a robust goodness of fit test statistic with application to ordinal regression models we propose a goodness of fit test statistic for linear regression with heterogeneous variance which is asymptotically chi square if the given model is correct the test statistic is computed as a quadratic form of observed minus predicted responses we apply the method to a linear regression for an ordinal categorical response the wheezing status of a child no wheeze wheeze with cold wheeze apart from cold as a function of maternal smoking and city of residence

on a class of second order anti periodic boundary value problems of concern is the existence and uniqueness of anti periodic solutions to a class of abstract nonlinear second order differential equations the investigation relies on the theory of m accretive operators in banach or hilbert spaces examples of boundary value problems for ordinary and partial differential equations are also discussed

simulated maximum likelihood estimation for discrete choices using transformed simulated frequencies many existing methods of simulated likelihood for discrete choice models require additive errors that have normal or extreme value distributions this paper focuses on a situation where the model does not admit such additive errors so that the popular method of ghk or logit estimation is not applicable this paper proposes a new method of simulated likelihood that is free from simulation bias for each finite number of simulations and yet flexible enough to accommodate various model specifications beyond those of additive normal or logit errors the method begins with the likelihood function involving simulated frequencies and finds a transform of the likelihood function that identifies the true parameter for each finite simulation number the transform is explicit containing no unknowns that demand an additional step of estimation the estimator achieves the efficiency of mle when the simulation number increases fast enough this paper presents and discusses results from monte carlo simulation studies of the new method

a hybrid estimator in nonlinear and generalised linear mixed effects models summary a hybrid method that combines laplace s approximation and monte carlo simulations to evaluate integrals in the likelihood function is proposed for estimation of the parameters in nonlinear mixed effects models that assume a normal parametric family for the random effects simulations show that these parametric estimates of fixed effects are close to the nonparametric estimates even though the mixing distribution is far from the assumed normal parametric family an asymptotic theory of this hybrid method for parametric estimation without requiring the true mixing distribution to belong to the assumed parametric family is developed to explain these results this hybrid method and its asymptotic theory are also extended to generalised linear mixed effects models

simulation of multivariate normal rectangle probabilities and their derivatives theoretical and computational results abstract an extensive literature in econometrics and in numerical analysis has considered the problem of evaluating the multiple integral p b a b n v dv e v 1 v b where v is a m dimensional normal vector with mean covariance matrix and density n v and 1 v b is an indicator for the event b v a a leading case of such an integral is the negative orthant probability where b v v the problem is computationally difficult except in very special cases the multinomial probit mnp model used in econometrics and biometrics has cell probabilities that are negative orthant probabilities with and depending on unknown parameters and in general on covariates estimation of this model requires for each trial parameter vector and each observation in a sample evaluation of p b and of its derivatives with respect to and this paper surveys monte carlo techniques that have been developed for approximations of p b and its linear and logarithmic derivatives that limit computation while possessing properties that facilitate their use in iterative calculations for statistical inference the crude frequency simulator cfs normal importance sampling nis a kernel smoothed frequency simulator kfs stern s decomposition simulator sds the geweke hajivassiliou keane simulator ghk a parabolic cylinder function simulator pcf deak s chi squared simulator dcs an acceptance rejection simulator ars the gibbs sampler simulator gss a sequentially unbiased simulator sus and an approximately unbiased simulator aus we also discuss gauss and fortran implementations of these algorithms and present our computational experience with them we find that ghk is overall the most reliable method

testing single index restrictions with a focus on average derivatives abstract this paper considers a situation where the violation of a single index restriction is a concern only to the extent that it causes bias to the estimates of the average derivatives we propose a method to construct tests that concentrate their asymptotic powers upon only such interesting alternatives the test has a 1 2 limiting distribution under the null hypothesis and even accommodates the case where the parameter estimates have a convergence rate slower than n as in the case of maximum score estimation the testing procedure can be easily modified when the main interest lies in average increment effects of binary covariates multivariate average derivatives or linear restrictions other than those of average derivatives results from monte carlo experiments show that the asymptotic theory is a reasonable approximation of the finite sample distributions and an application of our methods to female labor market participation illustrates the importance of this non omnibus approach

the method of simulated scores for the estimation of ldv models the method of simulated scores mss is presented for estimating limited dependent variables models ldv with flexible correlation structure in the unobservables we propose simulators that are continuous in the unknown parameter vectors and hence standard optimization methods can be used to compute the mss estimators that employ these simulators the first continuous method relies on a recursive conditioning of the multivariate normal density through a cholesky triangularization of its variance covariance matrix the second method combines results about the conditionals of the multivariate normal distribution with gibbs resampling techniques we establish consistency and asymptotic normality of the mss estimators and derive suitable rates at which the number of simulations must rise if biased simulators are used

quasi aitken estimation for heteroskedasticity of unknown form abstract two modifications to standard aitken estimation of heteroskedastic linear regressions are suggested 1 different criteria for calculating unknown parameters to yield more efficient estimates to the coefficients when the form of heteroskedasticity is misspecified and 2 a different way of estimating the covariance matrix of the feasible aitken estimator a sampling experiment provides evidence that the first adjustment could provide more efficient estimates though not clearly more so than alternatives the second adjustment is crucial for reliable inference even when the form of heteroskedasticity is correctly specified

a statistical analysis of brain morphology using wild bootstrapping methods for the analysis of brain morphology including voxel based morphology and surface based morphometries have been used to detect associations between brain structure and covariates of interest such as diagnosis severity of disease age iq and genotype the statistical analysis of morphometric measures usually involves two statistical procedures 1 invoking a statistical model at each voxel or point on the surface of the brain or brain subregion followed by mapping test statistics e g t test or their associated p values at each of those voxels 2 correction for the multiple statistical tests conducted across all voxels on the surface of the brain region under investigation we propose the use of new statistical methods for each of these procedures we first use a heteroscedastic linear model to test the associations between the morphological measures at each voxel on the surface of the specified subregion e g cortical or subcortical surfaces and the covariates of interest moreover we develop a robust test procedure that is based on a resampling method called wild bootstrapping this procedure assesses the statistical significance of the associations between a measure of given brain structure and the covariates of interest the value of this robust test procedure lies in its computationally simplicity and in its applicability to a wide range of imaging data including data from both anatomical and functional magnetic resonance imaging fmri simulation studies demonstrate that this robust test procedure can accurately control the family wise error rate we demonstrate the application of this robust test procedure to the detection of statistically significant differences in the morphology of the hippocampus over time across gender groups in a large sample of healthy subjects

