projected score methods for approximating conditional scores summary this paper extends the projected score methods of small mcleish 1989 it is shown that the conditional score function may be approximated with arbitrarily small stochastic error in terms of a natural basis for the space of centred likelihood ratios the utility of using this basis is established by identifying a u statistic representation theorem and a class of expectation identities for the basis elements making higher order asymptotics more tractable the results are applied to a canonical exponential family model where it is shown that the projected scores with estimated nuisance parameters can provide an accurate approximation to the conditional score function

a minimax approach to consistency and efficiency for estimating equations the consistency of estimating equations has been studied in the main along the lines of cramer s classical argument which only asserts the existence of consistent solutions the statement similar to that of doob and wald which identifies the consistent solutions has not yet been established the obstacle is that the solutions of estimating equations cannot in general be defined as the maximum of likelihood functions in this paper we demonstrate that the consistent solutions can be identified as the minimax of a function r whose properties resemble those of a log likelihood ratio but which exists in a much wider context furthermore since we do not need r to be differentiable the minimax is consistent even when the estimating equation does not exist in this respect the minimax is a new estimator we first convey the idea by focusing on the quasi likelihood estimate and then indicate its full generality by providing a set of sufficient conditions for consistency and studying a number of important cases efficiency will also be verified

cubic column relations in truncated moment problems for the truncated moment problem associated to a complex sequence 2n ij i j z i j 2n to have a representing measure it is necessary for the moment matrix m n to be positive semidefinite and for the algebraic variety v to satisfy rankm n cardv as well as a consistency condition the riesz functional vanishes on every polynomial of degree at most 2n that vanishes on v in previous work with l fialkow and h m moller the first named author proved that for the extremal case rankm n cardv positivity and consistency are sufficient for the existence of a representing measure in this paper we solve the truncated moment problem for cubic column relations in m 3 of the form z3 itz uz u t r we do this by checking consistency for u t in the open cone determined by 0 u t 2 u we first prove that the algebraic variety has exactly 7 points and rankm 3 7 we then apply the above mentioned result to obtain a concrete computable necessary and sufficient condition for the existence of a representing measure

orthogonal locally ancillary estimating functions for matched pair studies and errors in covariates summary we propose an estimating function method for two related applications matched pair studies and studies with errors in covariates under a functional model where a mismeasured unknown scalar covariate is treated as a fixed nuisance parameter our method addresses the severe inferential problem that is posed by an abundance of nuisance parameters in these two applications we propose orthogonal locally ancillary estimating functions for these two applications that depend on merely the mean model and partial modelling of the variances of the observations and observed mismeasured covariate if applicable and we achieve first order bias correction of inferences under a small dispersion and large sample size asymptotic simulation results confirm that the estimator proposed is largely improved over that using a regular profile estimating function we apply the approach proposed to a length of hospital stay study with a mismeasured covariate

matrix sum of squares relaxations for robust semi definite programs we consider robust semi definite programs which depend polynomially or rationally on some uncertain parameter that is only known to be contained in a set with a polynomial matrix inequality description on the basis of matrix sum of squares decompositions we suggest a systematic procedure to construct a family of linear matrix inequality relaxations for computing upper bounds on the optimal value of the corresponding robust counterpart with a novel matrix version of putinar s sum of squares representation for positive polynomials on compact semi algebraic sets we prove asymptotic exactness of the relaxation family under a suitable constraint qualification if the uncertainty region is a compact polytope we provide a new duality proof for the validity of putinar s constraint qualification with an a priori degree bound on the polynomial certificates finally we point out the consequences of our results for constructing relaxations based on the so called full block s procedure which allows to apply recently developed tests in order to computationally verify the exactness of possibly small sized relaxations

a deviance function for the quasi likelihood method summary we introduce a deviance function that can be used in conjunction with the quasilikelihood method the need for such functions arises when the quasi log likelihood function is not uniquely defined the deviance is obtained by projecting a pair of centred likelihood ratios onto the direct sum of two hilbert spaces spanned by the observations locally at the null and the alternative hypotheses the deviance function is equivalent to the quasi log likelihood ratio provided that the latter is uniquely defined like the quasilog likelihood ratio it is invariant antisymmetric and linear in the observations it can be defined for both independent and dependent observations in certain situations when the quasi score has multiple roots the confidence set based on the deviance is better than that based on the score test the deviance also induces a divergence measure between two sets of moments which resembles jeffreys divergence between two probability measures

strict positivstellensatze for matrix polynomials with scalar constraints abstract we extend krivine s strict positivstellensatz for usual real multivariate polynomials to symmetric matrix polynomials with scalar constraints the proof is an elementary computation with schur complements analogous extensions of schmudgen s and putinar s strict positivstellensatz were recently proved by hol and scherer using methods from optimization theory

