edgeworth correction by bootstrap in autoregressions we prove that the distribution of least squares estimates in autoregressions can be bootstrapped with accuracy o n 1 2 a s thereby improving the normal approximation error of o n

inference on impulse response functions in structural var models skepticism toward traditional identifying assumptions based on exclusion restrictions has led to a surge in the use of structural var models in which structural shocks are identified by restricting the sign of the responses of selected macroeconomic aggregates to these shocks researchers commonly report the vector of pointwise posterior medians of the impulse responses as a measure of central tendency of the estimated response functions along with pointwise 68 percent posterior error bands it can be shown that this approach cannot be used to characterize the central tendency of the structural impulse response functions we propose an alternative method of summarizing the evidence from sign identified var models designed to enhance their practical usefulness our objective is to characterize the most likely admissible model s within the set of structural var models that satisfy the sign restrictions we show how the set of most likely structural response functions can be computed from the posterior mode of the joint distribution of admissible models both in the fully identified and in the partially identified case and we propose a highest posterior density credible set that characterizes the joint uncertainty about this set our approach can also be used to resolve the long standing problem of how to conduct joint inference on sets of structural impulse response functions in exactly identified var models we illustrate the differences between our approach and the traditional approach for the analysis of the effects of monetary policy shocks and of the effects of oil demand and oil supply shocks

on the defining equations of the tangent cone of a numerical semigroup ring abstract let a a 1 a r be a sequence of positive integers and let h a denote the semigroup generated by a 1 a r for an integer k 0 we denote by a k the shifted sequence a 1 k a r k fix a field k we show that for all k 0 the tangent cone of the semigroup ring k h a k is cohen macaulay and that it has the same betti numbers as k h a k itself as a consequence we show that the number of defining equations of the tangent cone of a numerical semigroup ring is bounded by a value depending only on the width of the semigroup where the width of a numerical semigroup is defined to be the difference of the largest and the smallest element in the minimal generating set of the semigroup we also provide a conjectured upper bound of the above number of equations and we verify it in some cases

using empirical likelihood methods to obtain range restricted weights in regression estimators for surveys design weights in surveys are often adjusted to accommodate auxiliary information and to meet pre specified range restrictions typically via some ad hoc algorithmic adjustment to a generalised regression estimator in this paper we present a simple solution to this problem using empirical likelihood methods or generalised regression we first develop algorithms for computing empirical likelihood estimators and model calibrated empirical likelihood estimators the first algorithm solves the computational problem of the empirical likelihood method in general both in survey and non survey settings and theoretically guarantees its convergence the second exploits properties of the model calibration method and is particularly simple the algorithms are adapted for handling benchmark constraints and pre specified range restrictions on the weight adjustments

understanding unit rooters a helicopter tour skewed asymptotically while the likelihood and hence the posterior pdf remains symmetric we show that no single prior can rationalize treating p values as probabilities in these models and we display examples of the sample dependent priors that would do so we argue that these results imply at a minimum that the usual test statistics and covariance matrices for autoregressions which characterize the likelihood shape in dynamic models just as in static regression models should be reported without any corrections for the special unit root distribution theory even if the corrected classical p values are reported as well

inference in linear time series models with some unit roots this paper considers estimation and hypothesis testing in linear time series models when some or all of the variables have unit roots our motivating example is a vector autoregression with some unit roots in the companion matrix which might include polynomials in time as regressors in the general formulation the variable might be integrated or cointegrated of arbitrary orders and might have drifts as well we show that parameters that can be written as coefficients on mean zero nonintegrated regressors have jointly normal asymptotic distributions converging at the rate t 2 in general the other coefficients including the coefficients on polynomials in time will have nonnormal asymptotic distributions the results provide a formal characterization of which t or f tests such as granger causality tests will be asymptotically valid and which will have nonstandard limiting distributions

predictive ability with cointegrated variables in this paper we outline conditions under which the diebold and mariano dm j bus econom statist 13 1995 253 test for predictive ability can be extended to the case of two forecasting models each of which may include cointegrating relations when allowing for parameter estimation error we show that in the cases where either the loss function is quadratic or the length of the prediction period p grows at a slower rate than the length of the regression period r the standard dm test can be used on the other hand in the case of a generic loss function if p r as t 0 then the asymptotic normality result of west econometrica 64 1996 1067 no longer holds we also extend the data snooping technique of white econometrica 68 2000 1097 for comparing the predictive ability of multiple forecasting models to the case of cointegrated variables in a series of monte carlo experiments we examine the impact of both short run and cointegrating vector parameter estimation error on dm data snooping and related tests our results suggest that size is reasonable for r and p greater than 50 and power improves with p as expected furthermore the additional cost in terms of size distortion due to the estimation of the cointegrating relations is not substantial we illustrate the use of the tests in a nonnested cointegration framework by forming prediction models for industrial production which include two interest rate variables prices and either m1 m2 or m3

