new approximate formula has been developed for a quick estimate of energy eigenvalues for any angular momentum state

an extension of hawkes theorem on the hausdorff dimension of a galton watson tree let be the genealogical tree of a supercritical multitype galton watson process and let be the limit set of i e the set of all infinite self avoiding paths called ends through that begin at a vertex of the first generation the limit set is endowed with the metric d 2 n where n n is the index of the first generation where and differ to each end is associated the infinite sequence of types of the vertices of let be the space of all such sequences for any ergodic shift invariant probability measure on define to be the set of all generic sequences i e the set of all sequences such that each finite sequence v occurs in with limiting frequency v where v is the set of all that begin with the word v then the hausdorff dimension of 1 in the metric d is

nonparametric estimation of a bivariate survival function in the presence of censoring a new family of estimators of a bivariate survival function based on censored vectors is obtained from a decomposition of the bivariate survival function these estimators are uniformly consistent under bivariate censoring and are self consistent under univariate censoring

likelihood ratio tests for monotone functions we study the problem of testing for equality at a fixed point in the setting of nonparametric estimation of a monotone function the likelihood ratio test for this hypothesis is derived in the particular case of interval censoring or current status data and its limiting distribution is obtained the limiting distribution is that of the integral of the difference of the squared slope processes corresponding to a canonical version of the problem involving brownian motion t 2 and greatest convex minorants thereof inversion of the family of tests yields pointwise confidence intervals for the unknown distribution function we also study the behavior of the statistic under local and fixed alternatives 1 introduction we shall consider likelihood ratio tests and the corresponding confidence intervals in a class of problems involving nonparametric estimation of a monotone function the problem in each case involves testing the null hypothesis h 0 that the monotone function has a particular value at afixed point t 0 in the domain of the function of course with each testing problem there is a related problem of finding confidence intervals here are some examples of the problems we have in mind

efficient estimation in the bivariate censoring model and repairing npmle the npmle in the bivariate censoring model is not consistent the problem is caused by the singly censored observations in this paper we prove that the npmle based on interval censoring the singly censored observations is ecient for this reduced data and moreover if we let the width of the interval converge to zero slowly enough then the npmle is also ecient for the original data we are able to determine a lower bound for the rate at which the bandwidth should converge to zero the eciency proof uses the general identity which holds for npmle of a linear parameter in convex models as proved in van der laan 1993a

solving the schrodinger equation with use of 1 n perturbation theory the large n expansion provides a powerful new tool for solving the schrodinger equation in this paper we present simple recursion formulas which facilitate the calculation we do some numerical calculations which illustrate the speed and accuracy of the technique

a universal one sided law of the iterated logarithm we prove that the lim inf of suitably normalized sums of i i d nonnegative and nondegenerate random variables can with probability 1 only be a constant between 2 1 2 and 0 moreover we show that each value within this range is attainable by an appropriate choice of the underlying common distribution function

asymptotic iteration method for eigenvalue problems an asymptotic iteration method for solving second order homogeneous linear differential equations of the form y 0 x y s0 x y is introduced where 0 x 0 and s0 x are c functions applications to schrodinger type problems including some with highly singular potentials are presented