high order asymptotic expansions for robust tests summary this paper provides high order asymptotic expansions for the m ratio and the signedroot m ratio robust test statistics which allow one to compute accurate approximations to significance or critical values using the edgeworth approximation the bartlett correction the variance correction or the saddlepoint approximation specific results are obtained for the linear regression model with the huber m estimator a monte carlo study illustrates the numerical accuracy of these approximations with respect to the usual first order approximations

asymptotic properties of weighted m estimators for variable probability samples i provide a systematic treatment of the asymptotic properties of weighted m estimators under variable probability stratified sampling the characterization of the sampling scheme and representation of the objective function allow for a straightforward analysis simple consistent asymptotic variance matrix estimators are proposed for a large class of problems when stratification is based on exogenous variables i show that the unweighted m estimator is more efficient than the weighted estimator under a generalized conditional information matrix equality when population frequencies are known a more efficient weighting is possible i also show how the results carry over to multinomial sampling

parameters of interest nuisance parameters and orthogonality conditions an application to autoregressive error component models abstract in general when one considers a set of orthogonality conditions the parameters can be divided into parameters of interest that the econometrician wants to estimate the coefficient of the lagged endogenous variable in the case of an autoregressive error component arec model for instance and nuisance parameters most of the second order terms in an arec model we demonstrate that the elimination of such nuisance parameters using their empirical counterpart does not entail an efficiency loss when only the parameters of interest are estimated applications of our results to both autoregressive error component models and time varying individual fixed effects models are discussed at length they show the nature of the efficiency losses when some orthogonality conditions are left aside

gmm redundancy results for general missing data problems we consider questions of efficiency and redundancy in the gmm estimation problem in which we have two sets of moment conditions where two sets of parameters enter into one set of moment conditions while only one set of parameters enters into the other we then apply these results to a selectivity problem in which the first set of moment conditions is for the model of interest and the second set of moment conditions is for the selection process we use these results to explain the counterintuitive result in the literature that under an ignorability assumption that justifies gmm with weighted moment conditions weighting using estimated probabilities of selection is better than weighting using the true probabilities we also consider estimation under an exogeneity of selection assumption such that both the unweighted and the weighted moment conditions are valid and we show that when weighting is not needed for consistency it is also not useful for efficiency

topologies on homeomorphism spaces of certain metric spaces a new proof for the equivalence of several topologies on homeomorphism groups over certain metric spaces x is given which is based on the metric of x

estimating exposure effects by modelling the expectation of exposure conditional on confounders in order to estimate the causal effects of one or more exposures or treatments on an outcome of interest one has to account for the effect of confounding factors which both covary with the exposures or treatments and are independent predictors of the outcome in this paper we present regression methods which in contrast to standard methods adjust for the confounding effect of multiple continuous or discrete covariates by modelling the conditional expectation of the exposures or treatments given the confounders in the special case of a univariate dichotomous exposure or treatment this conditional expectation is identical to what rosenbaum and rubin have called the propensity score they have also proposed methods to estimate causal effects by modelling the propensity score our methods generalize those of rosenbaum and rubin in several ways first our approach straightforwardly allows for multivariate exposures or treatments each of which may be continuous ordinal or discrete second even in the case of a single dichotomous exposure our approach does not require subclassification or matching on the propensity score so that the potential for residual confounding i e bias due to incomplete matching is avoided third our approach allows a rather general formalization of the idea that it is better to use the estimated propensity score than the true propensity score even when the true score is known the additional power of our approach derives from the fact that we assume the causal effects of the exposures or treatments can be described by the parametric component of a semiparametric regression model to illustrate our methods we reanalyze the effect of current cigarette smoking on the level of forced expiratory volume in one second in a cohort of 2 713 adult white males we compare the results with those obtained using standard methods

efficient estimation of panel data models with sequential moment restrictions abstract this paper considers asymptotically efficient estimation of the panel data models with sequential moment restrictions in an environment with i i d observations it is shown that the gmm estimator with an increasing set of instruments attains the semiparametric efficiency bound of the model the estimator considered here allows for the heteroscedasticity of the unknown form