empirical likelihood and general estimating equations for some time so called empirical likelihoods have been used heuristically for purposes of nonparametric estimation owen showed that empirical likelihood ratio statistics for various parameters theta f of an unknown distribution f have limiting chi square distributions and may be used to obtain tests or confidence intervals in a way that is completely analogous to that used with parametric likelihoods our objective in this paper is twofold first to link estimating functions or equations and empirical likelihood second to develop methods of combining information about parameters we do this by assuming that information about f and theta is available in the form of unbiased estimating functions empirical likelihoods for parameters are developed and shown to have properties similar to those for parametric likelihood efficiency results for estimates of both theta and f are obtained the methods are illustrated on several problems and areas for future investigation are noted

empirical likelihood ratio confidence intervals for a single functional summary the empirical distribution function based on a sample is well known to be the maximum likelihood estimate of the distribution from which the sample was taken in this paper the likelihood function for distributions is used to define a likelihood ratio function for distributions it is shown that this empirical likelihood ratio function can be used to construct confidence intervals for the sample mean for a class of m estimates that includes quantiles and for differentiable statistical functionals the results are nonparametric extensions of wilks s 1938 theorem for parametric likelihood ratios the intervals are illustrated on some real data and compared in a simulation to some bootstrap confidence intervals and to intervals based on student s t statistic a hybrid method that uses the bootstrap to determine critical values of the likelihood ratio is introduced

a model calibration approach to using complete auxiliary information from survey data suppose that the finite population consists of n identifiable units associated with the ith unit are the study variable yi and a vector of auxiliary variables xi the values x1 x2 xn are known for the entire population i e complete but yi is known only if the ith unit is selected in the sample one of the fundamental questions is how to effectively use the complete auxiliary information at the estimation stage in this article a unified model assisted framework has been attempted using a proposed model calibration technique the proposed model calibration estimators can handle any linear or nonlinear working models and reduce to the conventional calibration estimators of deville and sarndal and or the generalized regression estimators in the linear model case the pseudoempirical maximum likelihood estimator of chen and sitter when used in this setting gives an estimator that is asymptotically equivalent to the model calibration estimator but with positive weights some existing estimators

a general bahadur representation of m estimators and its application to linear regression with nonstochastic designs we obtain strong bahadur representations for a general class of m estimators that satisfies sigma i psi x i theta o delta n where the x i s are independent but not necessarily identically distributed random variables the results apply readily to m estimators of regression with nonstochastic designs more specifically we consider the minimum l p distance estimators bounded influence gm estimators and regression quantiles under appropriate design conditions the error ratesobtained for the first order approximations are sharp in these cases we also provide weaker and more easily verifiable conditions that suffice for an error rate that is suboptimal but strong enough for deriving the asymptotic distribution of m estimators in a wide variety of problems

median regression models for longitudinal data with dropouts summary recently median regression models have received increasing attention when continuous responses follow a distribution that is quite different from a normal distribution usual mean regression models may fail to produce efficient estimators whereas median regression models may perform satisfactorily in this article we discuss using median regression models to deal with longitudinal data with dropouts weighted estimating equations are proposed to estimate the median regression parameters for incomplete longitudinal data where the weights are determined by modeling the dropout process consistency and the asymptotic distribution of the resultant estimators are established the proposed method is used to analyze a longitudinal data set arising from a controlled trial of hiv disease volberding et al 1990 the new england journal of medicine 322 941 949 simulation studies are conducted to assess the performance of the proposed method under various situations an extension to estimation of the association parameters is outlined

empirical likelihood for estimating equations with missing values we consider an empirical likelihood inference for parameters defined by general estimating equations when some components of the random observations are subject to missingness as the nature of the estimating equations is wide ranging we propose a nonparametric imputation of the missing values from a kernel estimator of the conditional distribution of the missing variable given the always observable variable the empirical likelihood is used to construct a profile likelihood for the parameter of interest we demonstrate that the proposed nonparametric imputation can remove the selection bias in the missingness and the empirical likelihood leads to more efficient parameter estimation the proposed method is further evaluated by simulation and an empirical study on a genetic dataset on recombinant inbred mice

cohen macaulayness of almost complete intersection tangent cones abstract we prove that associated graded rings of complete intersection monomial curves of codimension three are cohen macaulay if their defining ideals are generated by at most four elements

