nonparametric inference based on conditional moment inequalities abstract this paper develops methods of inference for nonparametric and semiparametric parameters defined by conditional moment inequalities and or equalities the parameters need not be identified confidence sets and tests are introduced the correct uniform asymptotic size of these procedures is established the false coverage probabilities and power of the cs s and tests are established for fixed alternatives and some local alternatives finite sample simulation results are given for a nonparametric conditional quantile model with censoring and a nonparametric conditional treatment effect model the recommended cs test uses a cramer von mises type test statistic and employs a generalized moment selection critical value

bootstrap inference in partially identified models defined by moment inequalities coverage of the identified set this paper introduces a novel bootstrap procedure to perform inference in a wide class of partially identi ed econometric models we consider econometric models de ned by nitely many weak moment inequalities y which encompass many applications of economic interest the objective of our inferential procedure is to cover the identi ed set with a prespeci ed probability z we compare our bootstrap procedure a competing asymptotic approximation and subsampling proce dures in terms of the rate at which they achieve the desired coverage level also known as the error in the coverage probability under certain conditions we show that our bootstrap procedure and the asymp totic approximation have the same order of error in the coverage probability which is smaller than the one obtained by using subsampling this implies that inference based on our bootstrap and asymptotic approximation should eventually be more precise than inference based on subsampling a monte carlo study con rms this nding in a small sample simulation

asymptotically efficient estimation of models defined by convex moment inequalities this paper examines the efficient estimation of partially identified models defined by moment inequalities that are convex in the parameter of interest in such a setting the identified set is itself convex and hence fully characterized by its support function we provide conditions under which despite being an infinite dimensional parameter the support function admits n consistent regular estimators a semiparametric efficiency bound is then derived for its estimation and it is shown that any regular estimator attaining it must also minimize a wide class of asymptotic loss functions in addition we show that the plug in estimator is efficient and devise a consistent bootstrap procedure for estimating its limiting distribution the setting we examine is related to an incomplete linear model studied in beresteanu and molinari 2008 and bontemps magnac and maurin 2012 which further enables us to establish the semiparametric efficiency of their proposed estimators for that problem

on the efficiency of newton and broyden numerical methods in the investigation of the regular polygon problem of n 1 bodies numerical methods of finding the roots of a system of non linear algebraic equations are treated in this paper this paper attempts to give an answer to the selection of the most efficient method in a complex problem of celestial dynamics the so called ring problem of n 1 bodies we apply newton and broyden s method to these problems and we investigate by means of their use the planar equilibrium points the five equilibrium zones which are symbolized by a 1 a 2 b c 2 and c 1 by order of appearance from the center o to the periphery of the imaginary circle on which the primaries lie t j kalvouridis a planar case of the n 1 body problem the ring problem astrophys space sci 260 3 1999 309 325 and the attracting regions of the system the efficiency of these methods is studied through a comparative process the obtained results are demonstrated in figures and are discussed

adjacency spectra of random and complete hypergraphs abstract we present progress on the problem of asymptotically describing the adjacency eigenvalues of random and complete uniform hypergraphs there is a natural conjecture arising from analogy with random matrix theory that connects these spectra to that of the all ones hypermatrix several of the ingredients along a possible path to this conjecture are established and may be of independent interest in spectral hypergraph hypermatrix theory in particular we provide a bound on the spectral radius of the symmetric bernoulli hyperensemble and show that the spectrum of the complete k uniform hypergraph for k 2 3 is close to that of an appropriately scaled all ones hypermatrix

gmm with many moment conditions this paper provides a first order asymptotic theory for generalized method of moments gmm estimators when the number of moment conditions is allowed to increase with the sample size and the moment conditions may be weak examples in which these asymptotics are relevant include instrumental variable iv estimation with many possibly weak or uninformed instruments and some panel data models that cover moderate time spans and have correspondingly large numbers of instruments under certain regularity conditions the gmm estimators are shown to converge in probability but not necessarily to the true parameter and conditions for consistent gmm estimation are given a general framework for the gmm limit distribution theory is developed based on epiconvergence methods some illustrations are provided including consistent gmm estimation of a panel model with time varying individual effects consistent limited information maximum likelihood estimation as a continuously updated gmm estimator and consistent iv structural estimation using large numbers of weak or irrelevant instruments some simulations are reported

higher order properties of gmm and generalized empirical likelihood estimators in an effort to improve the small sample properties of generalized method of moments gmm estimators a number of alternative estimators have been suggested these include empirical likelihood el continuous updating and exponential tilting estimators we show that these estimators share a common structure being members of a class of generalized empirical likelihood gel estimators we use this structure to compare their higher order asymptotic properties we find that gel has no asymptotic bias due to correlation of the moment functions with their jacobian eliminating an important source of bias for gmm in models with endogeneity we also find that el has no asymptotic bias from estimating the optimal weight matrix eliminating a further important source of bias for gmm in panel data models we give bias corrected gmm and gel estimators we also show that bias corrected el inherits the higher order property of maximum likelihood that it is higher order asymptotically efficient relative to the other bias corrected estimators