sample selection and information theoretic alternatives to gmm information theoretic alternatives to general method of moments gmm use over identifying moments to estimate the data generating distribution jointly with the parameters of interest this paper demonstrates how these estimates can be interpreted when the sample is not a random draw from the population of interest i make explicit the selection probability implied by the empirical likelihood and exponential tilting estimators two commonly used estimators in this class in addition i propose an alternative estimator that corresponds to a logisitic selection model the small sample properties of the estimators are demonstrated with a monte carlo experiment

semiparametric efficiency in multivariate regression models with missing data abstract we consider the efficiency bound for the estimation of the parameters of semiparametric models defined solely by restrictions on the means of a vector of correlated outcomes y when the data on y are missing at random we show that the semiparametric variance bound is the asymptotic variance of the optimal estimator in a class of inverse probability of censoring weighted estimators and that this bound is unchanged if the data are missing completely at random for this case we study the asymptotic performance of the generalized estimating equations gee estimators of mean parameters and show that the optimal gee estimator is inefficient except for special cases the optimal weighted estimator depends on unknown population quantities but for monotone missing data we propose an adaptive estimator whose asymptotic variance can achieve the bound

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

redundancy of moment conditions it is well known that adding moment conditions cannot decrease asymptotic efficiency however sometimes additional moment conditions are redundant in the sense that they do not increase the asymptotic efficiency of estimation for some or all of the parameters of interest this paper gives a general treatment of redundancy of moment conditions it provides necessary and sufficient conditions for redundancy in several equivalent forms the paper also provides interesting results for the case that there are three or more sets of moment conditions

a paradox concerning nuisance parameters and projected estimating functions summary this paper is concerned with a paradox associated with parameter estimation in the presence of nuisance parameters in a statistical model with unknown nuisance parameters the efficiency of an estimator of a parameter usually increases when the nuisance para meters are known however the opposite phenomenon can sometimes occur in this paper we elucidate the occurrence of this paradox by examining estimating functions in particular we focus on the projected estimating function which is defined by the projection of the score function on to a given estimating function a sufficient condition for the paradox to occur is the orthogonality of the two components of the projected estimating functions corresponding to parameters of interest and nuisance parameters in addition a numerical assessment is conducted in the context of a simple model to investigate the improvement of the asymptotic efficiency of estimators

inverse probability weighted estimation for general missing data problems abstract i study inverse probability weighted m estimation under a general missing data scheme examples include m estimation with missing data due to a censored survival time propensity score estimation of the average treatment effect in the linear exponential family and variable probability sampling with observed retention frequencies i extend an important result known to hold in special cases estimating the selection probabilities is generally more efficient than if the known selection probabilities could be used in estimation for the treatment effect case the setup allows a general characterization of a double robustness result due to scharfstein et al 1999 rejoinder journal of the american statistical association 94 1135 1146

improved instrumental variables and generalized method of moments estimators this paper considers gmm estimation with moment conditions of the usual type plus extra moment conditions that do not depend on the parameters the extra moment conditions improve efficiency if they are correlated with the original set of moment conditions this is true in linear and nonlinear models but in linear models we provide simple explicit formulas for the improved estimators our results may be useful in rational expectations models in which an equation s error is a forecast error and the extra moment conditions would be forecast errors in related variables

a symbolic algorithm for studying adjustments to the profile likelihood summary a symbolic algorithm for the derivation of the asymptotic expansion of the maximum likelihood estimate was first introduced by andrews stafford 1993 this algorithm is quite general and can be modified to give asymptotic expansions for the maximum likelihood estimate based on either the conditional modified or adjusted profile likelihoods expansions are obtained quickly with no chance for human clerical error as a result these three adjustments can be easily studied and some new statistical results are obtained bartlett corrections of each adjusted likelihood and asymptotic expansions for the first four moments of the maximum likelihood estimate based on the conditional and adjusted profile likelihoods are derived

on the role of the propensity score in efficient semiparametric estimation of average treatment effects in this paper the role of the propensity score in the efficient estimation of average treatment effects is examined under the assumption that the treatment is ignorable given some observed characteristics it is shown that the propensity score is ancillary for estimation of the average treatment effects the propensity score is not ancillary for estimation of average treatment effects on the treated it is suggested that the marginal value of the propensity score lies entirely in the dimension reduction efficient semiparametric estimators of average treatment effects and average treatment effects on the treated are shown to take the form of relevant sample averages of the data completed by the nonparametric imputation method it is shown that the projection on the propensity score is not necessary for efficient semiparametric estimation of average treatment effects on the treated even if the propensity score is known an application to the experimental data reveals that conditioning on the propensity score may even result in a loss of efficiency