bounded efficient and doubly robust estimation with inverse weighting consider estimating the mean of an outcome in the presence of missing data or estimating population average treatment effects in causal inference a doubly robust estimator remains consistent if an outcome regression model or a propensity score model is correctly specified we build on a previous nonparametric likelihood approach and propose new doubly robust estimators which have desirable properties in efficiency if the propensity score model is correctly specified and in boundedness even if the inverse probability weights are highly variable we compare the new and existing estimators in a simulation study and find that the robustified likelihood estimators yield overall the smallest mean squared errors

noninformative priors and frequentist risks of bayesian estimators of vector autoregressive models in this study we examine posterior properties and frequentist risks of bayesian estimators based on several noninformative priors in vector autoregressive var models we prove existence of the posterior distributions and posterior moments under a general class of priors using a variety of priors in this class we conduct numerical simulations of posteriors we find that in most examples bayesian estimators with a shrinkage prior on the var coefficients and the reference prior of yang and berger ann statist 22 1994 1195 on the var covariance matrix dominate mle bayesian estimators with the diffuse prior and bayesian estimators with the prior used in rats we also examine the informative minnesota prior and find that its performance depends on the nature of the data sample and on the tightness of the minnesota prior a tightly set minnesota prior is better when the data generating processes are similar to random walks but the shrinkage prior or constant prior can be better otherwise

asymptotic distributions of impulse responses step responses and variance decompositions of estimated linear dynamic models formulas are derived for computing asymptotic covariance matrices of sets of impulse responses step responses or variance decompositions of estimated dynamic simultane ous equations models in vector autoregressive moving average varma form com puted covariances would be used to test linear restrictions on sets of impulse responses step responses or variance decompositions the results unify and extend previous formulas to handle any model in varma form provide accurate computations based on analytic derivatives and provide insights into the structures of the asymptotic covariances following the lead of sims 1980 impulse responses step responses and variance decompositions have become standard tools for examining the empiri cal validity of economic theories or interpreting the forecasting and policy implications of estimated linear dynamic models runkle 1987 demonstrated the importance of accounting for sampling variability in these exercises he computed confidence bounds for impulse responses and variance decomposi tions derived from estimated vector autoregressive var models with sto chastic simulations and asymptotic normal approximations the two methods produced comparable results but the simulations took about 160 times longer to compute than normal approximations based on numerical derivatives whereas runkle only reported confidence bounds for individual impulse re sponses and decomposed variances of var models the present paper derives equations for computing full asymptotic covariance matrices of sets of impulse responses step responses or variance decompositions of estimated dynamic simultaneous equations models in autoregressive moving average form the computed covariances may be used to jointly test multiple linear restrictions or linear approximations of differentiable nonlinear restrictions on sets of im pulse responses step responses or variance decompositions of estimated mod els because the derived computational equations are based on analytical derivatives they have at least two advantages they lead to more accurate and often faster computations than those based on numerical derivatives they provide insight into the structures of asymptotic covariances of the estimated quantities by allowing for various identifying restrictions the paper extends and unifies results of schmidt 1973 evans and wells 1986 and liutkepohl 1988 1989

multiply robust estimation in regression analysis with missing data doubly robust estimators are widely used in missing data analysis they provide double protection on estimation consistency against model misspecifications however they allow only a single model for the missingness mechanism and a single model for the data distribution and the assumption that one of these two models is correctly specified is restrictive in practice for regression analysis with possibly missing outcome we propose an estimation method that allows multiple models for both the missingness mechanism and the data distribution the resulting estimator is consistent if any one of those multiple models is correctly specified and thus provides multiple protection on consistency this estimator is also robust against extreme values of the fitted missingness probability which for most doubly robust estimators can lead to erroneously large inverse probability weights that may jeopardize the numerical performance the numerical implementation of the proposed method through a modified newton ra

a general framework for quantile estimation with incomplete data quantile estimation has attracted significant research interests in recent years however there has been only a limited literature on quantile estimation in the presence of incomplete data in this paper we propose a general framework to address this problem our framework combines the two widely adopted approaches for missing data analysis the imputation approach and the inverse probability weighting approach via the empirical likelihood method the proposed method is capable of dealing with many different missingness settings we mainly study three of them i estimating the marginal quantile of a response that is subject to missingness while there are fully observed covariates ii estimating the conditional quantile of a fully observed response while the covariates are partially available and iii estimating the conditional quantile of a response that is subject to missingness with fully observed covariates and extra auxiliary variables the proposed method allows multiple models for both the missingness probability and the data distribution the resulting estimators are multiply robust in the sense that they are consistent if any one of these models is correctly specified the asymptotic distributions are established using the empirical process theory