estimating dynamic models of imperfect competition we describe a two step algorithm for estimating dynamic games under the assumption that behavior is consistent with markov perfect equilibrium in the first step the policy functions and the law of motion for the state variables are estimated in the second step the remaining structural parameters are estimated using the optimality conditions for equilibrium the second step estimator is a simple simulated minimum distance estimator the algorithm applies to a broad class of models including industry competition models with both discrete and continuous controls such as the ericson and pakes 1995 model we test the algorithm on a class of dynamic discrete choice models with normally distributed errors and a class of dynamic oligopoly models similar to that of pakes and mcguire 1994

cored hypergraphs power hypergraphs and their laplacian h eigenvalues abstract in this paper we introduce the class of cored hypergraphs and power hypergraphs and investigate the properties of their laplacian h eigenvalues from an ordinary graph one may generate a k uniform hypergraph called the kth power hypergraph of that graph power hypergraphs are cored hypergraphs but not vice versa sunflowers loose paths and loose cycles are power hypergraphs while squids generalized loose s paths and loose s cycles for 2 s k 2 are cored hypergraphs but not power graphs in general we show that the largest laplacian h eigenvalue of an even uniform cored hypergraph is equal to its largest signless laplacian h eigenvalue especially we find out these largest h eigenvalues for even uniform squids moreover we show that the largest laplacian h eigenvalue of an odd uniform squid loose path and loose cycle is equal to the maximum degree i e 2 we also compute the laplacian h spectra of the class of sunflowers when k is odd the laplacian h spectra of the loose cycle of size 3 and the loose path of length 3 are characterized as well

the zeros of special functions from a fixed point method a scheme for the computation of the zeros of special functions and orthogonal polynomials is developed we study the structure of the first order difference differential equations ddes satisfied by two fundamental sets of solutions of second order odes y n prime prime x a n x y n x 0 n being the order of the solutions and an x a family of continuous functions it is proved that with a convenient normalization of the solutions t pm1 z z pm mbox rm sign d arctan y n x z y n pm 1 x z are globally convergent iterations with fixed points z x n i x n i being the zeros of yn x d is one of the coefficients in the ddes and z x is a primitive of d the structure of the ddes is also used to set global bounds on the differences between adjacent zeros of functions of consecutive orders and to find iteration steps which guarantee that all the zeros inside a given interval can be found with certainty as an illustration we describe how to implement this scheme for the calculation of the zeros of arbitrary solutions of the bessel coulomb legendre hermite and laguerre equations

breaking the curse of dimensionality in conditional moment inequalities for discrete choice models abstract this paper studies inference of preference parameters in semiparametric discrete choice models when these parameters are not point identified and the identified set is characterized by a class of conditional moment inequalities exploring the semiparametric modeling restrictions we show that the identified set can be equivalently formulated by moment inequalities conditional on only two continuous indexing variables such formulation holds regardless of the covariate dimension thereby breaking the curse of dimensionality for nonparametric inference based on the underlying conditional moment inequalities we further apply this dimension reducing characterization approach to the monotone single index model and to a variety of semiparametric models under which the sign of conditional expectation of a certain transformation of the outcome is the same as that of the indexing variable

an algorithm for finding all solutions of a nonlinear system abstract let f x r k be a lipschitz continuous function on a compact subset x r d subdivision algorithms are described that can be used to find all solutions of the equation f x 0 that lie in x convergence is shown and numerical examples are presented modifications of the basic algorithm which speed convergence are given for the case of nondegenerate zeros of a vector field

bisection is optimal we seek an approximation to a zero of a continuous functionf a b such thatf a 0 andf b 0 it is known that the bisection algorithm makes optimal use ofn function evaluations i e yields the minimal error which is b a 2 n 1 see e g kung 2 traub and wozniakowski 5 proposed using more general information onf by permitting the adaptive evaluations ofn arbitrary linear functionals they conjectured 5 p 170 that the bisection algorithm remains optimal even if these general evaluations are permitted this paper affirmatively proves this conjecture in fact we prove optimality of the bisection algorithm even assuming thatf is infinitely many times differentiable on a b and has exactly one simple zero

a class of infinite sums and integrals in this paper closed form sums are given for various slowly convergent infinite series which arise essentially from the differentiation of dirichlet l series some associated integrations are also considered a small number of the results appear in standard tables but most seem to be new

