collision in a cross shaped domain a steady 2d navier stokes example demonstrating the importance of mass conservation in cfd abstract in the numerical simulation of the incompressible navier stokes equations different numerical instabilities can occur while instability in the discrete velocity due to dominant convection and instability in the discrete pressure due to a vanishing discrete ladyzhenskaya babuska brezzi lbb constant are well known instability in the discrete velocity due to a poor mass conservation at high reynolds numbers sometimes seems to be underestimated at least when using conforming galerkin mixed finite element methods like the taylor hood element the classical grad div stabilization for enhancing discrete mass conservation is often neglected in practical computations though simple academic flow problems showing the importance of mass conservation are well known these examples differ from practically relevant ones since specially designed force vectors are prescribed therefore we present a simple steady navier stokes problem in two space dimensions at reynolds number 1024 a colliding flow in a cross shaped domain where the instability of poor mass conservation is studied in detail and where no force vector is prescribed

stabilized finite element methods based on multiscale enrichment for the stokes problem this work concerns the development of stabilized finite element methods for the stokes problem considering nonstable different or equal order of velocity and pressure interpolations the approach is based on the enrichment of the standard polynomial space for the velocity component with multiscale functions which no longer vanish on the element boundary on the other hand since the test function space is enriched with bubble like functions a petrov galerkin approach is employed we use such a strategy to propose stable variational formulations for continuous piecewise linear in velocity and pressure and for piecewise linear piecewise constant interpolation pairs optimal order convergence results are derived and numerical tests validate the proposed methods

a conservative stable finite element method for stokes flow and nearly incompressible linear elasticity on rectangular grid abstract in this paper we discuss finite element methods for the incompressible stokes problem and the nearly incompressible linear elasticity problem specifically we present a finite element pair for the incompressible stokes problem which satisfies the discrete inf sup condition and the discrete korn s inequality and moreover which is element wise conservative the pair provides a locking free method for the nearly incompressible linear elasticity problem without reduced integration

second order locking free nonconforming elements for planar linear elasticity in this paper we present two nonconforming finite elements for the pure displacement planar elasticity problem both of them are locking free and have two order of convergence some numerical results attest the validity of our theoretical analysis

least squares methods for linear elasticity based on a discrete minus one inner product abstract the purpose of this paper is to develop and analyze least squares approximations for elasticity problems the major advantage of the least squares formulation is that it does not require that the classical ladyzhenskaya bab ska brezzi lbb condition be satisfied by employing least squares functionals which involve a discrete inner product which is related to the inner product in h 1 the sobolev space of order minus one on we develop a finite element method which is unconditionally stable for problems with traction type of boundary conditions and for almost and incompressible elastic media the use of such inner products applied to second order problems was proposed in an earlier paper by bramble lazarov and pasciak math comp 66 1997 935

misspecified structural change threshold and markov switching models sudden perturbations of a large amplitude occur frequently in macroeconomic and financial time series a usual practice is to test linearity against a permanent structural change however changes can also be captured by nonlinear stationary models such that threshold and markov switching models in this paper we show that tests designed for a threshold alternative have also power against parameter instability originating from structural change or markov switching models on the other hand it is shown that tests for structural change have no power if the data are generated by a markov switching or threshold model therefore it appears that testing the null of parameter stability against a threshold alternative is a robust way to detect parameter instability in economic and financial time series a monte carlo analysis based on several models studied in the literature illustrates how the tests perform in practice

convergence in the incompressible limit of finite element approximations based on the hu washizu formulation the classical hu washizu mixed formulation for plane problems in elasticity is examined afresh with the emphasis on behavior in the incompressible limit the classical continuous problem is embedded in a family of hu washizu problems parametrized by a scalar for which alpha lambda big mu corresponds to the classical formulation with and being the lame parameters uniform well posedness in the incompressible limit of the continuous problem is established for 1 finite element approximations are based on the choice of piecewise bilinear approximations for the displacements on quadrilateral meshes conditions for uniform convergence are made explicit these conditions are shown to be met by particular choices of bases for stresses and strains and include bases that are well known as well as newly constructed bases though a discrete version of the spherical part of the stress exhibits checkerboard modes it is shown that a independent a priori error estimate for the displacement can be established furthermore a independent estimate is established for the post processed stress the theoretical results are explored further through selected numerical examples