approximate power functions for some robust tests of regression coefficients approximate power functions for some robust tests of regression coefficients thomas j rothenberg department o f economics u n i v e r s i t y o f c a l i f o r n i a berkeley research papers i n economics no 84 1 summary edgeworth expansions are developed f o r the d i s t r i b u t i o n functions o f some t e s t s t a t i s t i c s i n the normal l i n e a r regression model where the e r r o r covariance matrix i s unknown tests based on generalized l e a s t squares estimates and also on ordinary l e a s t squares estimates are considered in both cases adjustments t o the asymptotic c r i t i c a l the approxi values are found and approximate l o c a l power c a l c u l a t e d and a u t o c o r r e l a t i o n mations are applied to a number o f examples i n c l u d i n g h e t e r o s c e d a s t i c i t y

multiple solutions for elliptic problems with asymmetric nonlinearity in this paper we establish the existence of multiple solutions for the semilinear elliptic problem u g x u in u 0on where rn is an open bounded domain with smooth boundary g r r is a function of class c1 such that g x 0 0 and which is asymptotically linear at infinity with jumping nonlinearities we use critical groups to distinguish the critical points

some new results on the existence of periodic solutions to a kind of rayleigh equation with a deviating argument abstract by employing the continuation theorem of coincidence degree theory we study a kind of rayleigh equation with a deviating argument as follows x t f x t g x t t p t some new results on the existence of periodic solutions are obtained

small sample performance of bias corrected sandwich estimators for cluster randomized trials with binary outcomes the sandwich estimator in generalized estimating equations gee approach underestimates the true variance in small samples and consequently results in inflated type i error rates in hypothesis testing this fact limits the application of the gee in cluster randomized trials crts with few clusters under various crt scenarios with correlated binary outcomes we evaluate the small sample properties of the gee wald tests using bias corrected sandwich estimators our results suggest that the gee wald z test should be avoided in the analyses of crts with few clusters even when bias corrected sandwich estimators are used with t distribution approximation the kauermann and carroll kc correction can keep the test size to nominal levels even when the number of clusters is as low as 10 and is robust to the moderate variation of the cluster sizes however in cases with large variations in cluster sizes the fay and graubard fg correction should be used instead furthermore we derive a formula to calculate the power and minimum total number of clusters one needs using the t test and kc correction for the crts with binary outcomes the power levels as predicted by the proposed formula agree well with the empirical powers from the simulations the proposed methods are illustrated using real crt data we conclude that with appropriate control of type i error rates under small sample sizes we recommend the use of gee approach in crts with binary outcomes because of fewer assumptions and robustness to the misspecification of the covariance structure copyright 2014 john wiley sons ltd

bias corrections for two step fixed effects panel data estimators this paper introduces large t bias corrected estimators for nonlinear panel data models with both time invariant and time varying heterogeneity these models include systems of equations with limited dependent variables and unobserved individual effects and sample selection models with unobserved individual effects our two step approach first estimates the reduced form by fixed effects procedures to obtain estimates of the time varying heterogeneity underlying the endogeneity selection bias we then estimate the primary equation by fixed effects including an appropriately constructed control variable from the reduced form estimates as an additional explanatory variable the fixed effects approach in this second step captures the time invariant heterogeneity while the control variable accounts for the time varying heterogeneity since either or both steps might employ nonlinear fixed effects procedures it is necessary to bias adjust the estimates due to the incidental parameters problem this problem is exacerbated by the two step nature of the procedure as these two step approaches are not covered in the existing literature we derive the appropriate correction thereby extending the use of large t bias adjustments to an important class of models simulation evidence indicates our approach works well in finite samples and an empirical example illustrates the applicability of our estimator

some heteroskedasticity consistent covariance matrix estimators with improved finite sample properties abstract we examine several modified versions of the heteroskedasticity consistent covariance matrix estimator of hinkley 1977 and white 1980 on the basis of sampling experiments which compare the performance of quasi t statistics we find that one estimator based on the jackknife performs better in small samples than the rest we also examine the finite sample properties of using modified critical values based on edgeworth approximations as proposed by rothenberg 1984 in addition we compare the power of several tests for heteroskedasticity and find that it may be wise to employ the jackknife heteroskedasticity consistent covariance matrix even in the absence of detected heteroskedasticity

analyzing two stage experiments in the presence of interference abstracttwo stage randomization is a powerful design for estimating treatment effects in the presence of interference that is when one individual s treatment assignment affects another individual s outcomes our motivating example is a two stage randomized trial evaluating an intervention to reduce student absenteeism in the school district of philadelphia in that experiment households with multiple students were first assigned to treatment or control then in treated households one student was randomly assigned to treatment using this example we highlight key considerations for analyzing two stage experiments in practice our first contribution is to address additional complexities that arise when household sizes vary in this case researchers must decide between assigning equal weight to households or equal weight to individuals we propose unbiased estimators for a broad class of individual and household weighted estimands with corresponding theoretical and estimated variances our second co

explaining individual response using aggregated data abstract empirical analysis of individual response behavior is sometimes limited due to the lack of explanatory variables at the individual level in this paper we put forward a new approach to estimate the effects of covariates on individual response where the covariates are unknown at the individual level but observed at some aggregated level this situation may for example occur when the response variable is available at the household level but covariates only at the zip code level we describe the missing individual covariates by a latent variable model which matches the sample information at the aggregate level parameter estimates can be obtained using maximum likelihood or a bayesian analysis we illustrate the approach estimating the effects of household characteristics on donating behavior to a dutch charity donating behavior is observed at the household level while the covariates are only observed at the zip code level