the determination of the location of the global maximum of a function in the presence of several local extrema the global maximum of a function can be determined by using information about the number of stationary points in the domain of interest this information is obtained by evaluating an integral that equals the exact number of stationary points of the function the integral is based on work by kronecker and picard at the end of the nineteenth century the numerical feasibility of the method is shown by two computed examples i e estimation problems from statistics and optical communication theory in these examples the global maximum of the likelihood function is obtained by using the total number of stationary points as revealed by the computed integral

consistent estimation of models defined by conditional moment restrictions in econometrics models stated as conditional moment restrictions are typically estimated by means of the generalized method of moments gmm the gmm estimation procedure can render inconsistent estimates since the number of arbitrarily chosen instruments is finite in fact consistency of the gmm estimators relies on additional assumptions that imply unclear restrictions on the data generating process this article introduces a new simple and consistent estimation procedure for these models that is directly based on the definition of the conditional moments the main feature of our procedure is its simplicity since its implementation does not require the selection of any user chosen number and statistical inference is straightforward since the proposed estimator is asymptotically normal in addition we suggest an asymptotically efficient estimator constructed by carrying out one newton raphson step in the direction of the efficient gmm estimator

inference on endogenously censored regression models using conditional moment inequalities under a quantile restriction randomly censored regression models can be written in terms of conditional moment inequalities we study the identified features of these moment inequalities with respect to the regression parameters where we allow for covariate dependent censoring endogenous censoring and endogenous regressors these inequalities restrict the parameters to a set we show regular point identification can be achieved under a set of interpretable sufficient conditions we then provide a simple way to convert conditional moment inequalities into unconditional ones while preserving the informational content our method obviates the need for nonparametric estimation which would require the selection of smoothing parameters and trimming procedures without the point identification conditions our objective function can be used to do inference on the partially identified parameter maintaining the point identification conditions we propose a quantile minimum distance estimator which converges at the parametric rate to the parameter vector of interest and has an asymptotically normal distribution a small scale simulation study and an application using drug relapse data demonstrate satisfactory finite sample performance

on the riemann property of angular lattice sums and the one dimensional limit of two dimensional lattice sums we consider a general class of two dimensional lattice sums consisting of complex powers s of inverse quadratic functions we consider two cases one where the quadratic function is negative definite and another more restricted case where it is positive definite in the former we use a representation due to h kober and consider the limit u where the lattice becomes ever more elongated along one period direction the one dimensional limit in the latter we use an explicit evaluation of the sum due to zucker and robertson in either case we show that the one dimensional limit of the sum is given in terms of 2 s if re s 1 2 and either 2 s 1 or 2 2 s if re s 1 2 in either case this leads to a riemann property of these sums in the one dimensional limit their zeros must lie on the critical line re s 1 2 we also comment on a class of sums that involve complex powers of the distance to points in a two dimensional square lattice and trigonometric functions of their angle we show that certain of these sums can have their zeros on the critical line but not in a neighbourhood of it others are identically zero on it while still others have no zeros on it

weighted ks statistics for inference on conditional moment inequalities abstract this paper proposes set estimators and conservative confidence regions for the identified set in conditional moment inequality models using kolmogorov smirnov statistics with a truncated inverse variance weighting with increasing truncation points the new weighting differs from those proposed in the literature in two important ways first this paper shows that estimators based on ks statistics with the proposed weighting function converge to the identified set at a faster rate than existing procedures based on bounded weight functions in a broad class of models this provides a theoretical justification for inverse variance weighting in this context and contrasts with analogous results for conditional moment equalities in which optimal weighting only affects the asymptotic variance the results on rates of convergence of set estimators are the first such results even for the existing procedures and involve developing the first general framework for determining consistency and rates of convergence for set estimators and confidence regions in this context second the new weighting changes the asymptotic behavior including the rate of convergence of the ks statistic itself requiring a new asymptotic theory in choosing the critical value a series of examples illustrate the broad applicability of the results

a convergence improving iterative method for computing periodic orbits near bifurcation points abstract the accurate computation of periodic orbits and the precise knowledge of their bifurcation properties are very important for studying the behavior of many dynamical systems of physical interest in this paper we present an iterative method for computing periodic orbits which has the advantage of improving the convergence of previous newton like schemes especially near bifurcation points this method is illustrated here on a conservative nonlinear mathieu equation for which a sequence of period doubling bifurcations is followed long enough to obtain accurate estimates of the two universal scaling constants as well as the universal rate by which the bifurcation values of a parameter q qk k 1 2 3 tend to their limiting value q k increases

nonparametric identification under discrete variation this paper provides weak conditions under which there is nonparametric interval identification of local features of a structural function that depends on a discrete endogenous variable and is nonseparable in latent variates the function delivers values of a discrete or continuous outcome and instruments may be discrete valued application of the analog principle leads to quantile regression based interval estimators of values and partial differences of structural functions the results are used to investigate the nonparametric identifying power of the quarter of birth instruments used in angrist and krueger s 1991 study of the returns to schooling

