model averaging based on leave subject out cross validation this paper develops a frequentist model averaging method based on the leave subject out cross validation this method is applicable not only to averaging longitudinal data models but also to averaging time series models which can have heteroscedastic errors the resulting model averaging estimators are proved to be asymptotically optimal in the sense of achieving the lowest possible squared errors both simulation study and empirical example show the superiority of the proposed estimators over their competitors

shrinkage for categorical regressors abstract this paper introduces a flexible regularization approach that reduces point estimation risk of group means stemming from e g categorical regressors quasi experimental data or panel data models the loss function is penalized by adding weighted squared l 2 norm differences between group location parameters and informative first stage estimates under quadratic loss the penalized estimation problem has a simple interpretable closed form solution that nests methods established in the literature on ridge regression discretized support smoothing kernels and model averaging methods we derive risk optimal penalty parameters and propose a plug in approach for estimation the large sample properties are analyzed in an asymptotic local to zero framework by introducing a class of sequences for close and distant systems of locations that is sufficient for describing a large range of data generating processes we provide the asymptotic distributions of the shrinkage estimators under different penalization schemes the proposed plug in estimator uniformly dominates the ordinary least squares estimator in terms of asymptotic risk if the number of groups is larger than three monte carlo simulations reveal robust improvements over standard methods in finite samples real data examples of estimating time trends in a panel and a difference in differences study illustrate potential applications

two stage testing procedures with independent filtering for genome wide gene environment interaction several two stage multiple testing procedures have been proposed to detect gene environment interaction in genome wide association studies in this article we elucidate general conditions that are required for validity and power of these procedures and we propose extensions of two stage procedures using the case only estimator of gene treatment interaction in randomized clinical trials we develop a unified estimating equation approach to proving asymptotic independence between a filtering statistic and an interaction test statistic in a range of situations including marginal association and interaction in a generalized linear model with a canonical link we assess the performance of various two stage procedures in simulations and in genetic studies from women s health initiative clinical trials

effects of error covariance structure on estimation of model averaging weights and predictive performance 1 when conducting model averaging for assessing groundwater conceptual model uncertainty the averaging weights are often evaluated using model selection criteria such as aic aicc bic and kic akaike information criterion corrected akaike information criterion bayesian information criterion and kashyap information criterion respectively however this method often leads to an unrealistic situation in which the best model receives overwhelmingly large averaging weight close to 100 which cannot be justified by available data and knowledge it was found in this study that this problem was caused by using the covariance matrix c of measurement errors for estimating the negative log likelihood function common to all the model selection criteria this problem can be resolved by using the covariance matrix cek of total errors including model errors and measurement errors to account for the correlation between the total errors an iterative two stage method was developed in the context of maximum likelihood inverse modeling to iteratively infer the unknown cek from the residuals during model calibration the inferred cek was then used in the evaluation of model selection criteria and model averaging weights while this method was limited to serial data using time series techniques in this study it can be extended to spatial data using geostatistical techniques the method was first evaluated in a synthetic study and then applied to an experimental study in which alternative surface complexation models were developed to simulate column experiments of uranium reactive transport it was found that the total errors of the alternative models were temporally correlated due to the model errors the iterative two stage method using cek resolved the problem that the best model receives 100 model averaging weight and the resulting model averaging weights were supported by the calibration results and physical understanding of the alternative models using cek obtained from the iterative two stage method also improved predictive performance of the individual models and model averaging in both synthetic and experimental studies

least squares model averaging this paper considers the problem of selection of weights for averaging across least squares estimates obtained from a set of models existing model average methods are based on exponential akaike information criterion aic and bayesian information criterion bic weights in distinction this paper proposes selecting the weights by minimizing a mallows criterion the latter an estimate of the average squared error from the model average fit we show that our new mallows model average mma estimator is asymptotically optimal in the sense of achieving the lowest possible squared error in a class of discrete model average estimators in a simulation experiment we show that the mma estimator compares favorably with those based on aic and bic weights the proof of the main result is an application of the work of li 1987

bayesian measures of model complexity and fit summary we consider the problem of comparing complex hierarchical models in which the number of parameters is not clearly defined using an information theoretic argument we derive a measure pd for the effective number of parameters in a model as the difference between the posterior mean of the deviance and the deviance at the posterior means of the parameters of interest in general pd approximately corresponds to the trace of the product of fisher s information and the posterior covariance which in normal models is the trace of the hat matrix projecting observations onto fitted values its properties in exponential families are explored the posterior mean deviance is suggested as a bayesian measure of fit or adequacy and the contributions of individual observations to the fit and complexity can give rise to a diagnostic plot of deviance residuals against leverages adding pd to the posterior mean deviance gives a deviance information criterion for comparing models which is related to other information criteria and has an approximate decision theoretic justification the procedure is illustrated in some examples and comparisons are drawn with alternative bayesian and classical proposals throughout it is emphasized that the quantities required are trivial to compute in a markov chain monte carlo analysis