stable cheapest nonconforming finite elements for the stokes equations we introduce two pairs of stable cheapest nonconforming finite element space pairs to approximate the stokes equations one pair has each component of its velocity field to be approximated by the p 1 nonconforming quadrilateral element while the pressure field is approximated by the piecewise constant function with globally two dimensional subspaces removed one removed space is due to the integral mean zero property and the other space consists of global checker board patterns the other pair consists of the velocity space as the p 1 nonconforming quadrilateral element enriched by a globally one dimensional macro bubble function space based on d s s y douglas santos sheen ye nonconforming finite element space the pressure field is approximated by the piecewise constant function with mean zero space eliminated we show that two element pairs satisfy the discrete inf sup condition uniformly and we investigate the relationship between them several numerical examples are shown to confirm the efficiency and reliability of the proposed methods

korn s inequalities for piecewise h 1 vector fields korn s inequalities for piecewise h 1 vector fields are established they can be applied to classical nonconforming finite element methods mortar methods and discontinuous galerkin methods

grad div stablilization for stokes equations in this paper a stabilizing augmented lagrangian technique for the stokes equations is studied the method is consistent and hence does not change the continuous solution we show that this stabilization improves the well posedness of the continuous problem for small values of the viscosity coefficient we analyze the influence of this stabilization on the accuracy of the finite element solution and on the convergence properties of the inexact uzawa method

nodal auxiliary space preconditioning in h curl and h div spaces in this paper we develop and analyze a general approach to preconditioning linear systems of equations arising from conforming finite element discretizations of h curl and h div elliptic variational problems the preconditioners exclusively rely on solvers for discrete poisson problems we prove mesh independent effectivity of the preconditioners by using the abstract theory of auxiliary space preconditioning the main tools are discrete analogues of so called regular decomposition results in the function spaces h curl and h div our preconditioner for h curl is similar to an algorithm proposed in r beck algebraic multigrid by component splitting for edge elements on simplicial triangulations tech rep sc 99 40 zib berlin germany 1999

analysis of time series subject to changes in regime abstract this paper introduces an em algorithm for obtaining maximum likelihood estimates of parameters for processes subject to discrete shifts in autoregressive parameters with the shifts themselves modeled as the outcome of a discrete valued markov process the simplicity of the em algorithm permits potential application of the approach to large vector systems

nonconforming finite element methods for the equations of linear elasticity in the adaptation of nonconforming finite element methods to the equations of elasticity with traction boundary conditions the main difficulty in the analysis is to prove that an appropriate discrete version of korn s second inequality is valid such a result is shown to hold for nonconforming piecewise quadratic and cubic finite elements and to be false for nonconforming piecewise linears optimal order error estimates uniform for poisson ratio v e 0 1 2 are then derived for the corresponding p2 and p3 methods this contrasts with the use of c finite elements where there is a deterioration in the convergence rate as v 1 2 for piecewise polynomials of degree 3 modifications of the continuous methods and the nonconforming linear method which also give uniform optimal order error estimates are discussed

neglected heterogeneity in moment condition models abstract the central concern of this paper is parameter heterogeneity in models specified by a number of unconditional or conditional moment conditions and thereby the provision of a framework for the development of apposite optimal m tests against its potential presence we initially consider the unconditional moment restrictions framework optimal m tests against moment condition parameter heterogeneity are derived with the relevant jacobian matrix obtained in terms of the second order own derivatives of the moment indicator in a leading case gmm and gel tests of specification based on generalized information matrix equalities appropriate for moment based models are described and their relation to optimal m tests against moment condition parameter heterogeneity examined a fundamental and important difference is noted between gmm and gel constructions the paper is concluded by a generalization of these tests to the conditional moment context and the provision of a limited set of simulation experiments to illustrate the efficacy of the proposed tests

a linear nonconforming finite element method for nearly incompressible elasticity and stokes flow abstract we introduce a new triangular element for nearly incompressible elasticity and incompressible fluid flow the method consists of conforming linear elements for one of the displacement or velocity for flows component and linear non conforming elements for the other component the element is proved to give an optimal approximation and this is also confirmed by several numerical examples

on mass conservation in least squares methods we compare three least squares finite element reformulations of the stokes equations paying particular attention to mass conservation the first problem we approximate has a simple analytical solution over a convex region even for this simple problem without special treatment of the conservation of mass term very poor numerical solutions may result sufficiently weighting this term leads to a dramatic improvement in the results over a range of test problems