a simple test for moment inequality models with an application to english auctions abstract testable predictions of many economic models involve inequality comparisons between transformations of nonparametric functionals we introduce an econometric test for these types of restrictions based on one sided l p statistics that adapt asymptotically to the contact sets without having to directly estimate them monte carlo experiments show that our test is less conservative than procedures based on least favorable configurations and has power comparable to other contact set based procedures as an application we test for interdependence of bidders valuations in ascending auctions using usfs timber auction data we reject the independent private values model in favor of a model of correlated private values

spectra of uniform hypergraphs abstract we present a spectral theory of uniform hypergraphs that closely parallels spectral graph theory a number of recent developments building upon classical work has led to a rich understanding of symmetric hyperdeterminants of hypermatrices a k a multidimensional arrays symmetric hyperdeterminants share many properties with determinants but the context of multilinear algebra is substantially more complicated than the linear algebra required to address spectral graph theory i e ordinary matrices nonetheless it is possible to define eigenvalues of a hypermatrix via its characteristic polynomial as well as variationally we apply this notion to the adjacency hypermatrix of a uniform hypergraph and prove a number of natural analogs of basic results in spectral graph theory open problems abound and we present a number of directions for further study

computing zeros on a real interval through chebyshev expansion and polynomial rootfinding robust polynomial rootfinders can be exploited to compute the roots on a real interval of a nonpolynomial function f x by the following i expand f as a chebyshev polynomial series ii convert to a polynomial in ordinary series of powers form and iii apply the polynomial rootfinder complex valued roots and real roots outside the target interval are discarded the expansion is most efficiently done by adaptive chebyshev interpolation with n equal to a power of two where n is the degree of the truncated chebyshev series all previous evaluations of f can then be reused when n is increased adaption stops when n is sufficiently large so that further increases produce no significant change in the interpolant we describe two conversion strategies the convert to powers method uses multiplication by mildly ill conditioned matrices to create a polynomial of degree n the degree doubling strategy defines a polynomial of larger degree 2n but is very well conditioned the convert to powers method although faster restricts n to moderate values this can always be accomplished by subdividing the target interval both these strategies allow simultaneous approximation of many roots on an interval whether simple or multiple for nonpolynomial f x

semismoothness of the maximum eigenvalue function of a symmetric tensor and its application abstract in this paper we examine the maximum eigenvalue function of an even order real symmetric tensor by using the variational analysis techniques we first show that the maximum eigenvalue function is a continuous and convex function on the symmetric tensor space in particular we obtain the convex subdifferential formula for the maximum eigenvalue function next for an mth order n dimensional symmetric tensor a we show that the maximum eigenvalue function is always th order semismooth at a for some rational number 0 in the special case when the geometric multiplicity is one we show that can be set as 1 2 m 1 n sufficient condition ensuring the strong semismoothness of the maximum eigenvalue function is also provided as an application we propose a generalized newton method to solve the space tensor conic linear programming problem which arises in medical imaging area local convergence rate of this method is established by using the semismooth property of the maximum eigenvalue function

generalized method of moments with many weak moment conditions using many moment conditions can improve efficiency but makes the usual generalized method of moments gmm inferences inaccurate two step gmm is biased generalized empirical likelihood gel has smaller bias but the usual standard errors are too small in instrumental variable settings in this paper we give a new variance estimator for gel that addresses this problem it is consistent under the usual asymptotics and under many weak moment asymptotics is larger than usual and is consistent we also show that the kleibergen 2005 lagrange multiplier and conditional likelihood ratio statistics are valid under many weak moments in addition we introduce a jackknife gmm estimator but find that gel is asymptotically more efficient under many weak moments in monte carlo examples we find that t statistics based on the new variance estimator have nearly correct size in a wide range of cases

on the z eigenvalues of the adjacency tensors for uniform hypergraphs abstract the adjacency matrices for graphs are generalized to the adjacency tensors for uniform hypergraphs and some fundamental properties for the adjacency tensor and its z eigenvalues of a uniform hypergraph are obtained in particular some bounds on the smallest and the largest z eigenvalues of the adjacency tensors for uniform hypergraphs are presented

on enclosing simple roots of nonlinear equations in this paper we present two efficient algorithms for enclosing a simple root of the nonlinear equation f x 0 in the interval a b they improve recent methods of alefeld and potra by achieving higher efficiency indices and avoiding the solution of a quadratic equation per iteration the efficiency indices of our methods are 1 5537 and 1 618 respectively we show that our second method is an optimal algorithm in some sense our numerical experiments show that the two methods of the present paper compare well with the above methods of alefeld and potra as well as efficient solvers of dekker brent and le the second method in this paper has the best behavior of all especially when the termination tolerance is small