variable selection for logistic regression using a prediction focused information criterion summary in biostatistical practice it is common to use information criteria as a guide for model selection we propose new versions of the focused information criterion fic for variable selection in logistic regression the fic gives depending on the quantity to be estimated possibly different sets of selected variables the standard version of the fic measures the mean squared error of the estimator of the quantity of interest in the selected model in this article we propose more general versions of the fic allowing other risk measures such as the one based on lp error when prediction of an event is important as is often the case in medical applications we construct an fic using the error rate as a natural risk measure the advantages of using an information criterion which depends on both the quantity of interest and the selected risk measure are illustrated by means of a simulation study and application to a study on diabetic retinopathy

model averaging in semiparametric estimation of treatment effects choosing the covariates and functional form of the propensity score is an important choice for estimating treatment effects this paper proposes a data driven way of averaging the estimators over candidate specifications to resolve the specification uncertainty in the propensity score weighting estimation of the att the proposed procedures minimize the estimated mse of the att estimator in a local asymptotic framework we formulate model averaging as a statistical decision problem in a limit experiment and derive an averaging scheme that is bayes optimal with respect to a given prior the averaging estimator outperforms selection estimators and the estimators in any of the candidate models in terms of bayes asymptotic mse our monte carlo studies illustrate the size of the mse gains we apply the averaging procedure to evaluate the effect of a labor market program

structural inference in transition measurement error models for longitudinal data we propose a new class of models transition measurement error models to study the effects of covariates and the past responses on the current response in longitudinal studies when one of the covariates is measured with error we show that the response variable conditional on the error prone covariate follows a complex transition mixed effects model the naive model obtained by ignoring the measurement error correctly specifies the transition part of the model but misspecifies the covariate effect structure and ignores the random effects we next study the asymptotic bias in naive estimator obtained by ignoring the measurement error for both continuous and discrete outcomes we show that the naive estimator of the regression coefficient of the error prone covariate is attenuated while the naive estimators of the regression coefficients of the past responses are generally inflated we then develop a structural modeling approach for parameter estimation using the maximum likelihood estimation method in view of the multidimensional integration required by full maximum likelihood estimation an em algorithm is developed to calculate maximum likelihood estimators in which monte carlo simulations are used to evaluate the conditional expectations in the e step we evaluate the performance of the proposed method through a simulation study and apply it to a longitudinal social support study for elderly women with heart disease an additional simulation study shows that the bayesian information criterion bic performs well in choosing the correct transition orders of the models

least squares forecast averaging abstract this paper proposes forecast combination based on the method of mallows model averaging mma the method selects forecast weights by minimizing a mallows criterion this criterion is an asymptotically unbiased estimate of both the in sample mean squared error mse and the out of sample one step ahead mean squared forecast error msfe furthermore the mma weights are asymptotically mean square optimal in the absence of time series dependence we show how to compute mma weights in forecasting settings and investigate the performance of the method in simple but illustrative simulation environments we find that the mma forecasts have low msfe and have much lower maximum regret than other feasible forecasting methods including equal weighting bic selection weighted bic aic selection weighted aic bates granger combination predictive least squares and granger ramanathan combination

inverse groundwater modeling for hydraulic conductivity estimation using bayesian model averaging and variance window 1 this study proposes a bayesian model averaging bma method to address parameter estimation uncertainty arising from nonuniqueness in parameterization methods bma is able to incorporate multiple parameterization methods for prediction through the law of total probability and to obtain an ensemble average of hydraulic conductivity estimates two major issues in applying bma to hydraulic conductivity estimation are discussed the first problem is using occam s window in usual bma applications to measure approximated posterior model probabilities occam s window only accepts models in a very narrow range tending to single out the best method and discard other good methods we propose a variance window to replace occam s window to cope with this problem the second problem is the kashyap information criterion kic in the approximated posterior model probabilities which tends to prefer highly uncertain parameterization methods by considering the fisher information matrix with sufficient amounts of observation data the bayesian information criterion bic is a good approximation and is able to avoid controversial results from using kic this study adopts multiple generalized parameterization gp methods such as the bma models to estimate spatially correlated hydraulic conductivity numerical examples illustrate the issues of using kic and occam s window and show the advantages of using bic and the variance window in bma application finally we apply bma to the hydraulic conductivity estimation of the 1500 foot sand in east baton rouge parish louisiana