higher energy solutions in the theory of phase transitions a variational approach abstract we establish the existence of a higher energy solution to the vector ginzburg landau equation with a triple well potential on a bounded and smooth domain on the plane this solution is obtained by a linking argument in implementing this variational approach we make several considerations on the dynamics of the negative gradient flow in particular we use the conley index to contruct a suitable one dimensional invariant set this solution has morse index two in the nondegenerate case we discuss its structure in connection with the so called triple junction configurations

likelihood approximation by numerical integration on sparse grids the calculation of likelihood functions of many econometric models requires the evaluation of integrals without analytical solutions approaches for extending gaussian quadrature to multiple dimensions discussed in the literature are either very specific or suffer from exponentially rising computational costs in the number of dimensions we propose an extension that is very general and easily implemented and does not suffer from the curse of dimensionality monte carlo experiments for the mixed logit model indicate the superior performance of the proposed method over simulation techniques

regression based methods for using control variates in monte carlo experiments abstract methods based on linear regression provide an easy way to use the information in control variates to improve the efficiency with which certain features of the distributions of estimators and test statistics are estimated in monte carlo experiments we propose a new technique that allows these methods to be used when the quantities of interest are quantiles we also propose new ways to obtain approximately optimal control variates in many cases of interest these methods seem to work well in practice and can greatly reduce the number of replications required to obtain a given level of accuracy

inference when a nuisance parameter is not identified under the null hypothesis many econometric testing problems involve nuisance parameters which are not identified under the null hypotheses this paper studies the asymptotic distribution theory for such tests the asymptotic distributions of standard test statistics are described as functionals of chi square processes in general the distributions depend upon a large number of unknown parameters we show that a transformation based upon a conditional probability measure yields an asymptotic distribution free of nuisance parameters and we show that this transformation can be easily approximated via simulation the theory is applied to threshold models with special attention given to the so called self exciting threshold autoregressive model monte carlo methods are used to assess the finite sample distributions the tests are applied to u s gnp growth rates and we find that potter s 1995 threshold effect in this series can be possibly explained by sampling variation

asymptotic normality of series estimators for nonparametric and semiparametric regression models this paper establishes the asymptotic normality of series estimators for nonparametric regression models gallant s fourier flexible form estimators trigonometric series estimators and polynomial series estimators are prime examples of the estimators covered by the results the results apply to a wide variety of estimates in the regression model under consideration including derivatives and integrals of the regression function the errors in the model may be homoskedastic or heteroskedastic the paper also considers series estimators for additive interactive regression semiparametric regression and semiparametric index regression models and shows them to be consistent and asymptotically normal copyright 1991 by the econometric society

covariance estimators for generalized estimating equations gee in longitudinal analysis with small samples generalized estimating equations gee is a general statistical method to fit marginal models for longitudinal data in biomedical studies the variance covariance matrix of the regression parameter coefficients is usually estimated by a robust sandwich variance estimator which does not perform satisfactorily when the sample size is small to reduce the downward bias and improve the efficiency several modified variance estimators have been proposed for bias correction or efficiency improvement in this paper we provide a comprehensive review on recent developments of modified variance estimators and compare their small sample performance theoretically and numerically through simulation and real data examples in particular wald tests and t tests based on different variance estimators are used for hypothesis testing and the guideline on appropriate sample sizes for each estimator is provided for preserving type i error in general cases based on numerical results moreover we develop a user friendly r package geesmv incorporating all of these variance estimators for public usage in practice

a comparison of the variance estimation methods for heteroscedastic nonlinear models heteroscedasticity is commonly encountered when fitting nonlinear regression models in practice we discuss eight different variance estimation methods for nonlinear regression models with heterogeneous response variances and present a simulation study to compare the performance of the eight methods in terms of estimating the standard errors of the fitted model parameters the simulation study suggests that when the true variance is a function of the mean model the power of the mean variance function estimation method and the transform both sides method are the best choices for estimating the standard errors of the estimated model parameters in general the wild bootstrap estimator and two modified versions of the standard sandwich variance estimator are reasonably accurate with relatively small bias especially when the heterogeneity is nonsystematic across values of the covariate furthermore we note that the two modified sandwich estimators are appealing choices in practice considering the computational advantage of these two estimation methods relative to the variance function estimation method and the transform both sides approach copyright 2016 john wiley sons ltd

periodic solutions for higher order neutraldifferential equations with several delays neutral differential equations arise in many practical problems and have importantapplications in physics and engineering this paper introduces fourier series method and inequality techniques to investigate periodic solutions for a class of higher order delayed neutral differential equations this method is different to some traditional methods such as critical point theory fixed point method and topological degree method which are applied to study periodic solution problem of neutral differential equations some new necessary and sufficient conditions are obtained ensuring the existence and uniqueness of periodic solutions in addition three examples are given to illustrate the theory

smooth unbiased multivariate probability simulators for maximum likelihood estimation of limited dependent variable models abstract we apply a new simulation method that solves the multidimensional probability integrals that arise in maximum likelihood estimation of a broad class of limited dependent variable models the simulation method has four key features the simulated choice probabilities are unbiased they are a continuous and differentiable function of the parameters of the model they are bounded between 0 and 1 and their computation takes an effort that is nearly linear in the dimension of the probability integral independent of the magnitudes of the true probabilities we also show that the new simulation method produces probability estimates with substantially smaller variance than those generated by acceptance rejection methods or by stern s 1992 method the simulated probabilities can therefore be used to revive the lerman and manski 1981 procedure of approximating the likelihood function using simulated choice probabilities by overcoming its computational disadvantages

on antiperiodic solutions for cohen grossberg shunting inhibitory neural networks with time varying delays and impulses in this letter a class of cohen grossberg shunting inhibitory neural networks with time varying delays and impulses is investigated sufficient conditions for the existence and exponential stability of antiperiodic solutions of such a class of neural networks are established our results are new and complementary to previously known results an example is given to illustrate the feasibility and effectiveness of our main results