homotopies for solving polynomial systems within a bounded domain abstract the problem considered in this paper is the computation of all solutions of a given polynomial system in a bounded domain proving rouche s theorem by homotopy continuation concepts yields a new class of homotopy methods the so called regional homotopy methods these methods rely on isolating a part of the system to be solved which dominates the rest of the system on the border of the domain as the dominant part has a sparser structure it is easier to solve it will be used as start system in the regional homotopy the paper further describes practical homotopy construction methods by presenting estimators to obtain bounds for polynomials over a bounded domain applications illustrate the usefulness of the approach

computing the zeros and turning points of solutions of second order homogeneous linear odes algorithms to compute the zeros and turning points of the solutions of second order odes y prime prime b x y prime a x y 0 are developed two fixed point methods are introduced the first method consists of fixed point iterations that are built from the first order differential system relating the problem function with a contrast function w the contrast function w has zeros interlaced with those of y and it is a solution of a different second order ode this method which generalizes previous findings j segura siam j numer anal 40 2002 pp 114 133 requires the evaluation of the ratio of functions y w the second method is based on fixed point iterations stemming from the second order ode it requires the computation of the logarithmic derivative y prime y both are quadratically convergent methods error bounds are provided the particular case of second order odes depending on one parameter y n prime prime b n x y n prime a n x y n x 0 with applications to the computation of the zeros and turning points of special functions is discussed in detail the combination of both methods provides algorithms for the efficient computation of the zeros and turning points of a broad family of special functions including hypergeometric and confluent hypergeometric functions of real parameters and variables jacobi laguerre and hermite polynomials are particular cases bessel airy coulomb and conical functions among others we provide numerical examples showing the efficiency of the methods

perron frobenius theorem for nonnegative multilinear forms and extensions we prove an analog of perron frobenius theorem for multilinear forms with nonnegative coefficients and more generally for polynomial maps with nonnegative coefficients we determine the geometric convergence rate of the power algorithm to the unique normalized eigenvector

moment inequalities and their application this paper provides conditions under which the inequality constraints generated by either single agent optimizing behavior or by the nash equilibria of multiple agent problems can be used as a basis for estimation and inference we also add to the econometric literature on inference in models defined by inequality constraints by providing a new specification test and methods of inference for the boundaries of the model s identified set two applications illustrate how the use of inequality constraints can simplify the problem of obtaining estimators from complex behavioral models of substantial applied interest

locating and computing all the simple roots and extrema of a function this paper describes and analyzes two algorithms for locating and computing with certainty all the simple roots of a twice continuously differentiable function f colon a b subset r to r and all the extrema of a three times continuously differentiable function in a b the first algorithm locates and computes all the simple roots or all the extrema while the second one is more efficient in the case where both simple roots and extrema are required this paper also gives analytical estimation of the expected complexity of the algorithms based on the distribution of the roots in a b here only the case of uniform distribution is examined which is also the approach to be followed when no statistical data are available for the function at hand the algorithms have been implemented and tested performance information for a well known bessel function is reported

on the construction of sufficient refinements for computation of topological degree let n p n r n be continuous wherep n is ann dimensional polyhedron with n n 0 0 0 anywhere onb p n the boundary ofp n thend n p n n the topological degree of n onp n relative to n is define ifb p n can be subdivided in a certain way sufficiently refined then a formula of stenger gives an easy method for computingd n p n n a class of algorithms is described in which each algorithm produces a sufficient refinement after a finite number of steps for any givenp n and n for whichd n p n n is defined

on perturbation of roots of homogeneous algebraic systems a problem concerning the perturbation of roots of a system of homogeneous algebraic equations is investigated the question of conservation and decomposition of a multiple root into simple roots are discussed the main theorem on the conservation of the number of roots of a deformed not necessarily homogeneous algebraic system is proved by making use of a homotopy connecting initial roots of the given system and roots of a perturbed system hereby we give an estimate on the size of perturbation that does not affect the number of roots further on we state the existence of a slightly deformed system that has the same number of real zeros as the original system in taking the multiplicities into account we give also a result about the decomposition of multiple real roots into simple real roots

efficiently computing many roots of a function we present a new bisection based method for counting and computing roots of a function in a given interval our method is focused on very large problems i e instances with the number of roots of the order of hundreds the method draws its power from the fact that the roots are expected to be many and is able to discover a large percentage of them very efficiently its main advantage apart from its efficiency is the fact that it requires only the sign of the function at a certain point and not its actual value also its simplicity makes it a suitable preprocessing step for reducing the size of the problem prior to more robust but also more demanding methods the algorithm is accompanied by a probabilistic analysis of its behavior which shows that a simple existence criterion like bolzano s rule can be a powerful tool in the zerofinding process