frequentist model average estimators the traditional use of model selection methods in practice is to proceed as if the final selected model had been chosen in advance without acknowledging the additional uncertainty introduced by model selection this often means underreporting of variability and too optimistic confidence intervals we build a general large sample likelihood apparatus in which limiting distributions and risk properties of estimators post selection as well as of model average estimators are precisely described also explicitly taking modeling bias into account this allows a drastic reduction in complexity as competing model averaging schemes may be developed discussed and compared inside a statistical prototype experiment where only a few crucial quantities matter in particular we offer a frequentist view on bayesian model averaging methods and give a link to generalized ridge estimators our work also leads to new model selection criteria the methods are illustrated with real data applications

adaptive regression by mixing adaptation over different procedures is of practical importance different procedures perform well under different conditions in many practical situations it is rather hard to assess which conditions are approximately satisfied so as to identify the best procedure for the data at hand thus automatic adaptation over various scenarios is desirable a practically feasible method named adaptive regression by mixing arm is proposed to convexly combine general candidate regression procedures under mild conditions the resulting estimator is theoretically shown to perform optimally in rates of convergence without knowing which of the original procedures work the best simulations are conducted in several settings including comparing a parametric model with nonparametric alternatives comparing a neural network with a projection pursuit in multidimensional regression and combining bandwidths in kernel regression the results clearly support the theoretical property of arm the arm algorithm assigns we

numerical solution of a singularly perturbed two point boundary value problem using equidistribution analysis of convergence abstract adaptive grid methods are becoming established as valuable computational techniques for the numerical solution of differential equations with near singular solutions adaptive methods are equally effective in approximating solutions of problems with boundary layers or interior layers see for example mulholland et al siam j sci comput 19 4 1998 1261 1289 much is now being done in developing error analyes for methods that are based on adaptivity in this paper we present a rigorous error analysis for the solution of a singularly perturbed two point boundary value problem on a grid that is constructed adaptively from a knowledge of the exact solution the discrete solutions are generated by an upwind finite difference scheme and the grid is formed by equidistributing a monitor function based on arc length an error analysis shows that the discrete solutions are uniformly convergent with respect to the perturbation parameter epsilon the epsilon uniform convergence is confirmed by numerical computations

semiparametric model averaging prediction for dichotomous response abstract model averaging has attracted abundant attentions in the past decades as it emerges as an impressive forecasting device in econometrics social sciences and medicine so far most developed model averaging methods focus only on either parametric models or nonparametric models with a continuous response in this paper we propose a semiparametric model averaging prediction smap method for a dichotomous response the idea is to approximate the unknown score function by a linear combination of one dimensional marginal score functions the weight parameters involved in the approximation are obtained by first smoothing the nonparametric marginal scores and then applying the parametric model averaging via a maximum likelihood estimation the proposed smap provides greater flexibility than parametric models while being more stable than a fully nonparametric approach theoretical properties are investigated in two practical scenarios i covariates are conditionally independent given the response and ii the conditional independence assumption does not hold in the first scenario we show that smap puts weight one to the true model and hence the model averaging estimators are consistent in the second scenario in which a true model may not exist smap is shown to be asymptotically optimal in the sense that its kullback leibler loss is asymptotically identical to that of the best but infeasible model averaging estimator empirical evidences from simulation studies and a real data analysis are presented to support and illustrate our methods

model averaging by jackknife criterion in models with dependent data abstract the past decade witnessed a literature on model averaging by frequentist methods for the most part the asymptotic optimality of various existing frequentist model averaging estimators has been established under i i d errors recently hansen and racine hansen b e racine j 2012 jackknife model averaging journal of econometrics 167 38 46 developed a jackknife model averaging jma estimator which has an important advantage over its competitors in that it achieves the lowest possible asymptotic squared error under heteroscedastic errors in this paper we broaden hansen and racine s scope of analysis to encompass models with i a non diagonal error covariance structure and ii lagged dependent variables thus allowing for dependent data we show that under these set ups the jma estimator is asymptotically optimal by a criterion equivalent to that used by hansen and racine a monte carlo study demonstrates the finite sample performance of the jma estimator in a variety of model settings

model averaging and weight choice in linear mixed effects models this article studies model averaging for linear mixed effects models we establish an unbiased estimator of the squared risk for the model averaging and use the estimator as a criterion for choosing weights the resulting model average estimator is proved to be asymptotically optimal under some regularity conditions simulation experiments show it is superior or comparable to estimators based on the final models selected by the commonly used methods and some existing averaging procedures the proposed procedure is applied to data from an aids clinic trial

regression and time series model selection in small samples summary a bias correction to the akaike information criterion aic is derived for regression and autoregressive time series models the correction is of particular use when the sample size is small or when the number of fitted parameters is a moderate to large fraction of the sample size the corrected method called aicc is asymptotically efficient if the true model is infinite dimensional furthermore when the true model is of finite dimension aicc is found to provide better model order choices than any other asymptotically efficient method applications to nonstationary autoregressive and mixed autoregressive moving average time series models are also discussed