an effective dichotomy for the counting constraint satisfaction problem bulatov proceedings of the 35 th international colloquium on automata languages and programming part 1 lecture notes in comput sci 5125 springer new york 2008 pp 646 661 gave a dichotomy for the counting constraint satisfaction problem csp a problem from csp is characterized by a constraint language gamma a fixed finite set of relations over a finite domain d an instance of the problem uses these relations to constrain an arbitrarily large finite set of variables bulatov showed that the problem of counting the satisfying assignments of instances of any problem from csp is either in polynomial time fp or is p complete his proof draws heavily on techniques from universal algebra and cannot be understood without a secure grasp of that field we give an elementary proof of bulatov s dichotomy based on succinct representations which we call frames of a class of highly structured relations which we call strongly rectangular we show that these are precisely the relation

a projected likelihood function for semiparametric models summary in a sequence of papers mcleish 1984 hutton nelson 1986 and godambe heyde 1987 have motivated the use of the quasi score function as an estimating function for semiparametric models by showing that it is the projection of the true score into a class of unbiased estimating functions thus the quasi score approximates the score within this class of functions in this paper we introduce a class of inference functions for independent observations that contain analogues of likelihood ratios under semiparametric assumptions this class of functions which contains the semiparametric estimating functions as a subspace is built through the use of tensor products for models with independent observations we consider the projection of the likelihood ratios into this class of inference functions it is shown that this projected likelihood is first order equivalent to the quasi likelihood

a simple and accurate method for approximate conditional inference applied to exponential family models summary a simple method for approximate conditional inference is described the methodology is applied to natural exponential family models where it is shown to provide accurate approximations to fully conditional estimates the approximation technique can be applied much more generally than in this particular class of models the technique depends only on the construction of certain projected scores derived from higher order likelihood derivatives and their covariances and so can be used in many problems where it is relevant to control for the estimation of nuisance parameters

hard constraint satisfaction problems have hard gaps at location 1 an instance of the maximum constraint satisfaction problem max csp is a finite collection of constraints on a set of variables and the goal is to assign values to the variables that maximises the number of satisfied constraints max csp captures many well known problems such as maxk sat and max cut and is consequently np hard thus it is natural to study how restrictions on the allowed constraint types or constraint language affect the complexity and approximability of max csp the pcp theorem is equivalent to the existence of a constraint language for which max csp has a hard gap at location 1 i e it is np hard to distinguish between satisfiable instances and instances where at most some constant fraction of the constraints are satisfiable all constraint languages for which the csp problem i e the problem of deciding whether all constraints can be satisfied is currently known to be np hard have a certain algebraic property we prove that any constraint language with this algebraic property makes max csp have a hard gap at location 1 which in particular implies that such problems cannot have a ptas unless p np we then apply this result to max csp restricted to a single constraint type this class of problems contains for instance max cut and max dicut assuming p np we show that such problems do not admit ptas except in some trivial cases our results hold even if the number of occurrences of each variable is bounded by a constant finally we give some applications of our results

simple relaxed conditional likelihood when the data are sparse but not exceedingly so we face a trade off between bias and precision that makes the usual choice between conducting either a fully unconditional inference or a fully conditional inference unduly restrictive we propose a method to relax the conditional inference that relies upon commonly available computer outputs in the rectangular array asymptotic setting the relaxed conditional maximum likelihood estimator has smaller bias than the unconditional estimator and smaller mean square error than the conditional estimator

non negative hereditary polynomials in a free algebra we prove a non negative stellensatz and a null stellensatz for a class of polynomials called hereditary polynomials in a free algebra

the l problem of moments in two dimensions abstract the classical solution to the l problem of moments due to n i akhiezer and m g krein is partially generalized to the two dimensional case a parallel between the apparently quite different one dimensional and two dimensional l problems of moments is established by means of the theories of the phase shift and of the principal function this point of view brings into the field the geometry of hilbert spaces and the analysis of the pairs of not necessarily commuting self adjoint operators acting on them

probleme des moments sur un compact de rn et decomposition de polynomes a plusieurs variables on donne la solution du probleme des moments pour un compact d interieur non vide quelconque de r n on utilise cette resolution pour obtenir un resultat de decomposition des polynomes strictement positifs sur un compact d interieur non vide

real algebraic geometry for matrices over commutative rings abstract we define and study preorderings and orderings on rings of the form m n r where r is a commutative unital ring we extend the artin lang theorem and krivine stengle stellensatze both abstract and geometric from r to m n r this problem has been open since the seventies when hilbert s 17th problem was extended from usual to matrix polynomials while the orderings of m n r are in one to one correspondence with the orderings of r this is not true for preorderings therefore our theory is not morita equivalent to the classical real algebraic geometry

classifying the complexity of constraints using finite algebras many natural combinatorial problems can be expressed as constraint satisfaction problems this class of problems is known to be np complete in general but certain restrictions on the form of the constraints can ensure tractability here we show that any set of relations used to specify the allowed forms of constraints can be associated with a finite universal algebra and we explore how the computational complexity of the corresponding constraint satisfaction problem is connected to the properties of this algebra hence we completely translate the problem of classifying the complexity of restricted constraint satisfaction problems into the language of universal algebra we introduce a notion of tractable algebra and investigate how the tractability of an algebra relates to the tractability of the smaller algebras which may be derived from it including its subalgebras and homomorphic images this allows us to reduce significantly the types of algebras which need to be classified using our results we also show that if the decision problem associated with a given collection of constraint types can be solved efficiently then so can the corresponding search problem we then classify all finite strictly simple surjective algebras with respect to tractability obtaining a dichotomy theorem which generalizes schaefer s dichotomy for the generalized satisfiability problem finally we suggest a possible general algebraic criterion for distinguishing the tractable and intractable cases of the constraint satisfaction problem

