one way communication complexity and the neciporuk lower bound on formula size in this paper the neciporuk method for proving lower bounds on the size of boolean formulas is reformulated in terms of one way communication complexity we investigate the settings of probabilistic formulas nondeterministic formulas and quantum formulas in all cases we can use results about one way communication complexity to prove lower bounds on formula size the main results regarding formula size are as follows we show a polynomial size gap between probabilistic quantum and deterministic formulas a near quadratic gap between the sizes of nondeterministic formulas with limited access to nondeterministic bits and nondeterministic formulas with access to slightly more such bits and a near quadratic lower bound on quantum formula size furthermore we give a polynomial separation between the sizes of quantum formulas with and without multiple read random inputs the lower bound methods for quantum and probabilistic formulas employ a variant of the neciporuk bound in terms of the vapnik chervonenkis dimension to establish our lower bounds we show optimal separations between one way and two way protocols for limited nondeterministic and quantum communication complexity and we show that zero error quantum one way communication complexity asymptotically equals deterministic one way communication complexity for total functions

simulated moments estimation of markov models of asset prices this paper provides a simulated moments estimator sme of the parameters of dynamic models in which the state vector follows a time homogeneous markov process conditions are provided for both weak and strong consistency as well as asymptotic normality various tradeoffs among the regularity conditions underlying the large sample properties of the sme are discussed in the context of an asset pricing model this paper provides conditions for the consistency and asymptotic normality of a simulated moments estimator sme of the parameters of asset pricing models with time homogeneous markov representations of the stochastic forc ing process sme s for economic models have been proposed by mcfadden 1989 and pakes and pollard 1989 for i i d environments and by lee and ingram 1991 for a time series environment the sme for time series models examined in this paper is as follows the state vector yt that determines asset prices is assumed to follow a time homogeneous markov process whose transi tion function depends on an unknown parameter vector 3 0 asset prices and possibly other relevant data are observed as f yt 0 for some given function f of the underlying state and parameter vector in parallel a simulated state process y is generated analytically or numerically from the economic model and corresponding simulated observations f yj3 13 are taken for a given parameter choice f3 the parameter is chosen so as to match moments that is to minimize the distance between sample moments of the data f y 8030 and those of the simulated series f yt f3 in a sense to be made precise the proposed sme extends the generalized method of moments gmm estimator hansen 1982 to a large class of asset pricing models for which the moment restrictions of interest do not have analytic representations in terms of observable variables and the unknown parameter vector we provide conditions on the transition function of yt and the observation function f under which the sme of 030 is consistent and characterize the normalized asymptotic distribu tion of the estimator for two reasons neither the regularity conditions underly ing hansen s 1982 analysis of gmm estimators for time series models without

error estimates for the approximation of semicoercive variational inequalities an abstract error estimate for the approximation of semicoercive variational inequalities is obtained provided a certain condition holds for the exact solution this condition turns out to be necessary as is demonstrated analytically and numerically the results are applied to the finite element approximation of poisson s equation with signorini boundary conditions and to the obstacle problem for the beam with no fixed boundary conditions for second order variational inequalities the condition is always satisfied whereas for the beam problem the condition holds if the center of forces belongs to the interior of the convex hull of the contact set applying the error estimate yields optimal order of convergence in terms of the mesh size h the numerical convergence rates observed are in good agreement with the predicted ones

saddlepoint approximations for regression models summary this paper uses the techniques of saddlepoint or tilted exponential approximation to develop an approximation to the small sample distribution of estimators defined by a system of estimating equations when observations are independently but not identically distributed this allows for the explicit treatment of models with explanatory variables the approximation is applied to l1 regression and the accuracy of the approximation is demonstrated in a case in which the usual normal approximation is very poor an analysis of the example indicates the usefulness of a new variant of exponential tilting in calculating the transformation measure a quantity which plays an important role in assuring the accuracy of the approximation

regression models for recurrent event data parametric random effects models with measurement error statistical methodology is presented for the statistical analysis of non linear measurement error models our approach is to provide adjustments for the usual maximum likelihood estimators their standard errors and associated significance tests in order to account for the presence of measurement error in some of the covariates we illustrate the technique with a mixed effects poisson regression model for recurrent event data applied to a randomized clinical trial for the prevention of skin tumours

an augmented lagrangian method for identifying discontinuous parameters in elliptic systems the identification of discontinuous parameters in elliptic systems is formulated as a constrained minimization problem combining the output least squares and the equation error method the minimization problem is then proved to be equivalent to the saddle point problem of an augmented lagrangian the finite element method is used to discretize the saddle point problem and the convergence of the discretization is also proved finally an uzawa algorithm is suggested for solving the discrete saddle point problem and is shown to be globally convergent