a latent variable approach to study gene environment interactions in the presence of multiple correlated exposures many existing cohort studies initially designed to investigate disease risk as a function of environmental exposures have collected genomic data in recent years with the objective of testing for gene environment interaction g e effects in environmental epidemiology interest in g e arises primarily after a significant effect of the environmental exposure has been documented cohort studies often collect rich exposure data as a result assessing g e effects in the presence of multiple exposure markers further increases the burden of multiple testing an issue already present in both genetic and environment health studies latent variable lv models have been used in environmental epidemiology to reduce dimensionality of the exposure data gain power by reducing multiplicity issues via condensing exposure data and avoid collinearity problems due to presence of multiple correlated exposures we extend the lv framework to characterize gene environment interaction in presence of multiple correlated exposures and genotype categories further similar to what has been done in case control g e studies we use the assumption of gene environment g e independence to boost the power of tests for interaction the consequences of making this assumption or the issue of how to explicitly model g e association has not been previously investigated in lv models we postulate a hierarchy of assumptions about the lv model regarding the different forms of g e dependence and show that making such assumptions may influence inferential results on the g e and g e parameters we implement a class of shrinkage estimators to data adaptively trade off between the most restrictive to most flexible form of g e dependence assumption and note that such class of compromise estimators can serve as a benchmark of model adequacy in lv models we demonstrate the methods with an example from the early life exposures in mexico city to neuro toxicants element study of lead exposure iron metabolism genes and birth weight

optimal designs for frequentist model averaging we consider the problem of designing experiments for estimating a target parameter in regression analysis when there is uncertainty about the parametric form of the regression function a new optimality criterion is proposed that chooses the experimental design to minimize the asymptotic mean squared error of the frequentist model averaging estimate necessary conditions for the optimal solution of a locally and bayesian optimal design problem are established the results are illustrated in several examples and it is demonstrated that bayesian optimal designs can yield a reduction of the mean squared error of the model averaging estimator by up to 45

optimal model averaging estimation for generalized linear models and generalized linear mixed effects models abstractconsidering model averaging estimation in generalized linear models we propose a weight choice criterion based on the kullback leibler kl loss with a penalty term this criterion is different from that for continuous observations in principle but reduces to the mallows criterion in the situation we prove that the corresponding model averaging estimator is asymptotically optimal under certain assumptions we further extend our concern to the generalized linear mixed effects model framework and establish associated theory numerical experiments illustrate that the proposed method is promising

factor profiled sure independence screening we propose a method of factor profiled sure independence screening for ultrahigh dimensional variable selection the objective of this method is to identify nonzero components consistently from a sparse coefficient vector the new method assumes that the correlation structure of the high dimensional data can be well represented by a set of low dimensional latent factors which can be estimated consistently by eigenvalue eigenvector decomposition the estimated latent factors should then be profiled out from both the response and the predictors such an operation referred to as factor profiling produces uncorrelated predictors therefore sure independence screening can be applied subsequently and the resulting screening result is consistent for model selection a major advantage that standard sure independence screening does not share we refer to the new method as factor profiled sure independence screening numerical studies confirm its outstanding performance copyright 2012 oxford university press

time varying model averaging abstract structural changes often occur in economics and finance due to changes in preferences technologies institutional arrangements policies crises etc improving forecast accuracy of economic time series with structural changes is a long standing problem model averaging aims at providing an insurance against selecting a poor forecast model all existing model averaging approaches in the literature are designed with constant non time varying combination weights little attention has been paid to time varying model averaging which is more realistic in economics under structural changes this paper proposes a novel model averaging estimator which selects optimal time varying combination weights by minimizing a local jackknife criterion it is shown that the proposed time varying jackknife model averaging tvjma estimator is asymptotically optimal in the sense of achieving the lowest possible local squared error loss in a class of time varying model averaging estimators under a set of regularity assumptions the tvjma estimator is t h consistent a simulation study and an empirical application highlight the merits of the proposed tvjma estimator relative to a variety of popular estimators with constant model averaging weights and model selection

model averaging prediction for time series models with a diverging number of parameters abstract an important problem with the model averaging approach is the choice of weights in this paper a generalized mallows model averaging gmma criterion for choosing weights is developed in the context of an infinite order autoregressive ar process the gmma method adapts to the circumstances in which the dimensions of candidate models can be large and increase with the sample size the gmma method is shown to be asymptotically optimal in the sense of achieving the lowest out of sample mean squared prediction error mspe for both the independent realization and the same realization predictions which as a byproduct solves a conjecture put forward by hansen 2008 that the well known mallows model averaging criterion from hansen 2007 is asymptotically optimal for predicting the future of a time series the rate of the gmma based weight estimator tending to the optimal weight vector minimizing the independent realization mspe is derived as well both simulation experiment and real data analysis illustrate the merits of the gmma method in the prediction of an ar process