a geometric structure preserving discretization scheme for incompressible linearized elasticity abstract in this paper we present a geometric discretization scheme for incompressible linearized elasticity we use ideas from discrete exterior calculus dec to write the action for a discretized elastic body modeled by a simplicial complex after characterizing the configuration manifold of volume preserving discrete deformations we use hamilton s principle on this configuration manifold the discrete euler lagrange equations are obtained without using lagrange multipliers the main difference between our approach and the mixed finite element formulations is that we simultaneously use three different discrete spaces for the displacement field we explicitly derive the governing equations for the two dimensional case where the discrete spaces for the displacement field are constructed by p 1 polynomials over primal meshes for incompressibility constraint p 0 polynomials over dual meshes for the kinetic energy and p 1 polynomials over support volumes for the elastic energy and the discrete space of the pressure field is constructed by p 0 polynomials over primal meshes we test the efficiency and robustness of this geometric scheme using some numerical examples in particular we do not see any volume locking and or checkerboarding of pressure in our numerical examples this suggests that our choice of discrete solution spaces is compatible

bootstrapping the gmm overidentification test under first order underidentification abstract the main contribution of this paper is to study the applicability of the bootstrap to estimating the distribution of the standard test of overidentifying restrictions of hansen 1982 when the model is globally identified but the rank condition fails to hold lack of first order local identification an important example for which these conditions are verified is the popular test of common conditionally heteroskedastic features proposed by engle and kozicki 1993 as dovonon and renault 2013b show the jacobian matrix for this model is identically zero at the true parameter value resulting in a highly nonstandard limiting distribution that complicates the computation of critical values we first show that the standard gmm bootstrap fails to consistently estimate the distribution of the overidentification restrictions test under lack of first order identification we then propose a new bootstrap method that is asymptotically valid in this context the modification consists of adding an additional term that recenters the bootstrap moment conditions in a way as to ensure that the bootstrap jacobian matrix is zero when evaluated at the gmm estimate

simple tests of alternative specifications in stochastic frontier models abstract previous work on stochastic production frontiers has generated a family of models of varying degrees of complexity since this family is nested in the sense that the more general models contain the less general we can test the restrictions that distinguish the model in this paper we provide tests of these restrictions based on the results of estimating the simpler restricted models some of our tests are lm tests however in other cases the lm test fails so we provide alternative simple tests

stabilized finite element methods for the generalized oseen problem the numerical solution of the non stationary incompressible navierstokes model can be split into linearized auxiliary problems of oseen type we present in a unique way different stabilization techniques of finite element schemes on isotropic meshes first we describe the state of the art for the classical residual based supg pspg method then we discuss recent symmetric stabilization techniques which avoid some drawbacks of the classical method these methods are closely related to the concept of variational multiscale methods which seems to provide a new approach to large eddy simulation finally we give a critical comparison of these methods

mean field theory of directed polymers with random complex weights we show that for the problem of directed polymers on a tree with i i d random complex weights on each bond three possible phases can exist the phase of a particular system is determined by the distribution of the random weights for each of these three phases we give the expression of the free energy per unit length in the limit of infinitely long polymers our proofs require several hypotheses on the distribution most importantly that the amplitude and the phase of each complex weight be statistically independent the main steps of our proofs use bounds on noninteger moments of the partition function and self averaging properties of the free energy we illustrate our results by some examples and discuss possible generalizations to a larger class of distributions to random energy models and to the finite dimensional case we note that our results are not in agreement with the predictions of a recent replica approach to a similar problem

linear finite element methods for planar linear elasticity a linear nonconforming conforming displacement finite element method for the pure displacement pure traction problem in two dimensional linear elasticity for a homogeneous isotropic elastic material is considered in the case of a convex polygonal configuration domain error estimates in the energy l sup 2 norm are obtained the convergence rate does not deteriorate for nearly incompressible material furthermore the convergence analysis does not rely on the theory of saddle point problems 22 refs

testing for common conditionally heteroskedastic factors this paper proposes a test for common conditionally heteroskedastic ch features in asset returns following engle and kozicki 1993 the common ch features property is expressed in terms of testable overidentifying moment restrictions however as we show these moment conditions have a degenerate jacobian matrix at the true parameter value and therefore the standard asymptotic results of hansen 1982 do not apply we show in this context that hansen s 1982 j test statistic is asymptotically distributed as the minimum of the limit of a certain random process with a markedly nonstandard distribution if two assets are considered this asymptotic distribution is a fifty fifty mixture of and where h is the number of moment conditions as opposed to a with more than two assets this distribution lies between the and p denotes the number of parameters these results show that ignoring the lack of first order identification of the moment condition model leads to oversized tests with a possibly increasing overrejection rate with the number of assets a monte carlo study illustrates these findings proquest denotes formulae symbols omitted