accurate and robust tests for indirect inference in this paper we propose accurate parameter and over identification tests for indirect inference under the null hypothesis the new tests are asymptotically 2 distributed with a relative error of order n 1 they exhibit better finite sample accuracy than classical tests for indirect inference which have the same asymptotic distribution but an absolute error of order n 1 2 robust versions of the tests are also provided we illustrate their accuracy in nonlinear regression poisson regression with overdispersion and diffusion models

semiparametric regression models for repeated events with random effects and measurement error abstract statistical methodology is presented for the regression analysis of multiple events in the presence of random effects and measurement error omitted covariates are modeled as random effects our approach to parameter estimation and significance testing is to start with a naive model of semiparametric poisson process regression and then to adjust for random effects and any possible covariate measurement error we illustrate the techniques with data from a randomized clinical trial for the prevention of recurrent skin tumors

approximations for densities of sufficient estimators abstract a simple method of obtaining asymptotic expansions for the densities of sufficient estimators is described it is an extension of the one developed by o barndorff nielsen and d r cox 1979 for exponential families a series expansion in powers of n 1 is derived of which the first term has an error of order n 1 which can effectively be reduced to n 3 2 by renormalization the results obtained are similar to those given by h e daniels s 1954 saddlepoint method but the derivations are simpler a brief treatment of approximations to conditional densities is given theorems are proved which extend the validity of the multivariate edgeworth expansion to parametric families of densities of statistics which need not be standardized sums of independent and identically distributed vectors these extensions permit the treatment of problems arising in time series analysis the technique is used by j durbin 1980 to obtain approximations to the densities of partial serial correlation coefficients

saddlepoint approximations in resampling methods summary saddlepoint approximations are shown to be easy to use and accurate in a variety of simple bootstrap and randomization applications examples include mean estimation ratio estimation two sample comparisons and autoregressive estimation

target estimation for bias and mean square error reduction given a statistical functional t and a parametric family of distributions a bias reduced functional t is defined by setting the expected value of the statistic equal to the observed value under certain regularity conditions this new statistic called the target estimator will have smaller bias and mean square error than the original estimator the theoretical aspects are analyzed by using higher order von mises expansions several examples are given including m estimates of location and scale the procedure is applied to an autoregressive model the errors in variables model and the logistic regression model a comparison with the jackknife and the bootstrap estimators is also included

the density of multivariate m estimates when a unique m estimate exists its density is obtained as a corollary to a more general theorem which asserts that under mild conditions the intensity function of the point process of solutions of the estimating equations exists and is given by the density of the estimating function standardized by multiplying it by the inverse of its derivative we apply the results to give a result for huber s proposal 2 applied to regression and scale estimates we also give a saddlepoint approximation for the density and use this to give approximations for tail areas for smooth functions of the m estimates

duality methods for solving variational inequalities abstract methods of maximal monotone operators are used in order to study from a general point of view duality numerical algorithms for solving variational inequalities with classical algorithms such as uzawa s method for the standard and augmented lagrangian this paper presents some new algorithms which appear to have very good numerical performances

augmented lagrangian sqp methods in hilbert spaces and application to control in the coefficients problems augmented lagrangian methods for optimization problems in hilbert spaces with second order updates of the lagrangian are analyzed the relationship and the differences to sqp methods are pointed out the convergence results depend on the second order sufficient optimality condition applications to parameter estimation problems are given and illustrated by numerical results these illustrate the global behavior of the method

on the empirical saddlepoint approximation summary the properties of the saddlepoint approximation are investigated when the required cumulant generating function is obtained empirically properties of the empirical moment generating function and empirical cumulant generating function and derivatives of these processes which are needed for this study are derived first in particular their uniform consistency moment structure and weak convergence to normality are established a numerical investigation exploring use of the empirical saddlepoint approximation as a tool in density estimation is discussed briefly

sequential monte carlo samplers summary we propose a methodology to sample sequentially from a sequence of probability distributions that are defined on a common space each distribution being known up to a normalizing constant these probability distributions are approximated by a cloud of weighted random samples which are propagated over time by using sequential monte carlo methods this methodology allows us to derive simple algorithms to make parallel markov chain monte carlo algorithms interact to perform global optimization and sequential bayesian estimation and to compute ratios of normalizing constants we illustrate these algorithms for various integration tasks arising in the context of bayesian inference