does modeling a structural break improve forecast accuracy mean square forecast error loss implies a bias variance trade off that suggests that structural breaks of small magnitude should be ignored in this paper we provide a test to determine whether modeling a structural break improves forecast accuracy the test is near optimal even when the date of a local to zero break is not consistently estimable the results extend to forecast combinations that weight the post break sample and the full sample forecasts by our test statistic in a large number of macroeconomic time series we find that structural breaks that are relevant for forecasting occur much less frequently than existing tests indicate

a robust effect size index effect size indices are useful tools in study design and reporting because they are unitless measures of association strength that do not depend on sample size existing effect size indices are developed for particular parametric models or population parameters here we propose a robust effect size index based on m estimators this approach yields an index that is very generalizable because it is unitless across a wide range of models we demonstrate that the new index is a function of cohen s d formula see text and standardized log odds ratio when each of the parametric models is correctly specified we show that existing effect size estimators are biased when the parametric models are incorrect e g under unknown heteroskedasticity we provide simple formulas to compute power and sample size and use simulations to assess the bias and standard error of the effect size estimator in finite samples because the new index is invariant across models it has the potential to make communication and comprehension of effect size uniform across the behavioral sciences

residual bootstrap tests in linear models with many regressors abstract this paper is concerned with bootstrap hypothesis testing in linear regression models with many regressors i show that bootstrap f l r and l m tests are asymptotically valid even when the numbers of estimated parameters and tested restrictions are not asymptotically negligible fractions of the sample size one of the conditions for these results is that the regressors come from an asymptotically balanced design depending on the number of restrictions tested and on the errors distribution violation of that condition might render the bootstrap tests asymptotically invalid in that case i propose bootstrapping calhoun s 2011 g statistic or modified versions of the l r and l m statistics and show that these procedures remain asymptotically valid monte carlo simulations indicate that the bootstrap tests often outperform asymptotic ones however analyzing the approximate third cumulant of the f statistic reveals that the bootstrap test does not generally provide the usual higher order asymptotic refinements nevertheless it is found that the bootstrap third cumulant partially matches the population third cumulant which might explain the bootstrap s good finite sample performances especially when the errors come from a symmetric distribution

on the power of tests for superexogeneity and structural invariance abstract this paper examines the finite sample power of tests of structural invariance and superexogeneity hypotheses in econometric models with contemporaneous conditioning variables we consider both direct parametric tests of superexogeneity as well as indirect procedures based on temporal stability tests for the parameters of interest our monte carlo analysis reveals that both types of tests may lack power in interesting classes of models an empirical illustration investigates the superexogeneity of the short term interest rate in a dynamic specification for the u s term structure

periodic solutions for 2kth order ordinary differential equations with resonance in this paper a non variational version of a max min principle is extended and some unique existence results are obtained for the periodic boundary value problem of the higher order ordinary differential equations under a resonant condition

improved heteroscedasticity consistent covariance matrix estimators summary the heteroscedasticity consistent covariance matrix estimator proposed by white 1980 is commonly used in practical applications and is implemented into a number of pieces of statistical software however although consistent it can display substantial bias in small to moderately large samples as shown by monte carlo simulations elsewhere this paper defines modified white estimators which are approximately bias free numerical results show that the modified estimators display much smaller bias than white s estimator in small samples we also show that the bias correction leads to some variance inflation in hypothesis testing based on heteroscedasticity consistent covariance matrix estimators numerical results suggest that tests based on the proposed bias corrected estimators typically display smaller size distortions

periodic solutions for nonlinear nth order differential equations with delays by applying the continuation theorem of coincidence degree theory we establish the existence of 2 periodic solutions for a class of nonlinear nth order differential equations with delays

a note on studentizing a test for heteroscedasticity abstract breusch and pagan 1979 have recently proposed a convenient test for heteroscedasticity in general linear models this note derives the asymptotic distribution of their test under sequences of contiguous alternatives to the null hypothesis of homoscedasticity the test is shown to possess asymptotically incorrect size nominal significance level except in the case of strictly gaussian disturbances a slight modification of the test is proposed which corrects this defect

subvector inference when the true parameter vector may be near or at the boundary abstract extremum estimators are not asymptotically normally distributed when the estimator satisfies the restrictions on the parameter space such as the non negativity of a variance parameter and the true parameter vector is near or at the boundary this possible lack of asymptotic normality makes it difficult to construct tests for testing subvector hypotheses that control asymptotic size in a uniform sense and have good local asymptotic power irrespective of whether the true parameter vector is at near or far from the boundary we propose a novel estimator that is asymptotically normally distributed even when the true parameter vector is near or at the boundary and the objective function is not defined outside the parameter space the proposed estimator allows the implementation of a new test based on the conditional likelihood ratio statistic that is easy to implement controls asymptotic size and has good local asymptotic power properties furthermore we show that the test enjoys certain asymptotic optimality properties when the parameter of interest is scalar in an application of the random coefficients logit model berry levinsohn and pakes 1995 to the european car market we find that for most parameters the new test leads to tighter confidence intervals than the two sided t test commonly used in practice

periodic solutions for a kind of lienard equation with a deviating argument abstract by means of continuation theorem of coincidence degree theory we study a kind of lienard equation with a deviating argument as follows x t f x t x t g x t t x t e t some new results on the existence of periodic solutions are obtained

alternative sampling methods for estimating multivariate normal probabilities we study the performance of alternative sampling methods for estimating multivariate normal probabilities through the ghk simulator the sampling methods are randomized versions of some quasi monte carlo samples halton niederreiter niederreiter xing sequences and lattice points and some samples based on orthogonal arrays latin hypercube orthogonal array and orthogonal array based latin hypercube samples in general these samples turn out to have a better performance than monte carlo and antithetic monte carlo samples improvements over these are large for low dimensional 4 and 10 cases and still significant for dimensions as large as 50

