computing gaussian likelihoods and their derivatives for general linear mixed models algorithms are described for computing the gaussian likelihood or restricted likelihood corresponding to a general linear mixed model included are arbitrary covariance structures for both the random effects and errors formulas are also given for the first and second derivatives of the likelihoods thus enabling a newton raphson implementation the algorithms make heavy use of the cholesky decomposition the sweep operator and the w transformation also described are the modifications needed for variance profiling fisher scoring and mivque 0 as well as the computational order of the procedures

robust estimation of mean squared error of small area estimators abstract a well known model due to fay and herriot for estimating small area domain means i is considered given i s it is assumed that the survey estimators y i are independent with means i and known variances di i 1 t further the i s are assumed to be independent with means x i and unknown variance a where x i is a vector of benchmark variables related to i and is a vector of regression parameters an empirical best linear unbiased prediction eblup estimator or an empirical linear bayes estimator t i y of i is obtained it is shown that an estimator of mean squared error mse of t t y derived by prasad and rao under normality of i and y i given i is robust with respect to nonnormality of the i s specifically it is shown that the prasad rao estimator of mse is correct to terms of order o t 1 for large t assuming only certain moment conditions on the i s and normally distributed survey errors results of a simulation study on the accur

finite dimensional approximation in infinite dimensional mathematical programming we consider the problem of approximating an optimal solution to a separable doubly infinite mathematical program p with lower staircase structure by solutions to the programs p n obtained by truncating after the first n variables and n constraints of p viewing the surplus vector variable associated with the nth constraint as a state and assuming that all feasible states are eventually reachable from any feasible state we show that the efficient set of all solutions optimal to all possible feasible surplus states for p n converges to the set of optimal solutions to p a tie breaking algorithm which selects a nearest point efficient solution for p n is shown for convex programs to converge to an optimal solution to p a stopping rule is provided for discovering a value of n sufficiently large to guarantee any prespecified level of accuracy the theory is illustrated by an application to production planning

on parametric bootstrap methods for small area prediction summary the particularly wide range of applications of small area prediction e g in policy making decisions has meant that this topic has received substantial attention in recent years the problems of estimating mean squared predictive error of correcting that estimator for bias and of constructing prediction intervals have been addressed by various workers although existing methodology is still restricted to a narrow range of models to overcome this difficulty we develop new bootstrap based methods which are applicable in very general settings for constructing bias corrected estimators of mean squared error and for computing prediction regions unlike existing techniques which are based largely on taylor expansions our bias corrected mean squared error estimators do not require analytical calculation they also have the property that they are non negative our prediction intervals have a high degree of coverage accuracy o n 3 where n is the number of areas if double bootstrap methods are employed the techniques do not depend on the form of the small area estimator and are applicable to general two level small area models where the variables at either level can be discrete or continuous and in particular can be non normal most importantly the new methods are simple and easy to apply

the estimation of the mean squared error of small area estimators abstract small area estimation has received considerable attention in recent years because of a growing demand for reliable small area statistics the direct survey estimators based only on the data from a given small area or small domain are likely to yield unacceptably large standard errors because of small sample size in the domain therefore alternative estimators that borrow strength from other related small areas have been proposed in the literature to improve the efficiency these estimators use models either implicitly or explicitly that connect the small areas through supplementary e g census and administrative data for example simple synthetic estimators are based on implicit modeling in this article three small area models of battese harter and fuller 1988 dempster rubin and tsutakawa 1981 and fay and herriot 1979 are investigated these models are all special cases of a general mixed linear model involving fixed and random effects and a small area mean can be expr

a matrix inequality and its statistical application abstract let a 1 a s be nonnegative definite matrices we prove that there are constants c i 1 i s depending on a 1 a s such that for all positive numbers x 1 x s a i c i x i 2 i j 1 s x j a j 2 1 i s where i represents the identity matrix the result is applied to statistical inference about mixed linear models to show the existence of moments of complex estimators including the maximum likelihood and restricted maximum likelihood estimators of the variance components and the empirical best linear unbiased estimators predictors of the fixed random effects

variance component testing in generalised linear models with random effects summary there is considerable interest in testing for overdispersion correlation and heterogeneity across groups in biomedical studies in this paper we cast the problem in the framework of generalised linear models with random effects we propose a global score test for the null hypothesis that all the variance components are zero this test is a locally asymptotically most stringent test and is robust in the special sense that the test does not require specifying the joint distribution of the random effects we also propose individual score tests and their approximations for testing the variance components separately both tests can be easily implemented using existing statistical software we illustrate these tests with an application to the study of heterogeneity of mating success across males and females in an experiment on salamander matings and evaluate their performance through simulation