inference for parameters defined by moment inequalities using generalized moment selection the topic of this paper is inference in models in which parameters are defined by moment inequalities and or equalities the parameters may or may not be identified this paper introduces a new class of confidence sets and tests based on generalized moment selection gms gms procedures are shown to have correct asymptotic size in a uniform sense and are shown not to be asymptotically conservative the power of gms tests is compared to that of subsampling m out of n bootstrap and plug in asymptotic pa tests the latter three procedures are the only general procedures in the literature that have been shown to have correct asymptotic size in a uniform sense for the moment inequality equality model gms tests are shown to have asymptotic power that dominates that of subsampling m out of n bootstrap and pa tests subsampling and m out of n bootstrap tests are shown to have asymptotic power that dominates that of pa tests

2d universality of period doubling bifurcations in 3d conservative reversible mappings abstract infinite sequences of period doubling bifurcations are known to occur generically i e with codimension 1 not only in dissipative 1d systems but also in 2d conservative systems described by area preserving mappings in this paper we study a 3d volume preserving reversible mapping and show that it does possess period 2 m m 1 2 orbits with stability intervals whose length decreases rapidly with increasing m varying one parameter of the system we find that these orbits always bifurcate out of one another with the usual stability exchange and universal properties of period doubling sequences of 2d conservative maps this raises the interesting question whether these 3d reversible maps possess an analytic integral which would render them essentially 2 dimensional

a rapid generalized method of bisection for solving systems of non linear equations a rapid generalized method of bisection for solving systems of non linear equations is presented in this paper based on the non zero value of the topological degree further while the method does not compute the topological degree it takes care of keeping its non zero value during the bisections and thus results in a fast bisection algorithm

asymptotically exact inference in conditional moment inequality models this paper derives the rate of convergence and asymptotic distribution for a class of kolmogorov smirnov style test statistics for conditional moment inequality models for parameters on the boundary of the identified set under general conditions using these results i propose tests that are more powerful than existing approaches for choosing critical values for this test statistic i quantify the power improvement by showing that the new tests can detect alternatives that converge to points on the identified set at a faster rate than those detected by existing approaches a monte carlo study confirms that the tests and the asymptotic approximations they use perform well in finite samples in an application to a regression of prescription drug expenditures on income with interval data from the health and retirement study confidence regions based on the new tests are substantially tighter than those based on existing methods

asymptotic efficiency in estimation with conditional moment restrictions abstract in this paper bounds on asymptotic efficiency are derived for a class of non parametric models the data are independent and identically distributed according to some unknown distribution f there is a given function of the data and a parameter the restrictions are that a conditional expectation of this function is zero at some point in the parameter space this point is to be estimated if f is assumed to be a multinomial distribution with known finite support then the problem becomes parametric and the bound can be obtained from the information matrix this bound turns out to depend only upon certain conditional moments and not upon the support of the distribution since a general f can be approximated by a multinomial distribution the multinomial bound applies to the general case

intersection bounds estimation and inference we develop a practical and novel method for inference on intersection bounds namely bounds defined by either the infimum or supremum of a parametric or nonparametric function or equivalently the value of a linear programming problem with a potentially infinite constraint set we show that many bounds characterizations in econometrics for instance bounds on parameters under conditional moment inequalities can be formulated as intersection bounds our approach is especially convenient for models comprised of a continuum of inequalities that are separable in parameters and also applies to models with inequalities that are nonseparable in parameters since analog estimators for intersection bounds can be severely biased in finite samples routinely underestimating the size of the identified set we also offer a median bias corrected estimator of such bounds as a by product of our inferential procedures we develop theory for large sample inference based on the strong approximation of a sequence of series or kernel based empirical processes by a sequence of penultimate gaussian processes these penultimate processes are generally not weakly convergent and thus are non donsker our theoretical results establish that we can nonetheless perform asymptotically valid inference based on these processes our construction also provides new adaptive inequality moment selection methods we provide conditions for the use of nonparametric kernel and series estimators including a novel result that establishes strong approximation for any general series estimator admitting linearization which may be of independent interest

improved newton s method without direct function evaluations for solving systems of nonlinear equations we have recently developed a newton s method to manage issues with inaccurate function values or problems with high computational cost in this work we introduce a modification of the above method reducing the total computational cost and improving in general its overall performance moreover the proposed version retains the quadratic convergence the good behavior over singular and ill conditioned cases of jacobian matrix and its capability to be ideal for imprecise function problems numerical results demonstrate the efficiency of the new proposed method

the inverse rank and product of tensors abstract in this paper we give some basic properties for the left right inverse rank and product of tensors the existence of order 2 left right inverses of tensors is characterized we obtain some equalities and inequalities on the tensor rank we also show that the rank of a uniform hypergraph is independent of the ordering of its vertices and the laplacian tensor and the signless laplacian tensor have the same rank for odd bipartite even uniform hypergraphs