inference for non random samples observational data are often analysed as if they had resulted from a controlled study and yet the tacit assumption of randomness can be crucial for the validity of inference we take some simple statistical models and supplement them by adding a parameter which reflects the degree of non randomness in the sample for a randomized study is known to be 0 we examine the profile log likelihood for and the sensitivity of inference to small non zero values of particular models cover the analysis of survey data with item non response the paired comparison t test and two group comparisons using observational data with covariates some practical examples are discussed allowing for sampling bias increases the uncertainty of estimation and weakens the significance of treatment effects sometimes substantially so

equivalence of finite element methods for problems in elasticity modifications of the morley method for the approximation of the biharmonic equation are obtained from various finite element methods applied to the equations of linear isotropic elasticity and the stationary stokes equations by elimination procedures analogous to those used in the continuous case problems with korn s first inequality for nonconforming p 1 elements and its implications for the approximation of the elasticity equations are also discussed

a locally conservative ldg method for the incompressible navier stokes equations in this paper a new local discontinuous galerkin method for the incompressible stationary navier stokes equations is proposed and analyzed four important features render this method unique its stability its local conservativity its high order accuracy and the exact satisfaction of the incompressibility constraint although the method uses completely discontinuous approximations a globally divergence free approximate velocity in h div is obtained by simple element by element post processing optimal error estimates are proven and an iterative procedure used to compute the approximate solution is shown to converge this procedure is nothing but a discrete version of the classical fixed point iteration used to obtain existence and uniqueness of solutions to the incompressible navier stokes equations by solving a sequence of oseen problems numerical results are shown which verify the theoretical rates of convergence they also confirm the independence of the number of fixed point iterations with respect to the discretization parameters finally they show that the method works well for a wide range of reynolds numbers

testing for regime switching we analyze use of a quasi likelihood ratio statistic for a mixture model to test the null hypothesis of one regime versus the alternative of two regimes in a markov regime switching context this test exploits mixture properties implied by the regime switching process but ignores certain implied serial correlation properties when formulated in the natural way the setting is nonstandard involving nuisance parameters on the boundary of the parameter space nuisance parameters identified only under the alternative or approximations using derivatives higher than second order we exploit recent advances by andrews 2001 and contribute to the literature by extending the scope of mixture models obtaining asymptotic null distributions different from those in the literature we further provide critical values for popular models or bounds for tail probabilities that are useful in constructing conservative critical values for regime switching tests we compare the size and power of our statistics to other useful tests for regime switching via monte carlo methods and find relatively good performance we apply our methods to reexamine the classic cartel study of porter 1983 and reaffirm porter s findings

stokes complexes and the construction of stable finite elements with pointwise mass conservation two families of conforming finite elements for the two dimensional stokes problem are developed guided by two discrete smoothed de rham complexes which we coin stokes complexes we show that the finite element pairs are inf sup stable and also provide pointwise mass conservation on very general triangular meshes

least squares finite element method for the stokes problem with zero residual of mass conservation in this paper the simulation of incompressible flow in two dimensions by the least squares finite element method lsfem in the vorticity velocity pressure version is studied in the lsfem the equations for continuity of mass and momentum and a vorticity equation are minimized on a discretization of the domain of interest a problem is these equations are minimized in a global sense thus this method may not enforce that div underline u 0 at every point of the discretization in this paper a modified lsfem is developed which enforces near zero residual of mass conservation i e div underline u is nearly zero at every point of the discretization this is accomplished by adding an extra restriction in the divergence free equation through the lagrange multiplier strategy in this numerical method the inf sup or say lbb condition is not necessary and the matrix resulting from applying the method on a discretization is symmetric the uniqueness of the solution and the application of the conjugate gradient method are also valid numerical experience is given in simulating the flow of a cylinder with diameter 1 moving in a narrow channel of width 1 5 results obtained by the lsfem show that mass is created or destroyed at different points in the interior of discretization the results obtained by the modified lsfem show the mass is nearly conserved everywhere

optimal test for markov switching parameters this paper proposes a class of optimal tests for the constancy of parameters in random coefficients models our testing procedure covers the class of hamilton s models where the parameters vary according to an unobservable markov chain but also applies to nonlinear models where the random coefficients need not be markov we show that the contiguous alternatives converge to the null hypothesis at a rate that is slower than the standard rate therefore standard approaches do not apply we use bartlett type identities for the construction of the test statistics this has several desirable properties first it only requires estimating the model under the null hypothesis where the parameters are constant second the proposed test is asymptotically optimal in the sense that it maximizes a weighted power function we derive the asymptotic distribution of our test under the null and local alternatives asymptotically valid bootstrap critical values are also proposed