constraint satisfaction problems solvable by local consistency methods we prove that constraint satisfaction problems without the ability to count are solvable by the local consistency checking algorithm this settles three equivalent conjectures feder vardi sicomp 98 bulatov lics 04 and larose zadori au 07

the complexity of quantified constraint satisfaction collapsibility sink algebras and the three element case the constraint satisfaction probem csp is a well acknowledged framework in which many combinatorial search problems can be naturally formulated the csp may be viewed as the problem of deciding the truth of a logical sentence consisting of a conjunction of constraints in front of which all variables are existentially quantified the quantified constraint satisfaction problem qcsp is the generalization of the csp where universal quantification is permitted in addition to existential quantification the general intractability of these problems has motivated research studying the complexity of these problems under a restricted constraint language which is a set of relations that can be used to express constraints this paper introduces collapsibility a technique for deriving positive complexity results on the qcsp in particular this technique allows one to show that for a particular constraint language the qcsp reduces to the csp we show that collapsibility applies to three known tractable cases of the qcsp that were originally studied using disparate proof techniques in different decades quantified 2 sat aspvall plass and tarjan in 1979 quantified horn sat karpinski kleine buning and schmitt in 1987 and quantified affine sat creignou khanna and sudan in 2001 this reconciles and reveals common structure among these cases which are describable by constraint languages over a two element domain in addition to unifying these known tractable cases we study constraint languages over domains of larger size

an evaluation of the power and conditionality properties of empirical likelihood empirical and dual likelihood are two of a growing array of artificial or approximate likelihoods currently in use in statistics a question of major interest focuses on the performance of these new constructs relative to ordinary parametric likelihoods we consider here two criteria local power and conditional properties looking at tests of a scalar parameter we show that there is no loss of efficiency in using a dual or empirical likelihood model to second order to third order either the artificial likelihood or the true likelihood test could be more efficient this is determined by the fisher information the distance between the null and the alternative hypotheses and the statistical curvature of the models conditionality properties are assessed by comparing empirical likelihood with quasilikelihood if there is overdispersion present or more generally an unknown amount of dispersion then there is little difference in the ancillaries for conditional inference based on empirical likelihood or on other possible likelihoods such as a quasilikelihood if the amount of dispersion is known however it is correct to base inference on the quasilikelihood

estimating dynamic panel data discrete choice models with fixed effects abstract this paper considers the estimation of dynamic binary choice panel data models with fixed effects it is shown that the modified maximum likelihood estimator mmle used in this paper reduces the order of the bias in the maximum likelihood estimator from o t 1 to o t 2 without increasing the asymptotic variance no orthogonal reparametrization is needed monte carlo simulations are used to evaluate its performance in finite samples where t is not large in probit and logit models containing lags of the endogenous variable and exogenous variables the estimator is found to have a small bias in a panel with eight periods a distinctive advantage of the mmle is its general applicability estimation and relevance of different policy parameters of interest in this kind of models are also addressed

a simple method for the adjustment of profile likelihoods summary we propose a simple adjustment for profile likelihoods the aim of the adjustment is to alleviate some of the problems inherent in the use of profile likelihoods such as bias inconsistency and overoptimistic variance estimates the adjustment is applied to the profile loglikelihood score function at each parameter value so that its mean is zero and its variance is the negative expected derivative matrix of the adjusted score function for cases in which explicit calculation of the adjustments is difficult we give two methods to simplify their computation an automatic simulation method that requires as input only the profile loglikelihood and its first few derivatives first order asymptotic expressions some examples are provided and a comparison is made with the conditional profile log likelihood of cox and reid

asymptotic behavior of likelihood methods for exponential families when the number of parameters tends to infinity consider a sample of size n from a regular exponential family in p n dimensions let hat theta n denote the maximum likelihood estimator and consider the case where p n tends to infinity with n and where theta n is a sequence of parameter values in r p n moment conditions are provided under which hat theta n theta n o p sqrt p n n and hat theta n theta n overline x n o p p n n where overline x n is the sample mean the latter result provides normal approximation results when p 2 n n rightarrow 0 it is shown by example that even for a single coordinate of hat theta n theta n p 2 n n rightarrow 0 may be needed for normal approximation however if p 3 2 n n rightarrow 0 the likelihood ratio test statistic lambda for a simple hypothesis has a chi square approximation in the sense that 2 log lambda p n sqrt 2p n rightarrow d mathscr n 0 1

asymptotics for panel quantile regression models with individual effects this paper studies panel quantile regression models with individual fixed effects we formally establish sufficient conditions for consistency and asymptotic normality of the quantile regression estimator when the number of individuals n and the number of time periods t jointly go to infinity the estimator is shown to be consistent under similar conditions to those found in the nonlinear panel data literature nevertheless due to the non smoothness of the objective function we had to impose a more restrictive condition on t to prove asymptotic normality than that usually found in the literature the finite sample performance of the estimator is evaluated by monte carlo simulations