set identified linear models we analyze the identification and estimation of parameters satisfying the incomplete linear moment restrictions e z x y e z u z where z is a set of instruments and u z an unknown bounded scalar function we first provide empirically relevant examples of such a set up second we show that these conditions set identify where the identified set b is bounded and convex we provide a sharp characterization of the identified set not only when the number of moment conditions is equal to the number of parameters of interest but also in the case in which the number of conditions is strictly larger than the number of parameters we derive a necessary and sufficient condition of the validity of supernumerary restrictions which generalizes the familiar sargan condition third we provide new results on the asymptotics of analog estimates constructed from the identification results when b is a strictly convex set we also construct a test of the null hypothesis 0 2 b whose size is asymptotically correct and which relies on the minimization of the support function of the set b 0 results of some monte carlo experiments are presented

a reliable algorithm for computing the topological degree of a mapping in r2 abstract in this paper we present a method to reliably compute the topological degree of a mapping in the plane over a simple closed polygon the method is based on henrici s argument principle and computes the degree using the winding number concept in range arithmetic the algorithm is then applied to the root computation of a univariate polynomial the proposed algorithms are demonstrated with examples

confidence sets for partially identified parameters that satisfy a finite number of moment inequalities this paper proposes a computationally simple way to construct confidence sets for a parameter of interest in models comprised of moment inequalities building on results from the literature on multivariate one sided tests i show how to test the hypothesis that any particular parameter value is logically consistent with the maintained moment inequalities the associated test statistic has an asymptotic chi bar square distribution and can be inverted to construct an asymptotic confidence set for the parameter of interest even if that parameter is only partially identified critical values for the test are easily computed and a monte carlo study demonstrates implementation and finite sample performance

efficient instrumental variables estimation of nonlinear models this paper considers asymptotically efficient instrumental variables estimation of nonlinear models in an i i d environment the class of models includes nonlinear simultaneous equations models and other models of interest a problem in constructing efficient instrumental variables estimators for such models is that the optimal instruments involve a conditional expectation calculation of which can require functional form assumptions for the conditional distribution of endogenous variables as well as integration nonparametric methods provide a way of avoiding this difficulty here it is shown that nonparametric estimates of the optimal instruments can give asymptotically efficient instrumental variables estimators also ways of choosing the nonparametric estimate in applications are discussed two types of nonparametric estimates of the optimal instruments are considered each involves nonparametric regression one by nearest neighbor and the other by series approximation the finite sample properties of the estimators are considered in a small sampling experiment involving an endogenous dummy variable model

estimation and confidence regions for parameter sets in econometric models this paper develops a framework for performing estimation and inference in econometric models with partial identification focusing particularly on models characterized by moment inequalities and equalities applications of this framework include the analysis of game theoretic models revealed preference restrictions regressions with missing and corrupted data auction models structural quantile regressions and asset pricing models specifically we provide estimators and confidence regions for the set of minimizers i of an econometric criterion function q in applications the criterion function embodies testable restrictions on economic models a parameter value that describes an economic model satisfies these restrictions if q attains its minimum at this value interest therefore focuses on the set of minimizers called the identified set we use the inversion of the sample analog q n of the population criterion q to construct estimators and confidence regions for the identified set and develop consistency rates of convergence and inference results for these estimators and regions to derive these results we develop methods for analyzing the asymptotic properties of sample criterion functions under set identification

nonlinear equations and optimization abstract interval analysis has led to improved numerical algorithms both for the solution of nonlinear equations and for nonlinear optimization in fact interval analysis has made possible algorithms for solving a problem often predicted to be numerically insoluble the nonlinear global optimization problem in this paper we survey interval methods for solving nonlinear equations and optimization

inference on regressions with interval data on a regressor or outcome this paper examines inference on regressions when interval data are available on one variable the other variables being measured precisely let a population be characterized by a distribution p y x v v 0 v 1 where y r 1 x r k and the real variables v v 0 v 1 satisfy v 0 v v 1 let a random sample be drawn from p and the realizations of y x v 0 v 1 be observed but not those of v the problem of interest may be to infer e y x v or e v x this analysis maintains interval i monotonicity m and mean independence mi assumptions i p v 0 v v 1 1 m e y x v is monotone in v mi e y x v v 0 v 1 e y x v no restrictions are imposed on the distribution of the unobserved values of v within the observed intervals v 0 v 1 it is found that the immi assumptions alone imply simple nonparametric bounds on e y x v and e v x these assumptions invoked when y is binary and combined with a semiparametric binary regression model yield an identification region for the parameters that may be estimated consistently by a modified maximum score mms method the immi assumptions combined with a parametric model for e y x v or e v x yield an identification region that may be estimated consistently by a modified minimum distance mmd method monte carlo methods are used to characterize the finite sample performance of these estimators empirical case studies are performed using interval wealth data in the health and retirement study and interval income data in the current population survey