hybridized globally divergence free ldg methods part i the stokes problem we devise and analyze a new local discontinuous galerkin ldg method for the stokes equations of incompressible fluid flow this optimally convergent method is obtained by using an ldg method to discretize a vorticity velocity formulation of the stokes equations and by applying a new hybridization to the resulting discretization one of the main features of the hybridized method is that it provides a globally divergence free approximate velocity without having to construct globally divergence free finite dimensional spaces only elementwise divergence free basis functions are used another important feature is that it has significantly less degrees of freedom than all other ldg methods in the current literature in particular the approximation to the pressure is only defined on the faces of the elements on the other hand we show that as expected the condition number of the schur complement matrix for this approximate pressure is of order h 2 in the mesh size h finally we present numerical experiments that confirm the sharpness of our theoretical a priori error estimates

irregular n2sls and lasso estimation of the matrix exponential spatial specification model abstract in this paper we consider estimation of the matrix exponential spatial specification model with the durbin and endogenous regressors we find that the nonlinear two stage least squares n2sls estimator is in general consistent and asymptotically normal however when the durbin and endogenous regressors are irrelevant the gradient vector of the n2sls criterion function has a singular covariance matrix with probability approaching one w p a 1 some components of the n2sls estimator have slower rates of convergence and their asymptotic distributions are nonstandard the distance difference and gradient test statistics which have irregular asymptotic distributions are derived to test for the irrelevance of the durbin and endogenous regressors as an alternative estimation and model selection approach we propose the adaptive group lasso which penalizes the coefficients of the durbin and endogenous explanatory variables we show that the estimator has the oracle properties so the true model can be selected w p a 1 and the estimator always has the n rate of convergence and asymptotic normal distribution we propose to select the tuning parameter for the adaptive group lasso by minimizing an information criterion

inference in second order identified models abstract we explore the local power properties of different test statistics for conducting inference in moment condition models that only identify the parameters locally to second order we consider the conventional wald and lm statistics and also the generalized anderson rubin gar statistic anderson and rubin 1949 dufour 1997 staiger and stock 1997 stock and wright 2000 klm statistic kleibergen 2002 kleibergen 2005 and the gmm extension of moreira 2003 gmm m conditional likelihood ratio statistic the gar klm and gmm m statistics are so called identification robust since their conditional limiting distribution is the same under first order weak and therefore also second order identification for inference about the model specification we consider the identification robust j statistic kleibergen 2005 and the gar statistic interestingly we find that the limiting distribution of the wald statistic under local alternatives not only depends on the distance to the null hypothesis but also on the convergence rate of the jacobian we specifically analyse two empirically relevant models with second order identification in the panel autoregressive model of order one our analysis indicates that the wald test of a unit root value of the autoregressive parameter has better power compared to the corresponding gar test which in turn dominates the klm gmm m and lm tests for the conditionally heteroskedastic factor model we compare kleibergen 2005 j and the gar statistics to hansen 1982 overidentifying restrictions test previously analysed in this context by dovonon and renault 2013 and find the power ranking depends on the sample size collectively our results suggest that tests with meaningful power can be conducted in second order identified models

a connection between scott vogelius and grad div stabilized taylor hood fe approximations of the navier stokes equations this article studies two methods for obtaining excellent mass conservation in finite element computations of the navier stokes equations using continuous velocity fields with a particular mesh construction the scott vogelius element pair has recently been shown to be inf sup stable and have optimal approximation properties while also providing pointwise mass conservation we present herein the first numerical tests of this element pair for the time dependent navier stokes equations we also prove that the limit of the grad div stabilized taylor hood solutions to the navier stokes problem converges to the scott vogelius solution as the stabilization parameter tends to infinity that is we provide theoretical justification that choosing the grad div parameter large does not destroy the solution numerical tests are provided which verify the theory and show how both scott vogelius and grad div stabilized taylor hood with large stabilization parameter elements can provide accurate results with excellent mass conservation for navier stokes approximations