identification of linear stochastic models with covariance restrictions abstract the purpose of this paper is to provide a systematic treatment of the problem of identification in systems of linear structural equations where some of the disturbances are uncorrelated

efficient quantile regression analysis with missing observations this article examines the problem of estimation in a quantile regression model when observations are missing at random under independent and nonidentically distributed errors we consider three approaches of handling this problem based on nonparametric inverse probability weighting estimating equations projection and a combination of both an important distinguishing feature of our methods is their ability to handle missing response and or partially missing covariates whereas existing techniques can handle only one or the other but not both we prove that our methods yield asymptotically equivalent estimators that achieve the desirable asymptotic properties of unbiasedness normality and n consistency because we do not assume that the errors are identically distributed our theoretical results are valid under heteroscedasticity a particularly strong feature of our methods under the special case of identical error distributions all of our proposed estimators achieve the semiparametric efficiency b

improving efficiency and robustness of the doubly robust estimator for a population mean with incomplete data considerable recent interest has focused on doubly robust estimators for a population mean response in the presence of incomplete data which involve models for both the propensity score and the regression of outcome on covariates the usual doubly robust estimator may yield severely biased inferences if neither of these models is correctly specified and can exhibit nonnegligible bias if the estimated propensity score is close to zero for some observations we propose alternative doubly robust estimators that achieve comparable or improved performance relative to existing methods even with some estimated propensity scores close to zero

error bands for impulse responses we show how correctly to extend known methods for generating error bands in reduced form var s to overidentified models we argue that the conventional pointwise bands common in the literature should be supplemented with measures of shape uncertainty and we show how to generate such measures we focus on bands that characterize the shape of the likelihood such bands are not classical confidence regions we explain that classical confidence regions mix information about parameter location with information about model fit and hence can be misleading as summaries of the implications of the data for the location of parameters because classical confidence regions also present conceptual and computational problems in multivariate time series models we suggest that likelihood based bands rather than approximate confidence bands based on asymptotic theory be standard in reporting results for this type of model

weighted quantile regression for analyzing health care cost data with missing covariates analysis of health care cost data is often complicated by a high level of skewness heteroscedastic variances and the presence of missing data most of the existing literature on cost data analysis have been focused on modeling the conditional mean in this paper we study a weighted quantile regression approach for estimating the conditional quantiles health care cost data with missing covariates the weighted quantile regression estimator is consistent unlike the naive estimator and asymptotically normal furthermore we propose a modified bic for variable selection in quantile regression when the covariates are missing at random the quantile regression framework allows us to obtain a more complete picture of the effects of the covariates on the health care cost and is naturally adapted to the skewness and heterogeneity of the cost data the method is semiparametric in the sense that it does not require to specify the likelihood function for the random error or the covariates we investigate the weighted quantile regression procedure and the modified bic via extensive simulations we illustrate the application by analyzing a real data set from a health care cost study copyright 2013 john wiley sons ltd

improving semiparametric estimation by using surrogate data summary the paper considers estimating a parameter that defines an estimating function u y x for an outcome variable y and its covariate x when the outcome is missing in some of the observations we assume that in addition to the outcome and the covariate a surrogate outcome is available in every observation the efficiency of existing estimators for depends critically on correctly specifying the conditional expectation of u given the surrogate and the covariate when the conditional expectation is not correctly specified which is the most likely scenario in practice the efficiency of estimation can be severely compromised even if the propensity function of missingness is correctly specified we propose an estimator that is robust against the choice of the conditional expectation via an empirical likelihood we demonstrate that the estimator proposed achieves a gain in efficiency whether the conditional score is correctly specified or not when the conditional score is correctly specified the estimator reaches the semiparametric variance bound within the class of estimating functions that are generated by u the practical performance of the estimator is evaluated by using simulation and a data set that is based on the 1996 us presidential election

hilbert function of local artinian level rings in codimension two abstract in this paper we characterize the possible hilbert functions of a local artinian level ring of given type and socle degree in the codimension two case in particular given an admissible numerical function h an effective method is given to produce a zero dimensional ideal i of r k x y such that r i is a local artinian level ring having h as hilbert function as a consequence we recover the well known result proved by briancon and iarrobino concerning the hilbert functions of a complete intersection of height two we extend notions of standard or grobner bases from the graded to the local algebra context the principal tools concern the concepts of enhanced standard basis and generic initial ideals which are defined for an ideal in the formal power series ring r k x 1 x n