semiparametric inference in generalized mixed effects models summary the paper presents a study of the generalized partially linear model including random effects in its linear part we propose an estimator that combines likelihood approaches for mixed effects models with kernel methods following the methodology of hardie and co workers we introduce a test for the hypothesis of a parametric mixed effects model against the alternative of a semiparametric mixed effects model the critical values are estimated by using a bootstrap procedure the asymptotic theory for the methods is provided as are the results of a simulation study these verify the feasibility and the excellent behaviour of the methods for samples of even moderate size the usefulness of the methodology is illustrated with an application in which the objective is to estimate forest coverage in galicia spain

dispersion matrix in balanced mixed anova models the inverse of the dispersion matrix plays an important role in mixed model analysis searle and henderson j amer statist assoc 74 1979 465 showed that for a general balanced mixed anova model such an inverse can be expression in the same form as the dispersion matrix i e a linear combination of kronecker products however the coefficients of such a linear combination are given in a way that it is not easy to see what each coefficient is in this paper we give explicit expressions for these coefficients an application of the result is considered

outlier robust small area estimation recently proposed outlier robust small area estimators can be substantially biased when outliers are drawn from a distribution that has a different mean from that of the rest of the survey data this naturally leads down to the idea of an outlier robust bias correction for these estimators in this paper we develop this idea and also propose two different analytical mean squared error estimators for the ensuring bias corrected outlier robust estimators simulations based on realistic outlier contaminated data show that the proposed bias correction often leads to more efficient estimators furthermore the proposed mean squared error estimators appear to perform well with a variety of outlier robust smal area estimators

exploring the information in p values for the analysis and planning of multiple test experiments a new methodology is proposed for estimating the proportion of true null hypotheses in a large collection of tests each test concerns a single parameter delta whose value is specified by the null hypothesis we combine a parametric model for the conditional cumulative distribution function cdf of the p value given delta with a nonparametric spline model for the density g delta of delta under the alternative hypothesis the proportion of true null hypotheses and the coefficients in the spline model are estimated by penalized least squares subject to constraints that guarantee that the spline is a density the estimator is computed efficiently using quadratic programming our methodology produces an estimate g delta of the density of delta when the null is false and can address such questions as when the null is false is the parameter usually close to the null or far away this leads us to define a falsely interesting discovery rate fidr a generalization of the false discovery rate we contrast the fidr approach to efron s 2004 journal of the american statistical association 99 96 104 empirical null hypothesis technique we discuss the use of g in sample size calculations based on the expected discovery rate edr our recommended estimator of the proportion of true nulls has less bias compared to estimators based upon the marginal density of the p values at 1 in a simulation study we compare our estimators to the convex decreasing estimator of langaas lindqvist and ferkingstad 2005 journal of the royal statistical society series b 67 555 572 the most biased of our estimators is very similar in performance to the convex decreasing estimator as an illustration we analyze differences in gene expression between resistant and susceptible strains of barley

reduced complexity decoders of long reed solomon codes based on composite cyclotomic fourier transforms long reed solomon rs codes are desirable for digital communication and storage systems due to their improved error performance but the high computational complexity of their decoders is a key obstacle to their adoption in practice as discrete fourier transforms dfts can evaluate a polynomial at multiple points efficient dft algorithms are promising in reducing the computational complexities of syndrome based decoders for long rs codes in this correspondence we first propose partial composite cyclotomic fourier transforms ccfts and then devise syndrome based decoders for long rs codes over large finite fields based on partial ccfts the new decoders based on partial ccfts achieve a significant saving of computational complexities for long rs codes in comparison to previous results based on horner s rule our hardware implementation for a 2720 2550 shortened rs code over gf 212 achieves much higher throughputs and better area time complexity

mean squared error of estimation or prediction under a general linear model abstract the problem considered is that of predicting a linear combination of the fixed and random effects of a mixed effects linear model more generally the problem considered is that of predicting an unobservable random variable from a set of observable random variables the best linear unbiased predictor depends on parameters which generally are unknown various exact or approximate expressions are given for the mean squared error mse of the predictor obtained by replacing the unknown parameters with estimates several estimators of the mse are investigated