asymptotic post selection inference for the akaike information criterion summaryignoring the model selection step in inference after selection is harmful in this paper we study the asymptotic distribution of estimators after model selection using the akaike information criterion first we consider the classical setting in which a true model exists and is included in the candidate set of models we exploit the overselection property of this criterion in constructing a selection region and we obtain the asymptotic distribution of estimators and linear combinations thereof conditional on the selected model the limiting distribution depends on the set of competitive models and on the smallest overparameterized model second we relax the assumption on the existence of a true model and obtain uniform asymptotic results we use simulation to study the resulting post selection distributions and to calculate confidence regions for the model parameters and we also apply the method to a diabetes dataset

estimation and accuracy after model selection classical statistical theory ignores model selection in assessing estimation accuracy here we consider bootstrap methods for computing standard errors and confidence intervals that take model selection into account the methodology involves bagging also known as bootstrap smoothing to tame the erratic discontinuities of selection based estimators a useful new formula for the accuracy of bagging then provides standard errors for the smoothed estimators two examples nonparametric and parametric are carried through in detail a regression model where the choice of degree linear quadratic cubic is determined by the cp criterion and a lasso based estimation problem

a mallows type model averaging estimator for the varying coefficient partially linear model in the last decade significant theoretical advances have been made in the area of frequentist model averaging fma however the majority of this work has emphasized parametric model setups this

parsimonious model averaging with a diverging number of parameters model averaging generally provides better predictions than model selection but the existing model averaging methods cannot lead to parsimonious models parsimony is an especially important propert

the risk inflation criterion for multiple regression a new criterion is proposed for the evaluation of variable selection procedures in multiple regression this criterion which we call the risk inflation is based on an adjustment to the risk essentially the risk inflation is the maximum increase in risk due to selecting rather than knowing the correct predictors a new variable selection procedure is obtained which in the case of orthogonal predictors substantially improves on aic c p and bic and is close to optimal in contrast to aic c p and bic which use dimensionality penalties of 2 2 and log n respectively this new procedure uses a penalty 2 log p where p is the number of available predictors for the case of nonorthogonal predictors bounds for the optimal penalty are obtained

weighted average least squares estimation of generalized linear models abstract the weighted average least squares wals approach introduced by magnus et al 2010 in the context of gaussian linear models has been shown to enjoy important advantages over other strictly bayesian and strictly frequentist model averaging estimators when accounting for problems of uncertainty in the choice of the regressors in this paper we extend the wals approach to deal with uncertainty about the specification of the linear predictor in the wider class of generalized linear models glms we study the large sample properties of the wals estimator for glms under a local misspecification framework and the finite sample properties of this estimator by a monte carlo experiment the design of which is based on a real empirical analysis of attrition in the first two waves of the survey of health aging and retirement in europe share

model selection an integral part of inference we argue that model selection uncertainty should be fully incorporated into statistical inference whenever estimation is sensitive to model choice and that choice is made with reference to the data we consider different philosophies for achieving this goal and suggest strategies for data analysis we illustrate our methods through three examples the first is a poisson regression of bird counts in which a choice is to be made between inclusion of one or both of two covariates the second is a line transect data set for which different models yield substantially different estimates of abundance the third is a simulated example in which truth is known

forecasting with factor augmented regression a frequentist model averaging approach abstract this paper considers forecast combination with factor augmented regression in this framework a large number of forecasting models are available varying by the choice of factors and the number of lags we investigate forecast combination across models using weights that minimize the mallows and the leave h out cross validation criteria the unobserved factor regressors are estimated by principle components of a large panel with n predictors over t periods with these generated regressors we show that the mallows and leave h out cross validation criteria are asymptotically unbiased estimators of the one step ahead and multi step ahead mean squared forecast errors respectively provided that n t however the paper does not establish any optimality properties for the methods in contrast to well known results in the literature this result suggests that the generated regressor issue can be ignored for forecast combination without restrictions on the relation between n and t simulations show that the mallows model averaging and leave h out cross validation averaging methods yield lower mean squared forecast errors than alternative model selection and averaging methods such as aic bic cross validation and bayesian model averaging we apply the proposed methods to the us macroeconomic data set in stock and watson 2012 and find that they compare favorably to many popular shrinkage type forecasting methods

a new approach with piecewise constant arguments to approximate and numerical solutions of oscillatory problems abstract this paper is devoted to the development of a novel approximate and numerical method for the solutions of linear and non linear oscillatory systems which are common in engineering dynamics the original physical information included in the governing equations of motion is mostly transferred into the approximate and numerical solutions therefore the approximate and numerical solutions generated by the present method reflect more accurately the characteristics of the motion of the systems furthermore the solutions derived are continuous everywhere with good accuracy and convergence in comparing with runge kutta method an approximate solution is developed for a linear oscillatory problem and compared with its corresponding exact solution a non linear oscillatory problem is also solved numerically and compared with the solutions of runge kutta method both the graphical and numerical comparisons are provided in the paper the accuracy of the approximate and numerical solutions can be controlled as desired by the number of terms in the taylor series and the value of a single parameter used in the present work formulae for numerical computation in solving various linear and non linear oscillatory problems by the new approach are provided in the paper