conditioning to reduce the sensitivity of general estimating functions to nuisance parameters summary a conditional method is presented that renders an estimating function insensitive to nuisance parameters the approach is a generalisation of the conditional score method to a general estimating function context and does not require complete specification of the probability model we exploit the informal relationship between general estimating functions and score functions to derive simple generalisations of sufficient and partially ancillary statistics referred to as g sufficient and g ancillary statistics respectively these two types of statistic are defined in a manner that does not require complete knowledge of the probability model and thus are more suitable for use with estimating functions if we condition on a g sufficient statistic for the nuisance parameters the resulting conditional estimating function is insensitive to nuisance parameters and in particular achieves the plug in unbiasedness property furthermore if the conditioning argument is also g ancillary for the parameters of interest then the conditional estimating function possesses an attractive optimality property

optimization of polynomials on compact semialgebraic sets a basic closed semialgebraic subset s of r n is defined by simultaneous polynomial inequalities g 1 ge 0 dotsc g m ge 0 we give a short introduction to lasserre s method for minimizing a polynomial f on a compact set s of this kind it consists of successively solving tighter and tighter convex relaxations of this problem which can be formulated as semidefinite programs we give a new short proof for the convergence of the optimal values of these relaxations to the infimum f ast of f on s which is constructive and elementary in the case where f possesses a unique minimizer x ast we prove that every sequence of nearly optimal solutions of the successive relaxations gives rise to a sequence of points in r n converging to x ast

eeboost a general method for prediction and variable selection based on estimating equations the modern statistical literature is replete with methods for performing variable selection and prediction in standard regression problems however simple models may misspecify or fail to capture important aspects of the data generating process such as missingness correlation and over underdispersion in this article we describe eeboost a strategy for variable selection and prediction which can be applied in high dimensional settings where inference for low dimensional parameters would typically be based on estimating equations the method is simple flexible and easily implemented using existing software the eeboost algorithm is obtained as a modification of the standard boosting or functional gradient descent technique we show that eeboost is closely related to a class of l1 constrained projected likelihood ratio minimizations and therefore produces similar variable selection paths to penalized methods without the need to apply constrained optimization algorithms the flexibility of eeboost is

matrix measures moment spaces and favard s theorem for the interval 0 1 and 0 abstract in this paper we study the moment spaces corresponding to matrix measures on compact intervals and on the nonnegative line 0 a representation for nonnegative definite matrix polynomials is obtained which is used to characterize moment points by properties of generalized hankel matrices we also derive an explicit representation of the orthogonal polynomials with respect to a given matrix measure which generalizes the classical determinant representation of the one dimensional case moreover the coefficients in the recurrence relations can be expressed explicitly in terms of the moments of the matrix measure these results are finally used to prove a refinement of the well known favard theorem for matrix measures which characterizes the domain of the underlying measure of orthogonality by properties of the coefficients in the recurrence relationships

maximum likelihood estimation of a survival function with a change point for truncated and interval censored data this paper considers estimation of a survival function when there exists a change point and the survival time of interest is defined as elapsed time between two related events furthermore there exists censoring on observations on the occurrences of both events and truncation on observations on the occurrence of the second event and thus the survival time of interest to obtain the maximum likelihood estimator of a survival function an em algorithm is developed when the survival function is completely unknown before the change point and known up to a vector of unknown parameters after the change point the idea is a generalization of that discussed in moeschberger and klein simulations and an example are used to evaluate and illustrate the algorithm

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

the complexity of the counting constraint satisfaction problem the counting constraint satisfaction problem csp h over a finite relational structure h can be expressed as follows given a relational structure g over the same vocabulary determine the number of homomorphisms from g to h in this article we characterize relational structures h for which csp h can be solved in polynomial time and prove that for all other structures the problem is p complete

on the unbiased asymptotic normality of quantile regression with fixed effects abstract nonlinear panel data models with fixed individual effects provide an important set of tools for describing microeconometric data in a large class of such models including probit proportional hazard and quantile regression to name just a few it is impossible to difference out the individual effects and inference is usually justified in a large n large t asymptotic framework however there is a considerable gap in the type of assumptions that are currently imposed in models with smooth score functions such as probit and proportional hazard and quantile regression in the present paper we show that this gap can be bridged and establish unbiased asymptotic normality for fixed effects quantile regression panels under conditions on n t that are very close to what is typically assumed in standard nonlinear panels our results considerably improve upon existing theory and show that quantile regression is applicable to the same type of panel data in terms of n t as other commonly used nonlinear panel data models thorough numerical experiments confirm our theoretical findings

projection as a method for increasing sensitivity and eliminating nuisance parameters l article analyse des methodes statistiques basees sur les fonctions d estimation qui sont des fonctions de parametres et de donnees a la fois destinees a permettre une inference sur un parametre inconnu une methode de projection peut etre utilisee pour la reduction de ces fonctions afin d accroitre la puissance d un test et l efficacite d un estimateur ponctuel ou bien de rendre une fonction d inference insensible a la valeur d un parametre de nuisance