amplification by read once formulas moore and shannon have shown that relays with arbitrarily high reliability can be built from relays with arbitrarily poor reliability valiant used similar methods to construct monotone read once formulas of size o n alpha 2 where alpha log sqrt 5 1 2 simeq 3 27 that amplify psi frac 1 n psi frac 1 n where psi sqrt 5 1 2 simeq0 62 to 2 n 1 2 n and deduced as a consequence the existence of monotone formulas of the same size that compute the majority of n bits boppana has shown that any monotone read once formula that amplifies p frac 1 n p frac 1 n to frac 1 4 frac 3 4 where 0we extend boppana s results in two ways we first show that his two lower bounds hold for general read once formulas not necessarily monotone that may even include exclusive or gates we are then able to join his two lower bounds together and show that any read once not necessarily monotone formula that amplifies p frac 1 n p frac 1 n to 2 n 1 2 n has size omega n alpha 2 this result does not follow from boppana s arguments and it shows that the amount of amplification achieved by valiant is the maximal achievable using read once formulas in a companion paper we construct monotone read once contact networks of size o n 2 99 that amplify frac 1 2 frac 1 n frac 1 2 frac 1 n to frac 1 4 frac 3 4 this shows that boppana s lower bound for the first amplification stage does not apply to contact networks even if they are required to be both monotone and read once

simulation estimation of time series models abstract a formal econometric treatment of the estimation of the parameters of a fully specified stochastic equilibrium model is proposed the method estimation by simulation yields an estimator which is shown to have an asymptotic normal distribution a goodness of fit test based on a chi square statistic is also derived

an information theoretic alternative to generalized method of moments estimation while optimally weighted gmm estimation has desirable large sample properties its small sample performance is poor in some applications we propose a computationally simple alternative for weakly dependent data generating mechanisms based on minimization of the kullback leibler information criterion conditions are derived under which the large sample properties of this estimator are similar to gmm i e the estimator will be consistent and asymptotically normal with the same asymptotic covariance matrix as gmm in addition we propose overidentifying and parametric restrictions tests as alternatives to analogous gmm procedures

saddlepoint approximations and tests based on multivariate m estimates we consider multidimensional m functional parameters defined by expectations of score functions associated with multivariate m estimators and tests for hypotheses concerning multidimensional smooth functions of these parameters we propose a test statistic suggested by the exponent in the saddlepoint approximation to the density of the function of the m estimates this statistic is analogous to the log likelihood ratio in the parametric case we show that this statistic is approximately distributed as a chi squared variate and obtain a lugannani rice style adjustment giving a relative error of order we propose an empirical exponential likelihood statistic and consider a test based on this statistic finally we present numerical results for three examples including one in robust regression

residual type a posteriori error estimates for elliptic obstacle problems a posteriori error estimators of residual type are derived for piecewise linear finite element approximations to elliptic obstacle problems an instrumental ingredient is a new interpolation operator which requires minimal regularity exhibits optimal approximation properties and preserves positivity both upper and lower bounds are proved and their optimality is explored with several examples sharp a priori bounds for the a posteriori estimators are given and extensions of the results to double obstacle problems are briefly discussed

general saddlepoint approximations with applications to l statistics abstract saddlepoint approximations are extended to general statistics the technique is applied to derive approximations to the density of linear combinations of order statistics including trimmed means a comparison with exact results shows the accuracy of these approximations even in very small sample sizes

asymptotically unbiased estimation in generalized linear models with random effects summary obtaining estimates that are nearly unbiased has proven to be difficult when random effects are incorporated into a generalized linear model in this paper we propose a general method of adjusting any conveniently defined initial estimates to result in estimates which are asymptotically unbiased and consistent the method is motivated by iterative bias correction and can be applied in principle to any parametric model a simulation based approach of implementing the method is described and the relationship of the method proposed with other sampling based methods is discussed results from a small scale simulation study show that the method proposed can lead to estimates which are nearly unbiased even for the variance components while the standard errors are only slightly inflated a new analysis of the famous salamander mating data is described which reveals previously undetected between animal variation among the male salamanders and results in better prediction of mating outcomes

estimating the rational expectations model of speculative storage a monte carlo comparison of three simulation estimators the non negativity constraint on inventories imposed on the rational expectations theory of speculative storage implies that the conditional mean and variance of commodity prices are non linear in lagged prices and have a kink at a threshold point in this paper the structural parameters of this model are estimated using three simulation based estimators in a monte carlo experiment the finite sample properties of the simulated methods of moments estimator of duffie and singleton 1993 econometrica 61 4 929 952 the indirect inference estimator of gourieroux et al 1993 journal of applied economterics 8 s85 s118 and the efficient method of moments estimator of gallant and tauchen 1996 econometric theory 12 657 681 are assessed exploiting the invariant distribution implied by the theory allows us to evaluate the error induced by simulations our results show that the estimators differ in their sensitivity to the sample size the number of simulations choice of auxiliary models and computation demands for some estimators the test for overidentifying restrictions exhibit significant size distortions in small samples overall while the simulation estimators have small bias they are less efficient than pseudo maximum likelihood pmle hence for the small sample sizes considered the simulation estimators are still inferior to the pmle estimates in a mean squared sense