combining minimax shrinkage estimators abstract when one estimates a multivariate normal mean the use of stein estimation entails both the grouping of coordinates and the selection of a set of targets toward which to shrink each group in this article we propose new minimax multiple shrinkage estimators that allow for multiple specifications of these aspects we provide examples that are evaluated on real and simulated data including an estimator that adaptively resolves the issue of combining possibly related estimation problems and an adaptive clustering shrinkage estimator the construction and properties of these estimators are shown to follow from the application of the multiple shrinkage results of george 1986a to general partitioned shrinkage estimators

sparsity oriented importance learning for high dimensional linear regression abstractwith now well recognized nonnegligible model selection uncertainty data analysts should no longer be satisfied with the output of a single final model from a model selection process regar

proof of concept and dose estimation with binary responses under model uncertainty this article suggests a unified framework for testing proof of concept poc and estimating a target dose for the benefit of a more comprehensive robust and powerful analysis in phase ii or similar clinical trials from a pre specified set of candidate models we choose the ones that best describe the observed dose response to decide which models if any significantly pick up a dose effect we construct the permutation distribution of the minimum p value over the candidate set this allows us to find critical values and multiplicity adjusted p values that control the familywise error rate of declaring any spurious effect in the candidate set as significant model averaging is then used to estimate a target dose popular single or multiple contrast tests for poc such as the cochran armitage dunnett or williams tests are only optimal for specific dose response shapes and do not provide target dose estimates with confidence limits a thorough evaluation and comparison of our approach to these tests reveal that its power is as good or better in detecting a dose response under various shapes with many more additional benefits it incorporates model uncertainty in poc decisions and target dose estimation yields confidence intervals for target dose estimates and extends to more complicated data structures we illustrate our method with the analysis of a phase ii clinical trial

weighted envelope estimation to handle variability in model selection summaryenvelope methodology can provide substantial efficiency gains in multivariate statistical problems but in some applications the estimation of the envelope dimension can induce selection volatility that may mitigate those gains current envelope methodology does not account for the added variance that can result from this selection in this article we circumvent dimension selection volatility through the development of a weighted envelope estimator theoretical justification is given for our estimator and the validity of the residual bootstrap for estimating its asymptotic variance is established a simulation study and real data analysis illustrate the utility of our weighted envelope estimator

jackknife model averaging for quantile regressions abstract in this paper we consider model averaging for quantile regressions qr when all models under investigation are potentially misspecified and the number of parameters is diverging with the sample size to allow for the dependence between the error terms and regressors in the qr models we propose a jackknife model averaging jma estimator which selects the weights by minimizing a leave one out cross validation criterion function and demonstrate its asymptotic optimality in terms of minimizing the out of sample final prediction error we conduct simulations to demonstrate the finite sample performance of our estimator and compare it with other model selection and averaging methods we apply our jma method to forecast quantiles of excess stock returns and wages

variance estimation in a model with gaussian sub models this article considers the problem of estimating the dispersion parameter in a gaussian model that is intermediate between a model where the mean parameter is fully known fixed and a model where the mean parameter is completely unknown one of the goals is to understand the implications of the two step process of first selecting a model among a finite number of submodels then estimating a parameter of interest after the model selection but using the same sample data the estimators are classified into global two step and weighted estimators whereas the global type estimators ignore the model space structure the two step estimators explore the structure adaptively and can be related to pretest estimators and the weighted estimators are motivated by the bayesian approach their performances are compared theoretically and through simulations using their risk functions based on a scale invariant quadratic loss function it is shown that in the variance estimation problem efficiency gains arise by ex

assessment and propagation of model uncertainty in most examples of inference and prediction the expression of uncertainty about unknown quantities y on the basis of known quantities x is based on a model m that formalizes assumptions about how x and y are related m will typically have two parts structural assumptions s such as the form of the link function and the choice of error distribution in a generalized linear model and parameters whose meaning is specific to a given choice of s it is common in statistical theory and practice to acknowledge parametric uncertainty about given a particular assumed structure s it is less common to acknowledge structural uncertainty about s itself a widely used approach involves enlisting the aid of x to specify a plausible single best choice s for s and then proceeding as if s were known to be correct in general this approach fails to assess and propagate structural uncertainty fully and may lead to miscalibrated uncertainty assessments about y given x when miscalibration occurs it will often result in understatement of inferential or predictive uncertainty about y leading to inaccurate scientific summaries and overconfident decisions that do not incorporate sufficient hedging against uncertainty in this paper i discuss a bayesian approach to solving this problem that has long been available in principle but is only now becoming routinely feasible by virtue of recent computational advances and examine its implementation in examples that involve forecasting the price of oil and estimating the chance of catastrophic failure of the us space shuttle