estimation of marginal effects in semiparametric selection models with binary outcomes this paper addresses the estimation of a semiparametric sample selection index model where both the selection rule and the outcome variable are binary since the marginal effects are often of primary interest and are difficult to recover in a semiparametric setting we focus on developing an estimator for the marginal effects this marginal effect estimator uses only observations where the selection probability is above a certain threshold a key innovation is that this high probability set is adaptive to the data we establish the large sample properties of the marginal effect estimator as well as those for an index estimator upon which it depends monte carlo studies show that these estimators perform well in finite samples

specification testing in markov switching time series models this paper develops a series of specification tests of markov switching time series models tests for omitted autocorrelation omitted arch misspecification of the markovian dynamics and omitted explanatory variables are proposed all of the tests can be constructed as a natural byproduct of the routine used to calculate the smoothed probability that a given observation came from a particular regime and do not require estimation of additional parameters the paper performs monte carlo analysis of the tests and briefly illustrates their use with an empirical application

on mitigating the analytical limitations of finely stratified experiments although attractive from a theoretical perspective finely stratified experiments such as paired designs suffer from certain analytical limitations that are not present in block randomized experiments with multiple treated and control individuals in each block in short when using a weighted difference in means to estimate the sample average treatment effect the traditional variance estimator in a paired experiment is conservative unless the pairwise average treatment effects are constant across pairs however in more coarsely stratified experiments the corresponding variance estimator is unbiased if treatment effects are constant within blocks even if they vary across blocks using insights from classical least squares theory we present an improved variance estimator that is appropriate in finely stratified experiments the variance estimator remains conservative in expectation but is asymptotically no more conservative than the classical estimator and can be considerably less conservative the magnitude of the improvement depends on the extent to which effect heterogeneity can be explained by observed covariates aided by this estimator a new test for the null hypothesis of a constant treatment effect is proposed these findings extend to some but not all superpopulation models depending on whether the covariates are viewed as fixed across samples

existence and uniqueness of periodic solutions for 2n 1 th order differential equations applying the schauder fixed point theorem and the continuation theorem of mawhin we give the existence and uniqueness results of periodic solutions for a class of 2n 1 th order nonlinear ordinary differential equations

statistical analysis of diffusion tensors in diffusion weighted magnetic resonance imaging data diffusion tensor imaging has been widely used to reconstruct the structure and orientation of fibers in biological tissues particularly in the white matter of the brain because it can track the effective diffusion of water along those fibers the raw diffusion weighted images from which diffusion tensors are estimated however inherently contain noise noise in the images produces uncertainty in the estimation of the tensors which are 3 3 positive definite matrices and of their derived quantities including eigenvalues eigenvectors and the fiber pathways that are reconstructed based on those tensor elements the aim of this article is to provide a comprehensive theoretical framework of statistical inference for quantifying the effects of noise on diffusion tensors on their eigenvalues and eigenvectors and on their morphological classification we propose a semiparametric model to account for noise in diffusion weighted images we then develop a one step weighted least squares estimate of the t

a method for smoothing simulated moments of discrete probabilities in multinomial probit models there has been much interest over the years in estimating discrete choice models however with the exception of multinomial logit models agents have been restricted to few choices so that multivariate integrals could be feasibly integrated numerically 2 pakes and pollard 1989 and mcfadden 1989 have independently developed the method of simulated moments msm to deal with estimating a wide class of models of which high order discrete choice models are a subset one of the problems a researcher must handle when using msm for a discrete choice model is how to smooth the discrete simulated random variables this is necessary to keep small the number of draws required to simulate derivatives and to simulate variation in the data geweke 1989 and mcfadden 1989 have suggested importance sampling methods and other methods to smooth the simulated variables for general error structures in this paper i present a factor analytic smoothing method that can be applied to probit problems although it is not as general as the methods geweke and mcfadden suggest it is easy to use and has clear intuition furthermore simulated probabilities will be unbiased will be bounded between zero and unity and will have smaller variances than unsmoothed probabilities always and smaller variances than importance sampling estimates for a large class of probabilities the second section of this paper develops notation defines the problem and presents the smoothing method and an algorithm to employ it the last section presents monte carlo comparisons of the smoothing method to the importance sampling method

universal donsker classes and metric entropy when mathfrak f is a universal donsker class then for independent indetically distributed i i d observation mathbf x 1 ldots mathbf x n with an unknown law p for any mathfrak f i in mathfrak f i 1 ldots m quad n 1 2 left mathfrak f 1 left mathbf x 1 right ldots mathfrak f i left mathbf x n right right 1 leq i leq m is asymptotically normal with mean vector n 1 2 left int mathfrak f i left mathbf x n right d mathbf p left x right right 1 leq i leq m and covariance matrix int mathfrak f i mathfrak f j d mathbf p int mathfrak f id mathbf p int mathfrak f jd mathbf p uniformly for mathfrak f i in mathfrak f then for certain statistics formed frome the mathfrak f i left mathbf x k right even where mathfrak f i may be chosen depending on the mathbf x k there will be asymptotic distribution as n rightarrow infty for example for mathbf x 2 statistics where f i are indicators of disjoint intervals depending suitably on mathbf x 1 ldots mathbf x n whose union is the real line mathbf x 2 quadratic forms have limiting distributions roy 1956 and watson 1958 which may however not be mathbf x 2 distributions and may depend on p chernoff and lehmann 1954 universal donsker classes of sets are up to mild measurability conditions just classes satisfying the vapnik cervonenkis comdinatorial conditions defined later in this section donsker the vapnik cervonenkis combinatorial conditions defined later in this section durst and dudley 1981 and dudley 1984 chapter 11 the use of such classes allows a variety of extensions of the roy watson results to general multidimensional sample spaces pollard 1979 and moore and subblebine 1981 vapnik and cervonenkis 1974 indicated application of their families of sets to classification pattern recognition problems more recently the classes have been applied to tree structured classifiacation breiman friedman olshen and stone 1984 chapter 12

resurrecting weighted least squares this paper shows how asymptotically valid inference in regression models based on the weighted least squares wls estimator can be obtained even when the model for reweighting the data is misspecified like the ordinary least squares estimator the wls estimator can be accompanied by heterokedasticty consistent hc standard errors without knowledge of the functional form of conditional heteroskedasticity first we provide rigorous proofs under reasonable assumptions second we provide numerical support in favor of this approach indeed a monte carly study demonstrates attractive finite sample properties compared to the status quo both in terms of estimation and making inference