the complexity of constraint satisfaction games and qcsp we study the complexity of two person constraint satisfaction games an instance of such a game is given by a collection of constraints on overlapping sets of variables and the two players alternately make moves assigning values from a finite domain to the variables in a specified order the first player tries to satisfy all constraints while the other tries to break at least one constraint the goal is to decide whether the first player has a winning strategy we show that such games can be conveniently represented by a logical form of quantified constraint satisfaction where an instance is given by a first order sentence in which quantifiers alternate and the quantifier free part is a conjunction of positive atomic formulas the goal is to decide whether the sentence is true while the problem of deciding such a game is pspace complete in general by restricting the set of allowed constraint predicates one can obtain infinite classes of constraint satisfaction games of lower complexity we use the quantified constraint satisfaction framework to study how the complexity of deciding such a game depends on the parameter set of allowed predicates with every predicate one can associate certain predicate preserving operations called polymorphisms we show that the complexity of our games is determined by the surjective polymorphisms of the constraint predicates we illustrate how this result can be used by identifying the complexity of a wide variety of constraint satisfaction games

congruence distributivity implies bounded width we show that a constraint language with compatible jonsson terms or equivalently associated with an algebra generating a congruence distributive variety defines a constraint satisfaction problem solvable by the local consistency checking algorithm

on constrained quasi likelihood estimation for maximum likelihood or least squares parameter estimation subject to a constrained parameter the standard approach is to use the method of lagrange multipliers in this paper it is shown that the same formal procedure applies very generally for constrained quasi likelihood estimation even though there is ordinarily no objective function to maximize or minimize this allows for optimal estimation in a wide range of problems where maximum likelihood cannot be used such as for semiparametric models the method is very similar to projecting the free estimator or projecting the equations which it solves

incidental versus random nuisance parameters let p vartheta eta vartheta eta in theta times h with theta subset mathbb r and h arbitrary be a family of mutually absolutely continuous probability measures on a measurable space x mathscr a the problem is to estimate vartheta based on a sample x 1 cdots x n from times n 1 p vartheta eta nu if eta 1 cdots eta n are independently distributed according to some unknown prior distribution gamma then the distribution of n 1 2 vartheta n vartheta under p n vartheta gamma p vartheta gamma being the gamma mixture of p vartheta eta eta in h cannot be more concentrated asymptotically than a certain normal distribution with mean 0 say n 0 sigma 2 0 vartheta gamma folklore says that such a bound is also valid if eta 1 cdots eta n are just unknown values of the nuisance parameter in this case the distribution cannot be more concentrated asymptotically than n 0 sigma 2 0 vartheta e n eta 1 cdots eta n where e n eta 1 cdots eta n is the empirical distribution of eta 1 cdots eta n the purpose of the present paper is to discuss to which extent this conjecture is true the results are summarized at the end of sections 1 and 3

jackknife and analytical bias reduction for nonlinear panel models fixed effects estimators of panel models can be severely biased because of the well known incidental parameters problem we show that this bias can be reduced by using a panel jackknife or an analytical bias correction motivated by large t we give bias corrections for averages over the fixed effects as well as model parameters we find large bias reductions from using these approaches in examples we consider asymptotics where t grows with n as an approximation to the properties of the estimators in econometric applications we show that if t grows at the same rate as n the fixed effects estimator is asymptotically biased so that asymptotic confidence intervals are incorrect but that they are correct for the panel jackknife we show t growing faster than n 1 3 suffices for correctness of the analytic correction a property we also conjecture for the jackknife

conditional score functions some optimality results summary the conditional score function has previously been shown to generate the optimal estimating equation for a parameter of interest when the conditioning statistic is complete and sufficient for the nuisance parameters godambe 1976 the present paper generalizes these results to partial likelihood factorizations and then examines the nature of the problem when the appropriate conditioning statistics depend on the parameter of interest in this case globally optimal estimating functions are impossible a weaker criterion of optimal weighting leads to a class of estimated conditional score functions which satisfy an information equality the purpose of this paper is to develop and justify an approach based on conditional likelihood to estimation in the presence of nuisance parameters the arguments focus on the case when there is a complete and sufficient statistic s 0 for the nuisance parameters when the parameter of interest 0 is fixed an extended notion of completeness will be presented in the context of partial likelihoods the problem takes on two separate natures depending on whether the statistic s 0 truly depends on 0 or not if s s 0 is free of 0 then the conditional likelihood of the data given s has been shown to generate optimal inferential procedures giving a best estimating equation godambe 1976 and a best asymptotically normal estimator in a mixture setting lindsay 1980 if s 0 depends on 0 then it will be shown that the best estimating equation also a conditional likelihood score will generally be only locally optimal leaving no single best strategy consideration of an optimal weighting strategy yields estimators which have second order properties similar to the conditional likelihoods of the first type