empirical saddlepoint approximations for multivariate m estimators summary in this paper we investigate the use of the empirical distribution function in place of the underlying distribution function f to construct an empirical saddlepoint approximation to the density fn of a general multivariate m estimator we obtain an explicit form for the error term in the approximation investigate the effect of renormalizing the estimator carry out some numerical comparisons and discuss the regression problem the sampling distribution of a statistic is a basic tool for evaluating the properties of the statistic and for constructing inference procedures based on the statistic the density of the sampling distribution may be used to compute moments to develop approximations or to compute the distribution function and hence various tail prob abilities which can be used to construct tests or confidence intervals when the exact density is intractable it may be possible to approximate it by a normal density an edgeworth expansion or a saddlepoint approximation daniels 1954 the saddle point approximation is derived from an asymptotic expansion but is often very accurate even in quite small samples to highlight this property hampel 1973 described these and related techniques as small sample asymptotic approximations as can be seen from equation 3 later the main property of this approximation and its major advantage over edgeworth expansions is that it is always non negative and the relative error is uniformly of order n 1 for recent reviews see reid 1988 barndorff nielsen and cox 1989 and field and ronchetti 1990 the construction of these approximations typically requires knowledge of the underlying distribution however just as we can estimate the variance of a normal approxima tion or estimate the terms in an edgeworth expansion we can estimate the saddle point approximation in this paper we examine the properties of an estimate of the saddlepoint approximation our investigation is motivated by three considerations firstly the estimated approximation may be useful for comparing various estimators on a given data set if the sampling distribution is non normal at least in principle we need to compare the whole distribution and comparisons based on the estimated asymptotic variance alone may be misleading secondly the estimated approximation may be useful for evaluating the quality of simpler approximations including the normal approxima tion finally it may be useful for developing inference procedures methods like

adaptive approximate bayesian computation sequential techniques can be adapted to the abc algorithm to enhance its efficiency for instance when sisson et al 2007 introduced the abc prc algorithm the goal was to improve upon existing abc mcmc algorithms marjoram et al 2003 while the abc prc method is based upon the theoretical developments of del moral et al 2006 the application to the setting of approximate bayesian computation induces a bias in the approximation to the posterior distribution of interest as we demonstrate in this paper via both theoretical reasoning and experimental results it is however possible to devise an alternative version based on genuine importance sampling arguments that we call abc pmc in connection with the population monte carlo method introduced in cappe et al 2004 furthermore the abc pmc algorithm is simpler than the abc prc algorithm in that it does not require any backward transition kernel and proposes an automatic scaling of the forward kernel in this paper we demonstrate the applicability of abc pmc and compare its performances with abc prc

on the relationship between empirical likelihood and empirical saddlepoint approximation for multivariate m estimators by comparing the expansions of the empirical log likelihood ratio and the empirical cumulant generating function calculated at the saddlepoint we investigate the relationship between empirical likelihood and empirical saddlepoint approximations this leads to a nonparametric approximation of the density of a multivariate m estimator based on the empirical likelihood and on the other hand it provides nonparametric confidence regions based on the empirical cumulant generating function some examples illustrate the use of the empirical likelihood in saddlepoint approximations and vice versa

optimal tests for parameter instability in the generalized method of moments framework this paper presents optimal tests for parameter instability in the gmm framework the new tests include tests that are optimal for both one sided and two sided alternatives one of the optimal tests for two sided alternatives is the gmm generalization of the test presented in andrews and ploberger 1994 for the likelihood framework the new tests include a class of optimal tests that direct the test s power to specific locations in the sample one of these optimal tests has the attractive feature of a normal distribution under the null hypothesis

robust indirect inference in this article we develop robust indirect inference for a variety of models in a unified framework we investigate the local robustness properties of indirect inference and derive the influence function of the indirect estimator as well as the level and power influence functions of indirect tests these tools are then used to design indirect inference procedures that are stable in the presence of small deviations from the assumed model although indirect inference was originally proposed for statistical models whose likelihood is difficult or even impossible to compute and or to maximize we use it here as a device to robustify the estimators and tests for models where this is not possible or is difficult with classical techniques such as m estimators examples from financial applications time series and spatial statistics are used for illustration