a method of simulated moments for estimation of discrete response models without numerical integration this paper proposes a simple modification of a conventional method of moments estimator for a discrete response model replacing response probabilities that require numerical integration with estimators obtained by monte carlo simulation this method of simulated moments msm does not require precise estimates of these probabilities for consistency and asymptotic normality relying instead on the law of large numbers operating across observations to control simulation error and hence can use simulations of practical size the method is useful for models such as high dimensional multinomial probit mnp where computation has restricted applications

a covariance estimator for gee with improved small sample properties summary in this paper we propose an alternative covariance estimator to the robust covariance estimator of generalized estimating equations gee hypothesis tests using the robust covariance estimator can have inflated size when the number of independent clusters is small resampling methods such as the jackknife and bootstrap have been suggested for covariance estimation when the number of clusters is small a drawback of the resampling methods when the response is binary is that the methods can break down when the number of subjects is small due to zero or near zero cell counts caused by resampling we propose a bias corrected covariance estimator that avoids this problem in a small simulation study we compare the bias corrected covariance estimator to the robust and jackknife covariance estimators for binary responses for situations involving 10 40 subjects with equal and unequal cluster sizes of 16 64 observations the bias corrected covariance estimator gave tests with sizes close to the nominal level even when the number of subjects was 10 and cluster sizes were unequal whereas the robust and jackknife covariance estimators gave tests with sizes that could be 2 3 times the nominal level the methods are illustrated using data from a randomized clinical trial on treatment for bone loss in subjects with periodontal disease

periodic solutions for nonlinearnth order ordinary differential equations abstract existence of periodic solutions ofnth order ordinary differential equations with nonlinear restoring termh x and time variable coefficients is studied without the assumption of boundedness ofh x andh x

moderation analysis using a two level regression model moderation analysis is widely used in social and behavioral research the most commonly used model for moderation analysis is moderated multiple regression mmr in which the explanatory variables of the regression model include product terms and the model is typically estimated by least squares ls this paper argues for a two level regression model in which the regression coefficients of a criterion variable on predictors are further regressed on moderator variables an algorithm for estimating the parameters of the two level model by normal distribution based maximum likelihood nml is developed formulas for the standard errors ses of the parameter estimates are provided and studied results indicate that when heteroscedasticity exists nml with the two level model gives more efficient and more accurate parameter estimates than the ls analysis of the mmr model when error variances are homoscedastic nml with the two level model leads to essentially the same results as ls with the mmr model most importantly the two level regression model permits estimating the percentage of variance of each regression coefficient that is due to moderator variables when applied to data from general social surveys 1991 nml with the two level model identified a significant moderation effect of race on the regression of job prestige on years of education while ls with the mmr model did not an r package is also developed and documented to facilitate the application of the two level model

some results on the glejser and koenker tests for heteroskedasticity abstract the glejser and koenker studentized lagrange multiplier tests for heteroskedasticity are considered asymptotic theory and monte carlo experiments are used to investigate the effects of nonnormality under null and alternative hypotheses and also the consequences of using an incorrect alternative a method for using test outcomes to select models of heteroskedasticity is critically examined glejser s test is not in general asymptotically valid under asymmetric disturbances koenker s test is asymptotically valid under asymmetric and other nonnormal disturbances but has estimated finite sample significance levels that are sometimes sensitive to skewness an adjusted version of koenker s statistic is discussed

empirically relevant critical values for hypothesis tests a bootstrap approach tests of statistical hypotheses can be based on either of two critical values the type i critical value or the size corrected critical value the former usually depends on unknown population parameters and cannot be evaluated exactly in applications but it can often be estimated very accurately by using the bootstrap the latter does not depend on unknown population parameters but is likely to yield a test with low power the critical values used in most monte carlo studies of the powers of tests are neither type i nor size corrected they are irrelevant to empirical research

a note on the efficiency of sandwich covariance matrix estimation the sandwich estimator also known as robust covariance matrix estimator heteroscedasticity consistent covariance matrix estimate or empirical covariance matrix estimator has achieved increasing use in the econometric literature as well as with the growing popularity of generalized estimating equations its virtue is that it provides consistent estimates of the covariance matrix for parameter estimates even when the fitted parametric model fails to hold or is not even specified surprisingly though there has been little discussion of properties of the sandwich method other than consistency we investigate the sandwich estimator in quasi likelihood models asymptotically and in the linear case analytically we show that under certain circumstances when the quasi likelihood model is correct the sandwich estimate is often far more variable than the usual parametric variance estimate the increased variance is a fixed feature of the method and the price that one pays to obtain consistency even when the p

a random effects model for ordered categorical data abstract this article presents a random effects model for multivariate and grouped univariate ordered categorical data the assumed family of distributions for the random effects adopts a wide variety of forms and shapes the model s likelihood has a closed expression and can be computed without recourse to numerical integration or gaussian quadrature a bivariate and a grouped univariate example are used to illustrate the proposed model

simulation based finite sample tests for heteroskedasticity and arch effects tests for heteroskedasticity in linear regressions are typically based on asymptotic approximations we show that the size of such tests can be perfectly controlledin nite samples through monte carlo test techniques with both gaussian and non gaussian heavy tailed disturbance distributions the procedures studied include standard heteroskedasticity tests e g glejser bartlett cochran hartley breusch pagan godfrey white szroeter as well as tests for arch type heterosked asticity sup type andcombinedtests are also proposedto allow for unknown breakpoints in the variance the fact that the proposedproced ures achieve size control andhave goodpower is demonstrated in a monte carlo simulation c 2003 elsevier b v all rights reserved