the moment problem on compact semi algebraic sets in this chapter we begin the study of the multidimensional moment problem the passage to dimensions d 2 brings new difficulties and unexpected phenomena in sect 3 2 we derived solvability criteria of the moment problem on intervals in terms of positivity conditions it seems to be natural to look for similar characterizations in higher dimensions as well this leads us immediately into the realm of real algebraic geometry and to descriptions of positive polynomials on semi algebraic sets in this chapter we treat this approach for basic closed compact semi algebraic subsets of mathbb r d it turns out that for such sets there is a close interaction between the moment problem and positivstellensatze for strictly positive polynomials

finsler s lemma for matrix polynomials abstract finsler s lemma characterizes all pairs of symmetric n n real matrices a and b which satisfy the property that v t a v 0 for every nonzero v r n such that v t b v 0 we extend this characterization to all symmetric matrices of real multivariate polynomials but we need an additional assumption that b is negative semidefinite outside some ball we also give two applications of this result to noncommutative real algebraic geometry which for n 1 reduce to the usual characterizations of positive polynomials on varieties and on compact sets

generalizations of ancillarity completeness and sufficiency in an inference function space on introduit l e ancillarite et l e exhaustivite complete des extensions naturelles des definitions d ancillarite et d exhaustivite complete a l espace des fonctions d estimation ou d inference

closure properties of constraints many combinatorial search problems can be expressed as constraint satisfaction problems and this class of problems is known to be np complete in general in this paper we investigate the subclasses that arise from restricting the possible constraint types we first show that any set of constraints that does not give rise to an np complete class of problems must satisfy a certain type of algebraic closure condition we then investigate all the different possible forms of this algebraic closure property and establish which of these are sufficient to ensure tractability as examples we show that all known classes of tractable constraints over finite domains can be characterized by such an algebraic closure property finally we describe a simple computational procedure that can be used to determine the closure properties of a given set of constraints this procedure involves solving a particular constraint satisfaction problem which we call an indicator problem

varieties with few subalgebras of powers the constraint satisfaction problem dichotomy conjecture of feder and vardi 1999 has in the last 10 years been profitably reformulated as a conjecture about the set sp fin a of subalgebras of finite cartesian powers of a finite universal algebra a one particular strategy advanced by dalmau in his doctoral thesis 2000 has confirmed the conjecture for a certain class of finite algebras a which among other things have the property that the number of subalgebras of a n is bounded by an exponential polynomial in this paper we characterize the finite algebras a with this property which we call having few subpowers and develop a representation theory for the subpowers of algebras having few subpowers our characterization shows that algebras having few subpowers are the finite members of a newly discovered and surprisingly robust maltsev class defined by the existence of a special term we call an edge term we also prove some tight connections between the asymptotic behavior of the number of subalgebras of a n and some related functions on the one hand and some standard algebraic properties of a on the other hand the theory developed here was applied to the constraint satisfaction problem dichotomy conjecture completing dalmau s strategy

projected partial likelihood and its application to longitudinal data summary an estimating equation which we call the projected partial score is introduced for longitudinal data analysis the estimating equation is obtained by projecting the partial likelihood score function onto the vector space spanned by a class of conditionally linear estimating equations we demonstrate that removing certain terms from the projection of the full likelihood score does not alter important inferential properties of the estimating equation and doing so is advantageous in handling missing data and time varying covariates within a prequential frame of reference it is shown that the estimating equation is optimal among the largest collection of estimating equations determined by the conditional moments furthermore the method possesses similar properties to generalised estimating equations in particular the correct conditional variance specification is necessary for efficiency but not for asymptotic consistency and distribution theory

approximate likelihood ratios for general estimating functions summary the method of estimating functions godambe 1991 is commonly used when one desires to conduct inference about some parameters of interest but the full distribution of the observations is unknown however this approach may have limited utility due to multiple roots for the estimating function a poorly behaved wald test or lack of a goodness of fit test this paper presents approximate likelihood ratios that can be used along with estimating functions when any of these three problems occurs we show that the approximate likelihood ratio provides correct large sample inference under very general circumstances including clustered data and misspecified weights in the estimating function two methods of constructing the approximate likelihood ratio one based on the quasi likelihood approach and the other based on the linear projection approach are compared and shown to be closely related in particular we show that quasi likelihood is the limit of the projection approach we illustrate the technique with two applications

biometrika centenary theory and general methodology note reprinted in biometrika one hundred years edited by d m titterington and d r cox oxford university press 11 50 reference stat article 2001 002doi 10 1093 biomet 88 1 13view record in web of science record created on 2006 04 21 modified on 2017 05 12

a simple algorithm for mal tsev constraints a mal tsev operation is a ternary operation varphi that satisfies the identities varphi x y y varphi y y x x constraint satisfaction problems involving constraints invariant under a mal tsev operation constitute an important class of constraint satisfaction problems which includes the affine satisfiability problem subgroup and near subgroup constraints and many others it is also known that any tractable case of the counting constraint satisfaction problem involves only mal tsev constraints the first algorithm solving the arbitrary constraint satisfaction problem with mal tsev constraints has been given by bulatov however this algorithm is very sophisticated and relies heavily on advanced algebraic machinery in this paper we give a different and much simpler algorithm for this type of constraint