tools for the symbolic computation of asymptotic expansions summary this paper describes a collection of procedures for the systematic computation of asymptotic expansions that are common in statistical theory and practice expansions of functions of sums of independent and identically distributed random variables the procedures permit the expansion of maximum likelihood estimates the associated deviance or drop in likelihood and more general functions of random variables with distributions involving one or more parameters the procedures are illustrated with examples involving general and specific laws much of statistical theory and practice is based on asymptotic expansions many programs are available to assist in the numerical evaluation of such expansions but there is a need for computational tools to assist in their derivation and symbolic evaluation heller 1991 shows how symbolic calculation may be used in a wide variety of statistical problems kendall 1988 1990 gives procedures for the symbolic computation of expressions in the analysis of the diffusion of euclidean shape silverman and young 1987 use computer algebra to evaluate criteria on which the decision to smooth a bootstrap distribution is based young and daniels 1990 apply symbolic computation to evaluate expressions in the assessment of bootstrap bias venables 1985 utilizes symbolic computation heavily to obtain expansions of maximum marginal likelihood estimates most notably fisher s a statistic barndorff nielsen and blasild 1986 describe procedures for the numerical calculation of bartlett factors in cases where cumulants of the likelihood function may be specified most of these references involve the evaluation of complicated formulae in particular cases and not the derivation of the formulae themselves here we give general procedures for both the derivation of formulae and their evaluation in specific cases the derivation of asymptotic expansions is typically a simple but laborious task consider for example the calculation of the expectation of the likelihood ratio test statistic for a one parameter family to order 1 n this may be accomplished in general

a note on the calculation of bartlett adjustments summary we describe for the numerical calculation of bartlett adjustments a method which may be of use when the cumulants of the log likelihood derivatives are easy to determine in one parametrization while the hypotheses to be tested are all linear in some other parametrization the method relies on the cartesian tensorial nature of those cumulants and should be particularly convenient in connection with statistical packages with structure similar to glim

model based direct adjustment abstract direct adjustment or standardization applies population weights to subclass means in an effort to estimate population quantities from a sample that is not representative of the population direct adjustment has several attractive features but when there are many subclasses it can attach large weights to small quantities of data often in a fairly erratic manner in the extreme direct adjustment can attach infinite weight to nonexistent data a noticeable inconvenience in practice this article proposes a method of model based direct adjustment that preserves the attractive features of conventional direct adjustment while stabilizing the weights attached to small subclasses the sample mean and conventional direct adjustment are both special cases of model based direct adjustment under two different extreme models for the subclass specific selection probabilities the discussion of this method provides some insights into the behavior of true and estimated propensity scores the estimated scores

a conditional saddlepoint approximation for testing problems a saddlepoint approximation is provided for the distribution function of one m statistic conditional on another m statistic many interesting statistics based on dependent quantities e g spacings multinomial frequencies rank differences can be expressed in terms of independent identically distributed random variables conditioned on their sum so that this conditional saddlepoint approximation yields accurate approximations for the distribution of such statistics this saddlepoint approximation can also be used in conditional testing where nuisance parameters are eliminated by conditioning on sufficient statistics

on parameter transformations and interval estimation summary parameterizations which reduce the asymptotic bias and skewness of various pivotal quantities arising in large sample theory are discussed for models depending on an unknown scalar parameter transformation formulae by which such parameterizations can be obtained are derived and these formulae extend those for one dimensional curved exponential families given by hougaard 1982 to assess the accuracy of normal approximations to the distributions of the pivots their second order properties are considered and comparisons with the signed square root of the likelihood ratio statistic are drawn

the bootstrap to smooth or not to smooth summary the bootstrap and smoothed bootstrap are considered as alternative methods of estimating properties of unknown distributions such as the sampling error of parameter estimates criteria are developed for determining whether it is advantageous to use the smoothed bootstrap rather than the standard bootstrap key steps in the argument leading to these criteria include the study of the estimation of linear functionals of distributions and the approximation of general functionals by linear functionals consideration of an example the estimation of the standard error in the variance stabilized sample correlation coefficient elucidates previously published simulation results and also illustrates the use of computer algebraic manipulation as a useful technique in asymptotic statistics finally the various approximations used are vindicated by a simulation study