block recursion and structural vector autoregressions in applications of structural var modeling finite sample properties may be difficult to obtain when certain identifying restrictions are imposed on lagged relationships as a result even though imposing some lagged restrictions makes economic sense lagged relationships are often left unrestricted to make statistical inference more convenient this paper develops block monte carlo methods to obtain both maximum likelihood estimates and exact bayesian inference when certain types of restrictions are imposed on the lag structure these methods are applied to two examples to illustrate the importance of imposing restrictions on lagged relationships

the central role of the propensity score in observational studies for causal effects abstract the results of observational studies are often disputed because of nonrandom treatment assignment for example patients at greater risk may be overrepresented in some treatment group this paper discusses the central role of propensity scores and balancing scores in the analysis of observational studies the propensity score is the estimated conditional probability of assignment to a particular treatment given a vector of observed covariates both large and small sample theory show that adjustment for the scalar propensity score is sufficient to remove bias due to all observed covariates applications include matched sampling on the univariate propensity score which is equal percent bias reducing under more general conditions than required for discriminant matching multivariate adjustment by subclassification on balancing scores where the same subclasses are used to estimate treatment effects for all outcome variables and in all subpopulations and visual representation of multivariate adjustment by a two dimensional plot author

estimation of regression coefficients when some regressors are not always observed abstract in applied problems it is common to specify a model for the conditional mean of a response given a set of regressors a subset of the regressors may be missing for some study subjects either by design or happenstance in this article we propose a new class of semiparametric estimators based on inverse probability weighted estimating equations that are consistent for parameter vector 0 of the conditional mean model when the data are missing at random in the sense of rubin and the missingness probabilities are either known or can be parametrically modeled we show that the asymptotic variance of the optimal estimator in our class attains the semiparametric variance bound for the model by first showing that our estimation problem is a special case of the general problem of parameter estimation in an arbitrary semiparametric model in which the data are missing at random and the probability of observing complete data is bounded away from 0 and then deriving a representation for the efficient score

multiple imputation in quantile regression summary we propose a multiple imputation estimator for parameter estimation in a quantile regression model when some covariates are missing at random the estimation procedure fully utilizes the entire dataset to achieve increased efficiency and the resulting coefficient estimators are root n consistent and asymptotically normal to protect against possible model misspecification we further propose a shrinkage estimator which automatically adjusts for possible bias the finite sample performance of our estimator is investigated in a simulation study finally we apply our methodology to part of the eating at american s table study data investigating the association between two measures of dietary intake

on the hilbert function of one dimensional local complete intersections abstract the sequences that occur as hilbert functions of standard graded algebras a are well understood by macaulay s theorem those that occur for graded complete intersections are elementary and were known classically however much less is known in the local case once the dimension of a is greater than zero or the embedding dimension is three or more using an extension to the power series ring r of grobner bases with respect to local degree orderings we characterize the hilbert functions h of one dimensional quadratic complete intersections a r i i f g of type 2 2 that is that are quotients of the power series ring r in three variables by a regular sequence f g whose initial forms are linearly independent and of degree 2 we also give a structure theorem up to analytic isomorphism of a for the minimal system of generators of i given the hilbert function more generally when the type of i is 2 b we are able to give some restrictions on the hilbert function in this case we can also prove that the associated graded algebra of a is cohen macaulay if and only if the hilbert function of a is strictly increasing

speed of adjustment in cointegrated systems this paper discusses summary measures for the speed of adjustment in possibly cointegrated vector autoregressive processes var in particular we propose long run half lives based on interim and total multipliers we discuss their relation with granger noncausality and other types of half life which are shown to convey different information except in the univariate ar 1 case we present likelihood based inference on long run half lives regarded as discrete functions of parameters in the var model it is shown how asymptotic confidence regions can be defined an empirical illustration concerning speed of adjustment to purchasing power parity is provided

intrinsic efficiency and multiple robustness in longitudinal studies with drop out intrinsic efficiency and multiple robustness are desirable properties in missing data analysis we establish both for estimating the mean of a response at the end of a longitudinal study with drop out the idea is to calibrate the estimated missingness probability at each visit using data from past visits we consider one working model for the missingness probability and multiple working models for the data distribution intrinsic efficiency guarantees that when the missingness probability is correctly modelled the multiple data distribution models combined with data prior to the end of the study are optimally accommodated to maximize efficiency the efficiency generally increases with the number of data distribution models except where one such model is correctly specified as well in which case all the proposed estimators attain the semiparametric efficiency bound multiple robustness ensures estimation consistency if the missingness probability model is misspecified but one data distribution model is correct our proposed estimators are all convex combinations of the observed responses and thus always fall within the parameter space