affine systems of equations and counting infinitary logic we study the definability of constraint satisfaction problems csps in various fixed point and infinitary logics we show that testing the solvability of systems of equations over a finite abelian group a tractable csp that was previously known not to be definable in datalog is not definable in the infinitary logic with finitely many variables and counting this implies that it is not definable in least fixed point logic or its extension with counting we relate definability of csps to their classification obtained from tame congruence theory of the varieties generated by the algebra of polymorphisms of the template structure in particular we show that if this variety admits either the unary or affine type the corresponding csp is not definable in the infinitary logic with counting

max ones generalized to larger domains we study a family of problems called maximum solution where the objective is to maximize a linear goal function over the feasible integer assignments to a set of variables subject to a set of constraints when the domain is boolean i e restricted to 0 1 the maximum solution problem is identical to the well studied max ones problem and the approximability is completely understood for all restrictions on the underlying constraints s khanna m sudan l trevisan and d p williamson siam j comput 30 2001 pp 1863 1920 we continue this line of research by considering domains containing more than two elements we present two main results a complete classification for the approximability of all maximal constraint languages over domains of cardinality at most 4 and a complete classification of the approximability of the problem when the set of allowed constraints contains all permutation constraints under the assumption that a conjecture due to szczepara minimal clones generated by groupoids ph d thesis universite de montreal montreal qc 1996 holds we give a complete classification for all maximal constraint languages these classes of languages are well studied in universal algebra and computer science they have for instance been considered in connection with machine learning and constraint satisfaction our results are proved by using algebraic results from clone theory and the results indicate that this approach is very powerful for classifying the approximability of certain optimization problems

on the algebraic structure of combinatorial problems abstract we describe a general algebraic formulation for a wide range of combinatorial problems including satisfiability graph colorability and graph isomorphism in this formulation each problem instance is represented by a pair of relational structures and the solutions to a given instance are homomorphisms between these relational structures the corresponding decision problem consists of deciding whether or not any such homomorphisms exist we then demonstrate that the complexity of solving this decision problem is determined in many cases by simple algebraic properties of the relational structures involved this result is used to identify tractable subproblems of satisfiability and to provide a simple test to establish whether a given set of boolean relations gives rise to one of these tractable subproblems

constraints consistency and closure although the constraint satisfaction problem is np complete in general a number of constraint classes have been identified for which some fixed level of local consistency is sufficient to ensure global consistency in this paper we describe a simple algebraic property which characterises all possible constraint types for which strong k consistency is sufficient to ensure global consistency for each k 2 we give a number of examples to illustrate the application of this result

the computational structure of monotone monadic snp and constraint satisfaction a study through datalog and group theory this paper starts with the project of finding a large subclass of np which exhibits a dichotomy the approach is to find this subclass via syntactic prescriptions while the paper does not achieve this goal it does isolate a class of problems specified by monotone monadic snp without inequality which may exhibit this dichotomy we justify the placing of all these restrictions by showing essentially using ladner s theorem that classes obtained by using only two of the above three restrictions do not show this dichotomy we then explore the structure of this class we show that all problems in this class reduce to the seemingly simpler class csp we divide csp into subclasses and try to unify the collection of all known polytime algorithms for csp problems and extract properties that make csp problems np hard this is where the second part of the title a study through datalog and group theory comes in we present conjectures about this class which would end in showing the dichotomy

reducing sensitivity to nuisance parameters in semiparametric models a quasi score method summary this paper proposes a semiparametric extension of the projected score method of waterman lindsay 1996 for the elimination of nuisance parameters the procedure addresses cases where only the mean and the variance of the response variable are specified and where the mean function involves both parameters of interest and nuisance parameters important applications of the semiparametric model include quasilikelihood models for matched designs and for measurement error models carroll stefanski 1990 as a result of the optimality and information unbiasedness of the quasi score function a second order quasi score basis of estimating functions for the nuisance parameter is derived second order locally ancillary estimating functions small mcleish 1994 pp 81 4 are then obtained by solving a simple linear system that corresponds to a true projection for canonical exponential family distributions asymptotic arguments and simulation work show that the impact of nuisance parameters is considerably reduced when adopting the proposed approach

constraint satisfaction problems over semilattice block mal tsev algebras abstract the dichotomy conjecture for the constraint satisfaction problem csp was recently settled independently by zhuk and the author the proofs of this conjecture are rather sophisticated and require deep understanding of the algebraic structure of csps this paper is a precursor of the author s proof of the dichotomy conjecture and represents its main ideas in a simpler and clearer form in a more restricted class of the csp there are two well known types of algorithms for solving csps local propagation and generating a basis of the solution space for several years the focus of the csp research has been on hybrid algorithms that somehow combine the two approaches in this paper we present a new method of such hybridization that allows us to solve certain csps that has been out of reach for a quite a while and eventually leads to resolving the dichotomy conjecture we apply this method to csps parametrized by a universal algebra an approach that has been very popular in the last decade or so specifically we consider a fairly restricted class of algebras we will call semilattice block mal tsev an algebra a is semilattice block mal tsev if it has a binary operation f a ternary operation m and a congruence such that the quotient a with operation f is a semilattice f is a projection on every block of and every block of is a mal tsev algebra with mal tsev operation m this means that the domain in such a csp is partitioned into blocks such that if the problem is considered on the quotient set a it can be solved by a simple constraint propagation algorithm on the other hand if the problem is restricted on individual blocks it can be solved by generating a basis of the solution space we show that the two methods can be combined in a highly nontrivial way and therefore the constraint satisfaction problem over a semilattice block mal tsev algebra is solvable in polynomial time