multiple nontrivial solutions of semilinear elliptic equations and their homotopy indexes abstract in this paper we consider the semilinear elliptic equation u x g u x x u x 0 x where r n is a bounded domain with smooth boundary and g r r is of class c 2 it is clear that solutions of this problem are equilibria of the local semiflow g generated by solutions of the semilinear parabolic equation u t t x u t x g u t x t 0 x u t x 0 t 0 x we study the existence of multiple nontrivial solutions of the above elliptic equation and their homotopy indices the homotopy index theory is an extension of conley s index theory to noncompact spaces due to rybakowski and gives some information about the local semiflow g

periodic solutions to neutral differential equation with deviating arguments by employing the continuation theorem of coincidence degree theory and some analysis techniques we study a kind of neutral differential equation with deviating arguments as follows x t cx t r f x t g x t t p t some new results on the existence of periodic solutions are obtained our result generalizes main work of wang app math lett 12 1999 41 44

anti periodic solutions for a class of nonlinear nth order differential equations with delays in this paper we use the leray schauder degree theory to establish new results on the existence and uniqueness of anti periodic solutions for a class of nonlinear nth order differential equations with delays of the form x n t f t x n 1 t g t x email protected t e t

heteroscedasticity robust inference in linear regression models with many covariates we consider inference in linear regression models that is robust to heteroscedasticity and the presence of many control variables when the number of control variables increases at the same rate as

the finite sample distributions of heteroskedasticity robust wald statistics the imhof procedure is used to calculate the exact finite sample distributions of alternative heteroskedasticity robust wald type tests of scalar linear hypothesis in the normal linear model the tests are distinguished by the variance estimator that they use these include the white eicker and jackknife estimators the calculations demonstrate the importance of regression design in determining the quality of conventional first order asymptotic approximations to the null distributions of the tests

simulation and the asymptotics of optimization estimators on demontre un theoreme de limite centrale general pour des estimateurs definis par minimisation de la longueur d une fonction de critere aleatoire a valeurs vectorielles aucune hypothese de regularite suffisante n est imposee sur la fonction de critere pour que les resultats puissent etre appliques a une classe d estimlateurs de simulation assez large des analyses completes de deux estimateurs de simulation l un introduit par pakes l autre par mcfadden illustrent l application des theoremes generaux

bayesian inference in econometric models using monte carlo integration methods for the systematic application of monte carlo integration with importance sampling to bayesian inference in econometric models are developed conditions under which the numerical approximation of a posterior moment converges almost surely to the true value as the number of monte carlo replications increases and the numerical accuracy of this approximation may be assessed reliably are set forth methods for the analytical verification of these conditions are discussed

some improvements in confidence intervals for standardized regression coefficients yuan and chan psychometrika 76 670 690 2011 doi 10 1007 s11336 011 9224 6 derived consistent confidence intervals for standardized regression coefficients under fixed and random score assumptions jones and waller psychometrika 80 365 378 2015 doi 10 1007 s11336 013 9380 y extended these developments to circumstances where data are non normal by examining confidence intervals based on browne s br j math stat psychol 37 62 83 1984 doi 10 1111 j 2044 8317 1984 tb00789 x asymptotic distribution free adf theory seven different heteroscedastic consistent hc estimators were investigated in the current study as potentially better solutions for constructing confidence intervals on standardized regression coefficients under non normality normal theory adf and hc estimators were evaluated in a monte carlo simulation findings confirmed the superiority of the hc3 mackinnon and white j econ 35 305 325 1985 doi 10 1016 0304 4076 85 90158 7 and hc5 cribari neto and da silva adv stat anal 95 129 146 2011 doi 10 1007 s10182 010 0141 2 interval estimators over jones and waller s adf estimator under all conditions investigated as well as over the normal theory method the hc5 estimator was more robust in a restricted set of conditions over the hc3 estimator some possible extensions of hc estimators to other effect size measures are considered for future developments

two step estimation of semiparametric censored regression models root n consistent estimators of the regression coecients in the linear censored regression model under conditional quantile restrictions on the error terms were pro posed by powell journal of econometrics 25 1984 303 325 32 1986a 143 155 while those estimators have desirable asymptotic properties under weak regularity conditions simulation studies have shown these estimators to exhibit a small sam ple bias in the opposite direction of the least squares bias for censored data this paper introduces two step estimators for these models which minimize convex objec tive functions and are designed to overcome thisnite sample bias the paper gives regularity conditions under which the proposed two step estimators are consistent and asymptotically normal a monte carlo study compares thenite sample behavior of the proposed methods with their one step counterparts 2001 elsevier science s a all rights reserved

quasi maximum likelihood estimation for conditional quantiles in this paper we derive the asymptotic distribution of a new class of quasi maximum likelihood estimators qmle based on a tick exponential family of densities we show that the tick exponential assumption is a necessary and sufficient condition for a qmle to be consistent for the parameters of a correctly specified model of a given conditional quantile hence the role of this family of densities in the conditional quantile estimation is analog to the role of the linear exponential family in the conditional mean estimation the tick exponential qmles are shown to be asymptotically normal with an asymptotic covariance matrix that has a novel form not seen in earlier work and which accounts for possible model misspecification for practical purposes we show that the maximization of the tick exponential quasi log likelihood can conveniently be carried out by using standard gradient based optimization techniques more importantly we provide a consistent estimator for the asymptotic covariance matrix based on the scores of the log likelihood which allows us to compute the conditional quantile confidence intervals

asymptotic theory and wild bootstrap inference with clustered errors abstract we study inference based on cluster robust variance estimators for regression models with clustered errors focusing on the wild cluster bootstrap we state conditions under which asymptotic and bootstrap tests and confidence intervals are asymptotically valid these conditions put limits on the rates at which the cluster sizes can increase as the number of clusters tends to infinity we also derive edgeworth expansions for the asymptotic and bootstrap test statistics simulation experiments illustrate the theoretical results and suggest that alternative variants of the wild cluster bootstrap may perform quite differently the edgeworth expansions explain the overrejection of asymptotic tests and shed light on the choice of auxiliary distribution and whether to use restricted or unrestricted estimates in the bootstrap data generating process