estimation in covariance components models abstract estimation techniques for linear covariance components models are developed and illustrated with special emphasis on explaining computational processes the estimation of fixed and random effects when the variances and covariances are known is presented in bayesian terms point estimates of the unknown variances and covariances are computed using the em algorithm for maximum likelihood estimation from incomplete data the techniques are illustrated with data on law schools field mice and professional football teams

resampling based empirical prediction an application to small area estimation best linear unbiased prediction is well known for its wide range of applications including small area estimation while the theory is well established for mixed linear models and under normality of the error and mixing distributions the literature is sparse for nonlinear mixed models under nonnormality of the error distribution or of the mixing distributions we develop a resampling based unified approach for predicting mixed effects under a generalized mixed model set up second order accurate nonnegative estimators of mean squared prediction errors are also developed given the parametric model the proposed methodology automatically produces estimators of the small area parameters and their mean squared prediction errors without requiring explicit analytical expressions for the mean squared prediction errors

small area estimation when auxiliary information is measured with error summary small area estimation methods typically combine direct estimates from a survey with pre dictions from a model in order to obtain estimates of population quantities with reduced mean squared error when the auxiliary information used in the model is measured with error using a small area estimator such as the fay herriot estimator while ignoring measurement error may be worse than simply using the direct estimator we propose a new small area estimator that accounts for sampling variability in the auxiliary information and derive its properties in particular show ing that it is approximately unbiased the estimator is applied to predict quantities measured in the u s national health and nutrition examination survey with auxiliary information from the u s national health interview survey

robust empirical bayes estimation of means from stratified samples abstract this article considers simultaneous estimation of means from several strata where each stratum contains a finite number of elements for example quite often estimates of annual incomes unemployment rates and so forth must be made simultaneously for many areas our results find application in small area estimation where very few samples are available from an individual area and an estimate of a certain area mean or simultaneous estimates of several area means can be improved by incorporating information from similar neighboring areas ghosh and meeden 1986 considered empirical bayes estimation of the finite population mean assuming a normal superpopulation model their estimators generalize automatically when one estimates a vector of stratum means this article relaxes the normality assumption and assumes instead that the posterior expectation of any stratum mean is a linear function of the sample observations this is the so called posterior linearity property as described in ericson 1

empirical bayes estimation of finite population means from complex surveys abstract estimation of finite population means is considered when samples are collected using a stratified sampling design finite populations for different strata are assumed to be realizations from different superpopulations the true means of the observations lie on a regression surface with random intercepts for different strata the true sampling variances are also different and random for different strata the strata are connected through two common prior distributions one for the intercepts and another for the sampling variances for all the strata the model is appropriate in two important survey situations first it can be applied to repeated surveys where the physical characteristics of the sampling units change slowly over time second the model is appropriate in small area estimation problems where a very few samples are available for any particular area empirical bayes estimators of the finite population means are shown to be asymptotically optimal in the sense of robbins the proposed emp

small area estimation with uncertain random effects random effects models play an important role in model based small area estimation random effects account for any lack of fit of a regression model for the population means of small areas on a set of explanatory variables in a recent article datta hall and mandal showed that if the random effects can be dispensed with via a suitable test then the model parameters and the small area means may be estimated with substantially higher accuracy the work of datta hall and mandal is most useful when the number of small areas m is moderately large for large m the null hypothesis of no random effects will likely be rejected rejection of the null hypothesis is usually caused by a few large residuals signifying a departure of the direct estimator from the synthetic regression estimator as a flexible alternative to the fay herriot random effects model and the approach in datta hall and mandal in this article we consider a mixture model for random effects it is reasonably expected that small areas wit

goodness of fit tests for mixed model diagnostics a simple goodness of fit test is proposed for checking distributional assumptions involved in a mixed linear model an estimated critical value of the test statistic is derived and is shown to be asymptotically correct under mild conditions as a special case the test may be applied to linear regression models to formally check distribution of the errors finite sample performance of the proposed test is examined and compared with that of a previously proposed test by simulations

a duality theory for infinite horizon optimization of concave input ouput processes a general concave horizon optimization model is analyzed with the help of a special convexity concept which combines both the usual convexity and the dynamic structure the axiomatic setup leads to a perfect symmetry between the primal and dual problems after introducing a particular dynamic feasibility hypothesis the following results are presented i boundedness of trajectories as a necessary condition for optimally ii the existence of primal and dual optimal trajectories iii approximation by finite horizon models and iv necessary and sufficient conditions for optimality