a dimension reducing method for unconstrained optimization abstract a new method for unconstrained optimization in r n is presented this method reduces the dimension of the problem in such a way that it can lead to an iterative approximate formula for the computation of n 1 components of the optimum while its remaining component is computed separately using the final approximations of the other components it converges quadratically to a local optimum and it requires storage of order n 1 n 1 besides it does not require a good initial guess for one component of the optimum and it does not directly perform gradient evaluations thus it can be applied to problems with imprecise gradient values moreover a procedure for transforming the matrix formed by our method into a symmetric as well as into a diagonal one is presented furthermore the proposed dimension reducing scheme using finite difference gradient and hessian is presented the methods have been implemented and tested performance information for well known test functions is reported

the number of eigenvalues of a tensor abstract eigenvectors of tensors as studied recently in numerical multilinear algebra correspond to fixed points of self maps of a projective space we determine the number of eigenvectors and eigenvalues of a generic tensor and we show that the number of normalized eigenvalues of a symmetric tensor is always finite we also examine the characteristic polynomial and how its coefficients are related to discriminants and resultants

energy of static electron lattices a new approach to evaluating the static lattice energy of any given wigner lattice is proposed the method is much simpler than the traditional one and the results are much faster to evaluate the new approach is applied to two dimensional triangular lattices where it is shown that the triangular lattice is more stable than the square lattice three dimensional hexagonal lattices are also investigated in addition the authors place many of their considerations on a rigorous mathematical footing

consistent estimation with a large number of weak instruments this paper analyzes the conditions under which consistent estimation can be achieved in instrumental variables iv regression when the available instruments are weak and the number of instruments k n goes to infinity with the sample size we show that consistent estimation depends importantly on the strength of the instruments as measured by r the rate of growth of the so called concentration parameter and also on k n in particular when k n the concentration parameter can grow even if each individual instrument is only weakly correlated with the endogenous explanatory variables and consistency of certain estimators can be established under weaker conditions than have previously been assumed in the literature hence the use of many weak instruments may actually improve the performance of certain point estimators more specifically we find that the limited information maximum likelihood liml estimator and the bias corrected two stage least squares b2sls estimator are consistent when k n r n 0 while the two stage least squares 2sls estimator is consistent only if k n r n 0 as n these consistency results suggest that liml and b2sls are more robust to instrument weakness than 2sls

consistent estimation with many moment inequalities in this paper we consider estimation of the identified set when the number of moment inequalities is large relative to sample size possibly infinite many applications in the recent literature on partially identified problems have this feature including dynamic games set identified iv models and parameters defined by a continuum of moment inequalities in particular conditional moment inequalities we provide a generic consistency result for criterion based estimators using an increasing number of unconditional moment inequalities we then develop more specific results for set estimation subject to conditional moment inequalities we first derive the fastest possible rate for estimating the sharp identification region under smoothness conditions on the conditional moment functions we also give rate conditions for inference under local alternatives

some logarithmic lattice sums in this paper we consider certain lattice sums which arise when a system of line charges set in a compensating jelly of opposite charge interact via the two dimensional logarithmic potential a lattice limit discontinuity phenomenon similar to that discovered by borwein et al 1989 in the three dimensional case is explored and a high precision asymptotic method is described for the numerical computation of two dimensional lattice limits

on eigenvalue problems of real symmetric tensors abstract we use variational methods to give a positive answer to a conjecture posed by liqun qi l qi eigenvalues of a real supersymmetric tensor j symbolic comput 40 2005 1302 1324 regarding the real eigenvalues of certain higher order tensors

computing real roots of a polynomial in chebyshev series form through subdivision with linear testing and cubic solves abstract an arbitrary polynomial of degree n f n x can always be represented as a truncated chebyshev polynomial series chebyshev form this representation is much better conditioned than the usual power form of a polynomial we describe a new method for finding the real roots of f n x in chebyshev form the canonical interval x 1 1 is subdivided into n s subintervals each is tested for zeros using the error bound for linear interpolation on zero possible intervals f n is approximated by a cubic polynomial whose roots are then found by the usual sixteenth century formulas

locating and computing arbitrarily distributed zeros the problem of locating and computing with certainty all the simple roots of a twice continuously differentiable function f colon a b subset r to r is studied when some additional information on the distribution of the roots in the interval is available the framework is the one proposed by siam j sci comput 17 1996 pp 1232 1248 where only the uniform case was examined this paper settles some of the problems posed there and generalizes some of its results by considering an arbitrary distribution of the roots in a b the theoretical results are accompanied by simulations in a number of problems of various size