statistical inference in the multinomial multiperiod probit model abstract statistical inference in multinomial multiperiod probit models has been hindered in the past by the high dimensional numerical integrations necessary to form the likelihood functions posterior distributions or moment conditions in these models we describe three alternative estimators implemented using simulation based approaches to inference that circumvent the integration problem posterior means computed using gibbs sampling and data augmentation gibbs simulated maximum likelihood sml estimation using the ghk probability simulator and method of simulated moment msm estimation using ghk we perform a set of monte carlo experiments to compare the sampling distributions of these estimators although all three estimators perform reasonably well some important differences emerge our most important finding is that holding simulation size fixed the relative and absolute performance of the classical methods especially sml gets worse when serial correlation in disturbances is strong in data sets with an ar 1 parameter of 0 50 the rmses for sml and msm based on ghk with 20 draws exceed those of gibbs by 9 and 0 respectively but when the ar 1 parameter is 0 80 the rmses for sml and msm based on 20 draws exceed those of gibbs by 79 and 37 respectively and the number of draws needed to reduce the rmses to within 10 of gibbs are 160 and 80 respectively also the sml estimates of serial correlation parameters exhibit significant downward bias thus while conventional wisdom suggests that 20 draws of ghk is enough to render the bias and noise induced by simulation negligible our results suggest that much larger simulation sizes are needed when serial correlation in disturbances is strong

monte carlo evaluation of multivariate normal probabilities abstract this paper extends research on the simulation of multivariate normal probabilities of high order dimension by developing a new family of simulators such simulators are useful for models with limited dependent variables including multinomial probit in panel studies spatial analysis and time series analysis the simulators are derived from a cholesky decomposition of the covariance matrix combined with a suitable choice of an importance sampling distribution the paper studies among others the impact of antithetical sampling the insights gained in this paper are of use in bayesian analysis as well in the evaluation of posterior densities

two step estimation of heteroskedastic sample selection models abstract this paper considers two step estimation of a sample selection model in which there is heteroskedasticity of unknown form in the latent errors we propose an estimator which uses recent developments in nonparametric regression estimation involving series approximations the estimator is shown to be consistent and asymptotically normally distributed under reasonable conditions a small monte carlo experiment demonstrates the usefulness of the estimator and highlights the bias inherent in the usual heckman 1979 estimator when there is heteroskedasticity

the wild bootstrap tamed at last the wild bootstrap is studied in the context of regression models with heteroskedastic disturbances we show that in one very specific case perfect bootstrap inference is possible and a substantial reduction in the error in the rejection probability of a bootstrap test is available much more generally however the version of the wild bootstrap with this desirable property is without the skewness correction afforded by the currently most popular version of the wild bootstrap simulation experiments show that this does not prevent the preferred version from having the smallest error in rejection probability in small and medium sized samples

inference in linear regression models with many covariates and heteroskedasticity the linear regression model is widely used in empirical work in economics statistics and many other disciplines researchers often include many covariates in their linear model specification in a

an optimized wild bootstrap method for evaluation of measurement uncertainties of dti derived parameters in human brain evaluation of measurement uncertainties or errors in diffusion tensor derived parameters is essential to quantify the sensitivity and specificity of these quantities as potential surrogate biomarkers for pathophysiological processes with diffusion tensor imaging dti computational methods such as the monte carlo simulation have provided insights into characterization of the measurement uncertainty in dti however due to the complexity of real brain data as well as different sources of variations during the image acquisition a robust estimator for dti measurement uncertainty in human brain is required recent studies have shown that wild bootstrap a novel nonparametric statistical method can potentially provide effective estimations of dti measurement uncertainties in human brain dti data in this study we further optimized the dti application of the wild bootstrap method for typical clinical applications we evaluated the validity of wild bootstrap utilizing numerical simulations with different combinations of dti protocol parameters and wild bootstrap experimental designs and quantitatively compared estimates of uncertainties from wild bootstrapping with those from monte carlo simulations our results demonstrate that a wild bootstrap implementation using at least 1000 wild bootstrap iterations with a type ii or type iii heteroskedasticity consistent covariance matrix estimator provides robust evaluations of most dti protocols

the moving blocks bootstrap and robust inference for linear least squares and quantile regressions abstract this paper presents the moving blocks bootstrap mbb as a new inference procedure for linear regression estimation which is robust to heteroskedasticity and autocorrelation of unknown forms the mbb covariance estimator is shown to provide heteroskedasticity and autocorrelation consistent hac standard errors for least squares ls and quantile regression qr coefficient estimators the mbb covariance estimator is shown to be asymptotically equivalent to the bartlett kernel estimator suggested by newey and west 1987 and the asymptotically optimal choice of the blocksize is discussed a monte carlo study is included showing that the mbb fares well in comparison to standard hac inference procedures considering strong mixing data generating processes the paper extends existing asymptotic results for the qr estimator the analogy to the ls case is stressed the paper analyzes both the cases of stochastic and of nonstochastic regressors and suggests two new grenander like conditions for the latter case the use of the mbb approach is illustrated for a practical example using a standard econometric package

marginal analysis of ordinal clustered longitudinal data with informative cluster size the issue of informative cluster size ics often arises in the analysis of dental data ics describes a situation where the outcome of interest is related to cluster size much of the work on modeling marginal inference in longitudinal studies with potential ics has focused on continuous outcomes however periodontal disease outcomes including clinical attachment loss are often assessed using ordinal scoring systems in addition participants may lose teeth over the course of the study due to advancing disease status here we develop longitudinal cluster weighted generalized estimating equations cwgee to model the association of ordinal clustered longitudinal outcomes with participant level health related covariates including metabolic syndrome and smoking status and potentially decreasing cluster size due to tooth loss by fitting a proportional odds logistic regression model the within teeth correlation coefficient over time is estimated using the two stage quasi least squares method the motivation for our work stems from the department of veterans affairs dental longitudinal study in which participants regularly received general and oral health examinations in an extensive simulation study we compare results obtained from cwgee with various working correlation structures to those obtained from conventional gee which does not account for ics our proposed method yields results with very low bias and excellent coverage probability in contrast to a conventional generalized estimating equations approach