malmem model averaging in linear measurement error models we develop model averaging estimation in the linear regression model where some covariates are subject to measurement error the absence of the true covariates in this framework makes the calculation of the standard residual based loss function impossible we take advantage of the explicit form of the parameter estimators and construct a weight choice criterion it is asymptotically equivalent to the unknown model average estimator minimizing the loss function when the true model is not included in the set of candidate models the method achieves optimality in terms of minimizing the relative loss whereas when the true model is included the method estimates the model parameter with root n rate simulation results in comparison with existing bayesian information criterion and akaike information criterion model selection and model averaging methods strongly favour our model averaging method the method is applied to a study on health

likelihood for statistically equivalent models in likelihood inference we usually assume that the model is fixed and then base inference on the corresponding likelihood function often however the choice of model is rather arbitrary and there may be other models which fit the data equally well we study robustness of likelihood inference over such statistically equivalent models and suggest a simple envelope likelihood to capture this aspect of model uncertainty robustness depends critically on how we specify the parameter of interest some asymptotic theory is presented illustrated by three examples

minimax multiple shrinkage estimation on presente des nouveaux estimateurs qui incorporent l information partielle a priori en utilisant les donnees comme emulation du comportement et du risque des estimateurs consideres les plus efficaces

post processing posterior predictive p values this article addresses issues of model criticism and model comparison in bayesian contexts and focuses on the use of the so called posterior predictive p values ppp these involve a general discrepancy or conflict measure and depend on the prior the model and the data they are used in statistical practice to quantify the degree of surprise or conflict in data and to compare different combinations of prior and model the distribution of such ppp values is far from uniform however as we demonstrate for different models making their interpretation and comparison a difficult matter we propose a natural calibration of the ppp values where the resulting cppp values are uniform on the unit interval under model conditions the cppp values which in general rely on a double simulation scheme for their computation may then be used to assess and compare different priors and models our methods also make it possible to compare parametric and nonparametric model specifications in that genuine measures of

optimal weight choice for frequentist model average estimators there has been increasing interest recently in model averaging within the frequentist paradigm the main benefit of model averaging over model selection is that it incorporates rather than ignores the uncertainty inherent in the model selection process one of the most important yet challenging aspects of model averaging is how to optimally combine estimates from different models in this work we suggest a procedure of weight choice for frequentist model average estimators that exhibits optimality properties with respect to the estimator s mean squared error mse as a basis for demonstrating our idea we consider averaging over a sequence of linear regression models building on this base we develop a model weighting mechanism that involves minimizing the trace of an unbiased estimator of the model average estimator s mse we further obtain results that reflect the finite sample as well as asymptotic optimality of the proposed mechanism a monte carlo study based on simulated and real data evaluates

model averaging for robust assessment of qt prolongation by concentration response analysis assessing the qt prolongation potential of a drug is typically done based on pivotal safety studies called thorough qt studies model based estimation of the drug induced qt prolongation at the estimated mean maximum drug concentration could increase efficiency over the currently used intersection union test however robustness against model misspecification needs to be guaranteed in pivotal settings the objective of this work was to develop an efficient fully prespecified model based inference method for thorough qt studies which controls the type i error and provides satisfactory test power this is achieved by model averaging the proposed estimator of the concentration response relationship is a weighted average of a parametric linear and a nonparametric monotonic i splines estimator with weights based on mean integrated square error the desired properties of the method were confirmed in an extensive simulation study which demonstrated that the proposed method controlled the type i error adequately and that its power was higher than the power of the nonparametric method alone the method can be extended from thorough qt studies to the analysis of qt data from pooled phase i studies

efficient shrinkage in parametric models this paper introduces shrinkage for general parametric models we show how to shrink maximum likelihood estimators towards parameter subspaces de ned by general nonlinear restrictions we derive the asymptotic distribution and risk of the generalized shrinkage estimator using a local asymptotic framework we show that if the shrinkage dimension exceeds two the asymptotic risk of the shrinkage estimator is strictly less than that of the mle this reduction holds globally in the parameter space we show that the reduction in asymptotic risk is substantial even for moderately large values of the parameters the formula simplify in a very convenient way in the context of high dimensional models we derive a simple bound for the asymptotic risk we also provide a new large sample minimax e ciency bound we use the concept of local asymptotic minimax bounds a generalization of the conventional asymptotic minimax bounds the