efficiency of projected score methods in rectangular array asymptotics summary the paper considers a rectangular array asymptotic embedding for multistratum data sets in which both the number of strata and the number of within stratum replications increase and at the same rate it is shown that under this embedding the maximum likelihood estimator is consistent but not efficient owing to a non zero mean in its asymptotic normal distribution by using a projection operator on the score function an adjusted maximum likelihood estimator can be obtained that is asymptotically unbiased and has a variance that attains the cram6r rao lower bound the adjusted maximum likelihood estimator can be viewed as an approximation to the conditional maximum likelihood estimator

a representation theorem for certain partially ordered commutative rings let a be a commutative ring with 1 let p subset a be a preordering of higher level i e 0 1 in p 1 not in p p p subset p p cdot p subset p and a 2n subset p for some n in n and let m subset a be an archimedean p module i e 1 in m 1 not in m m m subset m p cdot m subset m and forall a in a exists n in n n a in m we endow x m varphi in rm hom a r mid varphi m subset r with the weak topology with respect to all mappings widehat a x m rightarrow r widehat a varphi varphi a and consider the representation phi m a rightarrow cal c x m r a mapsto widehat a we find that x m is a non empty compact hausdorff space further we prove that

complexity of counting csp with complex weights we give a complexity dichotomy theorem for the counting constraint satisfaction problem csp in short with algebraic complex weights to this end we give three conditions for its tractability let f be any finite set of algebraic complex valued functions defined on an arbitrary finite domain we show that csp f is solvable in polynomial time if all three conditions are satisfied and is p hard otherwise our dichotomy theorem generalizes a long series of important results on counting problems and reaches a natural culmination a the problem of counting graph homomorphisms is the special case when f has a single symmetric binary function dyer and greenhill 2000 bulatov and grohe 2005 goldberg et al 2010 cai et al 2013 b the problem of counting directed graph homomorphisms is the special case when f has a single but not necessarily symmetric binary function dyer et al 2007 cai and chen 2010 c the unweighted form of csp is when all functions in f take values in 0 1 bulatov 2008 dyer and richerby 2013

locally ancillary quasi score models for errors in covariates we use the notion of locally ancillary estimating functions to develop a quasi score method for fitting regression models containing measurement error in the covariates suppose that interest is on the model e y u w for response y the observed data are y x w and x is a mismeasured surrogate for u we take a functional modeling approach treating the u as a fixed nuisance parameter beginning with quasi scores for the regression parameter and the unknown u we derive a bias corrected quasi score for the regression parameter that is second order locally ancillary for the nuisance u our method for this requires only the correct specification of the mean and variance functions for y and x in terms of u w and the regression parameter when an estimator for u is plugged into the corrected quasi score local approximations show that the bias is small we present simulations verifying this result and an example from child psychiatry both using log linear regression models

positivity and optimization for semi algebraic functions we describe algebraic certificates of positivity for functions belonging to a finitely generated algebra of borel measurable functions with particular emphasis on algebras generated by semi algebraic functions in this case the standard global optimization problem with constraints given by elements of the same algebra is reduced via a natural change of variables to the better understood case of polynomial optimization a collection of simple examples and numerical experiments complement the theoretical parts of the article

an optimal estimating equation based on the first three cumulants for improving on quasilikelihood estimation two types of quadratic estimating equations have been proposed one based on the edgeworth expansion the other on the generalisation of the quasi score the first requires that the skewness of observations has a small departure from the exponential family the second requires the knowledge of both skewness and kurtosis we introduce an optimal quadratic estimating equation applicable when the skewness is not small and the kurtosis is unknown apart from optimality the manner in which skewness is incorporated ensures that its misspecification does not affect n consistency and that the estimator enjoys an invariance property akin to that of the bias corrected maximum likelihood estimate simulations indicate a solid improvement in accuracy

empirical likelihood in missing data problems missing data is a ubiquitous problem in medical and social sciences it is well known that inferences based only on the complete data may not only lose efficiency but may also lead to biased results if the data is not missing completely at random mcar the inverse probability weighting method proposed by horvitz and thompson 1952 is a popular alternative when the data is not mcar the horvitz thompson method however is sensitive to the inverse weights and may suffer from loss of efficiency in this paper we propose a unified empirical likelihood approach to missing data problems and explore the use of empirical likelihood to effectively combine unbiased estimating equations when the number of estimating equations is greater than the number of unknown parameters one important feature of this approach is the separation of the complete data unbiased estimating equations from the incomplete data unbiased estimating equations the proposed method can achieve semiparametric efficiency if the probabilit