calibration estimators in survey sampling abstract this article investigates estimation of finite population totals in the presence of univariate or multivariate auxiliary information estimation is equivalent to attaching weights to the survey data we focus attention on the several weighting systems that can be associated with a given amount of auxiliary information and derive a weighting system with the aid of a distance measure and a set of calibration equations we briefly mention an application to the case in which the information consists of known marginal counts in a two or multi way table known as generalized raking the general regression estimator greg was conceived with multivariate auxiliary information in mind ordinarily this estimator is justified by a regression relationship between the study variable y and the auxiliary vector x but we note that the greg can be derived by a different route by focusing instead on the weights the ordinary sampling weights of the kth observation is 1 k where k is the inclusion probabilit

analysis of semiparametric regression models for repeated outcomes in the presence of missing data abstract we propose a class of inverse probability of censoring weighted estimators for the parameters of models for the dependence of the mean of a vector of correlated response variables on a vector of explanatory variables in the presence of missing response data the proposed estimators do not require full specification of the likelihood they can be viewed as an extension of generalized estimating equations estimators that allow for the data to be missing at random but not missing completely at random these estimators can be used to correct for dependent censoring and nonrandom noncompliance in randomized clinical trials studying the effect of a treatment on the evolution over time of the mean of a response variable the likelihood based parametric g computation algorithm estimator may also be used to attempt to correct for dependent censoring and nonrandom noncompliance but because of possible model misspecification the parametric g computation algorithm estimator in contrast with the proposed w

efficient and doubly robust imputation for covariate dependent missing responses in this article we study a well known response missing data problem missing data is an ubiquitous problem in medical and social science studies imputation is one of the most popular methods for dealing with missing data the most commonly used imputation that makes use of covariates is regression imputation in which the regression model can be parametric semiparametric or nonparametric parametric regression imputation is efficient but is not robust against misspecification of the regression model although nonparametric regression imputation such as nearest neighbor imputation and kernel regression imputation is model free it is not efficient especially if the dimension of covariate vector is high the well known problem of curse of dimensionality semiparametric regression imputation such as partially linear regression imputation can reduce the dimension of the covariate in nonparametric regression fitting but is not robust against misspecification of the linear component in the regression

minimal free resolution of a finitely generated module over a regular local ring abstract numerical invariants of a minimal free resolution of a module m over a regular local ring r n can be studied by taking advantage of the rich literature on the graded case the key is to fix suitable n stable filtrations m of m and to compare the betti numbers of m with those of the associated graded module gr m m this approach has the advantage that the same module m can be detected by using different filtrations on it it provides interesting upper bounds for the betti numbers and we study the modules for which the extremal values are attained among others the koszul modules have this behavior as a consequence of the main result we extend some results by aramova conca herzog and hibi on the rigidity of the resolution of standard graded algebras to the local setting

koszul homology and extremal properties of gin and lex for every homogeneous ideal i in a polynomial ring r and for every p dim r we consider the koszul homology h i p r i with respect to a sequence of p of generic linear forms the koszul betti number ijp r i is by definition the dimension of the degree j part of h i p r i in characteristic 0 we show that the koszul betti numbers of any ideal i are bounded above by those of the gin revlex gin i of i and also by those of the lex segment lex i of i we show that ijp r i ijp r gin i iff i is componentwise linear and that and ijp r i ijp r lex i iff i is gotzmann we also investigate the set gins i of all the gin of i and show that the koszul betti numbers of any ideal in gins i are bounded below by those of the gin revlex of i on the other hand we present examples showing that in general there is no j is gins i such that the koszul betti numbers of any ideal in gins i are bounded above by those of j

the leading ideal of a complete intersection of height two abstract let s n be a noetherian local ring and let i f g be an ideal in s generated by a regular sequence f g of length two assume that the associated graded ring gr n s of s with respect to n is a ufd we examine generators of the leading form ideal i of i in gr n s and prove that i is a perfect ideal of gr n s if i is 3 generated thus in this case letting r s i and m n i if gr n s is cohen macaulay then gr m r gr n s i is cohen macaulay as an application we prove that if r m is a one dimensional gorenstein local ring of embedding dimension 3 then gr m r is cohen macaulay if the reduction number of m is at most 4

empirical likelihood based inference under imputation for missing response data inference under kernel regression imputation for missing response data is considered an adjusted empirical likelihood approach to inference for the mean of the response variable is developed a nonparametric version of wilks theorem is proved for the adjusted empirical log likelihood ratio by showing that it has an asymptotic standard chi squared distribution and the corresponding empirical likelihood confidence interval for the mean is constructed with auxiliary information an empirical likelihood based estimator is defined and an adjusted empirical log likelihood ratio is derived asymptotic normality of the estimator is proved also it is shown that the adjusted empirical log likelihood ratio obeys wilks theorem a simulation study is conducted to compare the adjusted empirical likelihood and the normal approximation methods in terms of coverage accuracies and average lengths of confidence intervals based on biases and standard errors a comparision is also made by simulation between the empirical likelihood based estimator and related estimators our simulation indicates that the adjusted empirical likelihood method performs competitively and that the use of auxiliary information provides improved inferences