optimal model assessment selection and combination central to statistical theory and application is statistical modeling which typically involves choosing a single model or combining a number of models of different sizes and from different sources whereas model selection seeks a single best modeling procedure model combination combines the strength of different modeling procedures in this article we look at several key issues and argue that model assessment is the key to model selection and combination most important we introduce a general technique of optimal model assessment based on data perturbation thus yielding optimal selection in particular model selection and combination from a frequentist perspective we advocate model combination over a selected subset of modeling procedures because it controls bias while reducing variability hence yielding better performance in terms of the accuracy of estimation and prediction to realize the potential of model combination we develop methodologies for determining the optimal tuning parameter such

minimax estimation of the mean of a normal distribution when the parameter space is restricted if x is a n theta 1 random variable let rho m be the minimax risk for estimation with quadratic loss subject to theta leq m then rho m 1 pi 2 m 2 o m 2 we exhibit estimates which are asymptotically minimax to this order as well as approximations to the least favorable prior distributions the approximate least favorable distributions correct to order m 2 have density m 1 cos 2 big frac pi 2m s big s leq m rather than the naively expected uniform density on lbrack m m rbrack we also show how our results extend to estimation of a vector mean and give some explicit solutions

can one estimate the conditional distribution of post model selection estimators we consider the problem of estimating the conditional distribution of a post model selection estimator where the conditioning is on the selected model the notion of a post model selection estimator here refers to the combined procedure resulting from first selecting a model e g by a model selection criterion such as aic or by a hypothesis testing procedure and then estimating the parameters in the selected model e g by least squares or maximum likelihood all based on the same data set we show that it is impossible to estimate this distribution with reasonable accuracy even asymptotically in particular we show that no estimator for this distribution can be uniformly consistent not even locally this follows as a corollary to local minimax lower bounds on the performance of estimators for this distribution similar impossibility results are also obtained for the conditional distribution of linear functions e g predictors of the post model selection estimator

markov switching model selection using kullback leibler divergence abstract in markov switching regression models we use kullback leibler kl divergence between the true and candidate models to select the number of states and variables simultaneously specifically we derive a new information criterion markov switching criterion msc which is an estimate of kl divergence msc imposes an appropriate penalty to mitigate the over retention of states in the markov chain and it performs well in monte carlo studies with single and multiple states small and large samples and low and high noise we illustrate the usefulness of msc via applications to the u s business cycle and to media advertising

an asymptotic theory for model selection inference in general semiparametric problems summary hjort claeskens 2003 developed an asymptotic theory for model selection model averaging and subsequent inference using likelihood methods in parametric models along with associated confidence statements in this article we consider a semiparametric version of this problem wherein the likelihood depends on parameters and an unknown function and model selection averaging is to be applied to the parametric parts of the model we show that all the results of hjort claeskens hold in the semiparametric context if the fisher information matrix for parametric models is replaced by the semiparametric information bound for semiparametric models and if maximum likelihood estimators for parametric models are replaced by semiparametric efficient profile estimators our methods of proof employ le cam s contiguity lemmas leading to transparent results the results also describe the behaviour of semiparametric model estimators when the parametric component is misspecified and also have implications for pointwise consistent model selectors somekeywords akaike information criterion bayes information criterion efficient semiparametric estimation frequentist model averaging model averaging model selection profile likelihood semiparametric model

the focused information criterion a variety of model selection criteria have been developed of general and specific types most of these aim at selecting a single model with good overall properties for example formulated via average prediction quality or shortest estimated overall distance to the true model the akaike the bayesian and the deviance information criteria along with many suitable variations are examples of such methods these methods are not concerned however with the actual use of the selected model which varies with context and application the present article takes the view that the model selector should instead focus on the parameter singled out for interest in particular a model that gives good precision for one estimand may be worse when used for inference for another estimand we develop a method that for a given focus parameter estimates the precision of any submodel based estimator the framework is that of large sample likelihood inference using an unbiased estimate of limiting risk we propose a foc

least squares model averaging by mallows criterion abstract this paper is in response to a recent paper by hansen 2007 who proposed an optimal model average estimator with weights selected by minimizing a mallows criterion the main contribution of hansen s paper is a demonstration that the mallows criterion is asymptotically equivalent to the squared error so the model average estimator that minimizes the mallows criterion also minimizes the squared error in large samples we are concerned with two assumptions that accompany hansen s approach the first is the assumption that the approximating models are strictly nested in a way that depends on the ordering of regressors often there is no clear basis for the ordering and the approach does not permit non nested models which are more realistic from a practical viewpoint second for the optimality result to hold the model weights are required to lie within a special discrete set in fact hansen noted both difficulties and called for extensions of the proof techniques we provide an alternative proof which shows that the result on the optimality of the mallows criterion in fact holds for continuous model weights and under a non nested set up that allows any linear combination of regressors in the approximating models that make up the model average estimator these results provide a stronger theoretical basis for the use of the mallows criterion in model averaging by strengthening existing findings