estimating functions in indirect inference summary there are models for which the evaluation of the likelihood is infeasible in practice for these models the metropolis hastings acceptance probability cannot be easily computed this is the case for instance when only departure times from a g g 1 queue are observed and inference on the arrival and service distributions are required indirect inference is a method to estimate a parameter in models whose likelihood function does not have an analytical closed form but from which random samples can be drawn for fixed values of first an auxiliary model is chosen whose parameter can be directly estimated next the parameters in the auxiliary model are estimated for the original data leading to an estimate the parameter is also estimated by using several sampled data sets simulated from the original model for different values of the original parameter finally the parameter which leads to the best match to is chosen as the indirect inference estimate we analyse which properties an auxiliary model should have to give satisfactory indirect inference we look at the situation where the data are summarized in a vector statistic t and the auxiliary model is chosen so that inference on is drawn from t only under appropriate assumptions the asymptotic covariance matrix of the indirect estimators is proportional to the asymptotic covariance matrix of t and componentwise inversely proportional to the square of the derivative with respect to of the expected value of t we discuss how these results can be used in selecting good estimating functions we apply our findings to the queuing problem

a stabilized lagrange multiplier method for the finite element approximation of contact problems in elastostatics in this work we consider a stabilized lagrange or kuhn tucker multiplier method in order to approximate the unilateral contact model in linear elastostatics the particularity of the method is that no discrete inf sup condition is needed in the convergence analysis we propose three approximations of the contact conditions well adapted to this method and we study the convergence of the discrete solutions several numerical examples in two and three space dimensions illustrate the theoretical results and show the capabilities of the method

small sample confidence intervals abstract in this article we present a technique for constructing one or two sided confidence intervals which are second order correct in terms of coverage for either parametric or nonparametric problems the construction is valid in the presence of nuisance parameters the situation we consider is this there are p parameters and we want a confidence interval for some function of them possibly one of the parameters itself the p parameters are estimated by m estimates which means they are obtained as a solution of a system of equations maximum likelihood estimates are included as a special case the essential intermediate result given in equation 3 4 says that the estimated parameter of interest can be written as a mean up to order o p 1 n the representation 3 4 is attained when is a smooth function of the parameters and is the solution of a well behaved system of equations we avoid the use of pivots and strive to obtain accurate coverage confidence intervals are construc

on the shrinkage exponent for read once formulae we prove that the size of any read once de morgan formula reduces on average by a factor of at least p o 1 when all but a fraction p of the input variables are randomly assigned to 0 1 here llog2 5 1 3 27 this resolves in the affirmative a conjecture of paterson and zwick the bound is shown to be tight up to a polylogarithmic factor for all p n 1

small sample asymptotic distributions of m estimators of location summary asymptotic formulae for the distribution of m estimators i e maximum likelihood type estimators of location including the arithmetic mean are derived which numerical studies show to give relative errors for densities and tail areas of the order of magnitude of 1 down to sample sizes 3 and 4 even in the extreme tails the paper is the continuation of earlier work by the second author and is also closely related to daniels s work on the saddlepoint approximation the method consists in expanding the derivative of the logarithm of the unstandardized density of the estimator in powers of 1 n at each point using recentring by means of conjugate distributions this method yields a unified point of view for the comparison of other asymptotic methods namely saddlepoint method edgeworth expansion and large deviations approach which are also compared numerically

on the augmented lagrangian approach to signorini elastic contact problem the signorini problem describes the contact of a linearly elastic body with a rigid frictionless foundation it is transformed into a saddle point problem of some augmented lagrangian functional and then discretized by finite element methods optimal error estimates are obtained for general smooth domains which are not necessarily convex the key ingredient in the analysis is a discrete inf sup condition which guarantees the existence of the saddle point

approximation of degenerate parabolic problems using numerical integration a class of multidimensional degenerate parabolic equations is considered the two phase stefan problem and the porous medium equation are analyzed as models of singular parabolic equations nonstat

indirect estimation of large conditionally heteroskedastic factor models with an application to the dow 30 stocks we derive indirect estimators of conditionally heteroskedastic factor models in which the volatilities of common and idiosyncratic factors depend on their past unobserved values by calibrating the score of a kalman filter approximation with inequality constraints on the auxiliary model parameters we also propose alternative indirect estimators for large scale models and explain how to apply our procedures to many other dynamic latent variable models we analyse the small sample behaviour of our indirect estimators and several likelihood based procedures through an extensive monte carlo experiment with empirically realistic designs finally we apply our procedures to weekly returns on the dow 30 stocks