multiply robust imputation procedures for the treatment of item nonresponse in surveys summaryitem nonresponse in surveys is often treated through some form of imputation we introduce multiply robust imputation in finite population sampling this is closely related to multiple robustness which extends double robustness in practice multiple nonresponse models and multiple imputation models may be fitted each involving different subsets of covariates and possibly different link functions an imputation procedure is said to be multiply robust if the resulting estimator is consistent when all models but one are misspecified a jackknife variance estimator is proposed and shown to be consistent random and fractional imputation procedures are discussed a simulation study suggests that the proposed estimation procedures have low bias and high efficiency

some impossibility theorems in econometrics with applications to structural and dynamic models general characterizations of valid confidence sets and tests in problems which involve locally almost unidentified lau parameters are provided and applied to several econo metric models two types of inference problems are studied i inference about parame ters which are not identifiable on certain subsets of the parameter space and ii inference about parameter transformations with discontinuities when a lau parameter or parametric function has an unbounded range it is shown under general regularity conditions that any valid confidence set with level 1 a for this parameter must be unbounded with probability close to 1 a in the neighborhood of nonidentification subsets and will have a nonzero probability of being unbounded under any distribution compatible with the model no valid confidence set which is almost surely bounded does exist these properties hold even if identifying restrictions are imposed similar results also obtain for parameters with bounded ranges consequently a confidence set which does not satisfy this characterization has zero coverage probability level this will be the case in particular for wald type confidence intervals based on asymptotic standard errors furthermore wald type statistics for testing given values of a lau parameter cannot be pivotal functions i e they have distributions which depend on unknown nuisance param eters and even cannot be usefully bounded over the space of the nuisance parameters these results are applied to several econometric problems inference in simultaneous equations instrumental variables iv regressions linear regressions with autoregressive errors inference about long run multipliers and cointegrating vectors for example it is shown that standard asymptotically justified confidence intervals based on iv estimators such as two stage least squares and the associated standard errors have zero coverage probability and the corresponding t statistics have distributions which cannot be bounded by any finite set of distribution functions a result of interest for interpreting iv regressions with weak instruments furthermore expansion methods e g edgeworth expansions and bootstrap techniques cannot solve these difficulties finally in a number of cases where wald type methods are fundamentally flawed e g iv regressions with poor instruments it is observed that likelihood based methods e g likelihood ratio tests and confidence sets combined with projection techniques can easily yield valid tests and confidence sets

estimation with missing data beyond double robustness summary we propose an estimator that is more robust than doubly robust estimators based on weighting completecasesusingweightsotherthaninverseprobabilitywhenestimatingthepopulationmean of a response variable subject to ignorable missingness we allow multiple models for both the propensity score and the outcome regression our estimator is consistent if any of the multiple models is correctly specified such multiple robustness against model misspecification is a significant improvement over double robustness which allows only one propensity score model and one outcome regression model our estimator attains the semiparametric efficiency bound when one propensity score model and one outcome regression model are correctly specified without requiring knowledge of which models are correct

sheaf cohomology and free resolutions over exterior algebras in this paper we derive an explicit version of the bernstein gel fand gel fand bgg correspondence between bounded complexes of coherent sheaves on projective space and minimal doubly infinite free reso lutions over its koszul dual exterior algebra among the facts about the bgg correspondence that we derive is that taking homology of a complex of sheaves corresponds to taking the linear part of a resolution over the exterior algebra we explore the structure of free resolutions over an exterior algebra for example we show that such resolutions are eventually dominated by their linear parts in the sense that erasing all terms of degree 1 in the complex yields a new complex which is eventually exact as applications we give a construction of the beilinson monad which expresses a sheaf on projective space in terms of its cohomology by using sheaves of differential forms the explicitness of our version allows us to to prove two conjectures about the morphisms in the monad and we get an efficient method for machine computation of the cohomology ofsheaves we also construct all the monads for a sheaf that can be built from sums of line bundles and show that they are often characterized by numerical data let v be a finite dimensional vector space over a field k and let w v be the dual space in this paper we will study complexes and resolutions over the exterior algebra e v and their relation to modules over s sym w and sheaves on projective space p w in this paper we study the bernstein gel fand gel fand bgg correspondence 1978 usually stated as an equivalence between the derived category of bounded complexes of coherent sheaves on p w and the stable category of finitely generated graded modules over e its essential content is a functor r from complexes of graded s modules to complexes of graded e modules and its adjoint l for example if m imi is a graded s module regarded as a complex with just one term then the first and third authors are grateful to the nsf for partial support during the preparation of this paper the third author wishes to thank msri for its hospitality