a family of q_ k 1 k timesq_ k k 1 divergence free finite elements on rectangular grids it is shown that the conforming q k 1 k times q k k 1 q k mixed element is stable and provides optimal order of approximation for the stokes equations on rectangular grids for all k ge2 here q k 1 k stands for the space of continuous piecewise polynomials of degree k 1 or less in x direction and of degree k or less in y direction q k is the space of discontinuous polynomials of separated degree k or less with spurious modes filtered to be precise q k is the divergence of the discrete velocity space q k 1 k times q k k 1 polynomials of different degrees in x and y are used for different components of velocity so that the resulting finite element solution is also divergence free point wise this is the first divergence free element found on nontriangular grids an efficient iterative method is presented where the discrete pressure function is produced as a byproduct numerical tests are provided confirming the theory

a new family of efficient conforming mixed finite elements on both rectangular and cuboid meshes for linear elasticity in the symmetric formulation a new family of mixed finite elements is proposed for solving the classical hellinger reissner mixed problem of the elasticity equations for two dimensions the normal stress of the matrix valued stress field is approximated by an enriched brezzi douglas fortin marini element of order k the shear stress by the serendipity element of order k and the displacement field by an enriched discontinuous vector valued p k 1 element the degrees of freedom on each element of the lowest order element which is of first order are 10 plus 4 for three dimensions the normal stress is approximated by an enriched raviart thomas element of order k each component of the shear stress by a product space of the serendipity element space of two variables and the space of polynomials of degree leq k 1 with respect to the rest variable and the displacement field by an enriched discontinuous vector valued q k 1 element the degrees of freedom on each element of the lowest order element which is of fi

consistent local projection stabilized finite element methods this work establishes a formal derivation of local projection stabilized methods as a result of an enriched petrov galerkin strategy for the stokes problem the initial unstable mathbb p 1 times mathbb p l l 0 1 finite element space is enhanced with solutions of residual based local problems and then the static condensation procedure is applied to derive new methods the approach keeps degrees of freedom unchanged while giving rise to new stable and consistent methods for continuous and discontinuous approximation spaces for the pressure the resulting methods do not need the use of a macroelement grid structure and are parameter free the numerical analysis is carried out showing optimal convergence in natural norms and moreover two ways of rendering the velocity field locally mass conservative are proposed some numerics validate the theoretical results

mixed finite elements for elasticity there have been many efforts dating back four decades to develop stable mixed finite elements for the stress displacement formulation of the plane elasticity system this requires the development of a compatible pair of finite element spaces one to discretize the space of symmetric tensors in which the stress field is sought and one to discretize the space of vector fields in which the displacement is sought although there are number of well known mixed finite element pairs known for the analogous problem involving vector fields and scalar fields the symmetry of the stress field is a substantial additional difficulty and the elements presented here are the first ones using polynomial shape functions which are known to be stable we present a family of such pairs of finite element spaces one for each polynomial degree beginning with degree two for the stress and degree one for the displacement and show stability and optimal order approximation we also analyze some obstructions to the construction of such finite element spaces which account for the paucity of elements available

a stationary point for the stochastic frontier likelihood the likelihood function for the stochastic frontier model is shown to possess an unusual stationary point which may or may not be a maximum a condition is given to determine if the point is a maximum and the result is interpreted in the context of specification and estimation the specification and estimation of frontier or envelope functions has been a concern of econometricians for some time the subject became of widespread interest with the introduction of a particularly attractive formulation of the model the stochastic frontier of aigner love11 and schmidt als 1977 and meeusen and van den broeck 1977 in that model the random disturbance is composed of the sum of a symmetric and a one sided random variable this specification avoids the statistical problems of earlier formulations while providing a convenient measure of the relative inefficiency a recent survey may be found in f sund et al 1980 the distributional assumptions most commonly made are the zero mean normal distribution for the symmetric component and the same distribution truncated above zero for the one sized component with this assumption the likelihood equation is well defined and als propose maximum likelihood estimation this method has been widely applied for examples see 3 4 and 5 however recent experience with simulating the properties of competing estimators for this model has indicated that the likelihood function is not entirely well behaved we show that the likelihood function always has a stationary point at one particular set of parameter values 3 a condition is given when this point is a local maximum and when it is a

specification testing when score test statistics are identically zero we investigate the problem of previous specification testing when the score vector evaluated at the restricted mle is identically zero several econometric examples are provided a general test procedure which generalizes the geometric principle of the score test is proposed the wald and the likelihood ratio tests are also analyzed even under such irregularities the usual asymptotic distribution of the likelihood ratio statistics is still valid however the wald type statistics need to be modified the generalized score the likelihood ratio and the modified wald tests are shown to be asymptotically equivalent the asymptotic efficiency of the tests is derived