doubly robust estimation in missing data and causal inference models the goal of this article is to construct doubly robust dr estimators in ignorable missing data and causal inference models in a missing data model an estimator is dr if it remains consistent when either but not necessarily both a model for the missingness mechanism or a model for the distribution of the complete data is correctly specified because with observational data one can never be sure that either a missingness model or a complete data model is correct perhaps the best that can be hoped for is to find a dr estimator dr estimators in contrast to standard likelihood based or nonaugmented inverse probability weighted estimators give the analyst two chances instead of only one to make a valid inference in a causal inference model an estimator is dr if it remains consistent when either a model for the treatment assignment mechanism or a model for the distribution of the counterfactual data is correctly specified because with observational data one can never be sure that a model for the treatment assignment mechanism or a model for the counterfactual data is correct inference based on dr estimators should improve upon previous approaches indeed we present the results of simulation studies which demonstrate that the finite sample performance of dr estimators is as impressive as theory would predict the proposed method is applied to a cardiovascular clinical trial

the leading ideal of a complete intersection of height two part ii abstract let s n be a regular local ring and let i f g be an ideal in s generated by a regular sequence f g of length two let r s i and m n i as in s goto w heinzer m k kim the leading ideal of a complete intersection of height two j algebra 298 2006 238 247 we examine the leading form ideal i of i in the associated graded ring g gr n s if gr m r is cohen macaulay we describe precisely the hilbert series h gr m r in terms of the degrees of homogeneous generators of i and of their successive gcd s if d gcd f g is a prime element of gr n s that is regular on gr n s f d g d we prove that i is 3 generated and a perfect ideal if ht gr n s f g h 2 where h i is such that h is of minimal degree in i f g gr n s we prove i is 3 generated and a perfect ideal of gr n s so gr m r gr n s i is a cohen macaulay ring we give several examples to illustrate our theorems

empirical likelihood based inference in missing response problems and its application in observational studies summary the problem of missing response data is ubiquitous in medical and social science studies in the case of responses that are missing at random depending on some covariate information analyses focused only on the complete data may lead to biased results various debias methods have been extensively studied in the literature particularly the weighting method that was motivated by horvitz and thompson s estimators to improve efficiency robins rotnitzky and zhao proposed augmented estimating equations based on corrected complete case analyses a nice feature of the augmented method is its double robustness i e the estimator that is derived from the augmented method is asymptotically unbiased if either the underlying missing data mechanism or the underlying regression function is correctly specified furthermore the augmented estimator can achieve full efficiency if both the missing data mechanism and the regression function are correctly specified in general however it is very difficult to specify the regression function correctly especially when the dimension of covariates is high this is the so called curse of dimensionality problem the augmented estimator has much lower efficiency if the working regression model is not close to the true regression model in this paper the empirical likelihood method is employed to seek a constrained empirical likelihood estimation of mean response with the assumption that responses are missing at random the empirical likelihood based estimators enjoy the double robustness property moreover it is possible that the empirical likelihood based inference can produce asymptotically unbiased and efficient estimators even if the true regression function is not completely known simulation results indicate that the empirical likelihood based estimators are very robust to a misspecification of the propensity score and dominate other competitors in the sense of having smaller mean square errors methods that are developed in this paper have a nice application in observational causal inferences the propensity score is used to adjust for differences in pretreatment variables in the estimation of average treatment effects

tuning parameter selection in high dimensional penalized likelihood summary determining how to select the tuning parameter appropriately is essential in penalized likelihood methods for high dimensional data analysis we examine this problem in the setting of penalized likelihood methods for generalized linear models where the dimensionality of covariates p is allowed to increase exponentially with the sample size n we propose to select the tuning parameter by optimizing the generalized information criterion with an appropriate model complexity penalty to ensure that we consistently identify the true model a range for the model complexity penalty is identified in the generlized information criterion we find that this model complexity penalty should diverge at the rate of some power of log p depending on the tail probability behaviour of the response variables this reveals that using the akaike information criterion or bayes information criterion to select the tuning parameter may not be adequate for consistently identifying the true model on the basis of our theoretical study we propose a uniform choice of the model complexity penalty and show that the approach proposed consistently identifies the true model among candidate models with asymptotic probability 1 we justify the performance of the procedure proposed by numerical simulations and a gene expression data analysis

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