confidence intervals for the binomial parameter some new considerations several methods have been proposed to construct confidence intervals for the binomial parameter some recent papers introduced the mean coverage criterion to evaluate the performance of confidence intervals and suggested that exact methods because of their conservatism are less useful than asymptotic ones in these studies however exact intervals were always represented by the clopper pearson interval c p now we focus on sterne s interval which is also exact and known to be better than the c p in the two sided case introducing a computer intensive level adjustment procedure which allows constructing intervals that are exact in terms of mean coverage we demonstrate that sterne s interval performs better than the best asymptotic intervals even in the mean coverage context level adjustment improves the c p as well which with an appropriate level adjustment becomes equivalent to the mid p interval finally we show that the asymptotic behaviour of the mid p method is far poorer than is generally expected

stationary dual prices and depreciation this paper models a machine replacement and capacity expansion problem as an infinite horizon linear program we establish a strong duality result and show that stationary dual prices are optimal regardless of initial conditions these prices measure the economic value of owning vintage machinery and thus define depreciation schedules we present necessary and sufficient conditions for straight line depreciation

fast transform for decoding both errors and erasures of reed solomon codes over gf 2 sup m for 8 spl les m spl les 10 in this letter it is shown that a fast prime factor discrete fourier transform dft algorithm can be modified to compute fourier like transforms of long sequences of 2 sup m 1 points over gf 2 sup m where 8 spl les m spl les 10 using these transforms together with the berlekamp massey algorithm the complexity of the transform domain decoder for correcting both errors and erasures of the reed solomon codes of block length 2 sup m 1 over gf 2 sup m for 8 spl les m spl les 10 is reduced substantially from the previous time domain decoder a computer simulation verifies these new results

bayesian versus frequentist measures of error in small area estimation summary this paper compares analytically and empirically the frequentist and bayesian measures of error in small area estimation the model postulated is the nested error regression model which allows for random small area effects to represent the joint effect of small area characteristics that are not accounted for by the fixed regressor variables when the variance components are known then under a uniform prior for the regression coefficients and normality of the error terms the frequentist and the bayesian approaches yield the same predictors and prediction mean squared errors mses defined accordingly when the variance components are unknown it is common practice to replace the unknown variances by sample estimates in the expressions for the optimal predictors so that the resulting empirical predictors remain the same under the two approaches the use of this paradigm requires however modifications to the expressions of the prediction mse to account for the extra variability induced by the need to estimate the variance components the main focus of this paper is to review and compare the modifications to prediction mses proposed in the literature under the two approaches with special emphasis on the orders of the bias of the resulting approximations to the true mses some new approximations based on monte carlo simulation are also proposed and compared with the existing methods the advantage of these approximations is their simplicity and generality finite sample frequentist properties of the various methods are explored by a simulation study the main conclusions of this study are that the use of second order bias corrections generally yields better results in terms of the bias of the mse approximations and the coverage properties of confidence intervals for the small area means the bayesian methods are found to have good frequentist properties but they can be inferior to the frequentist methods the second order approximations under both approaches have however larger variances than the corresponding first order approximations which in most cases result in higher mses of the mse approximations

semiparametric regression of multidimensional genetic pathway data least squares kernel machines and linear mixed models we consider a semiparametric regression model that relates a normal outcome to covariates and a genetic pathway where the covariate effects are modeled parametrically and the pathway effect of multiple gene expressions is modeled parametrically or nonparametrically using least squares kernel machines lskms this unified framework allows a flexible function for the joint effect of multiple genes within a pathway by specifying a kernel function and allows for the possibility that each gene expression effect might be nonlinear and the genes within the same pathway are likely to interact with each other in a complicated way this semiparametric model also makes it possible to test for the overall genetic pathway effect we show that the lskm semiparametric regression can be formulated using a linear mixed model estimation and inference hence can proceed within the linear mixed model framework using standard mixed model software both the regression coefficients of the covariate effects and the lskm estimator of the genetic pathway effect can be obtained using the best linear unbiased predictor in the corresponding linear mixed model formulation the smoothing parameter and the kernel parameter can be estimated as variance components using restricted maximum likelihood a score test is developed to test for the genetic pathway effect model variable selection within the lskm framework is discussed the methods are illustrated using a prostate cancer data set and evaluated using simulations

small area inference for binary variables in the national health interview survey abstract the national health interview survey is designed to produce precise estimates of finite population parameters for the entire united states but not for small geographical areas or subpopulations our investigation concerns estimates of proportions such as the probability of at least one visit to a doctor within the past 12 months to include all sources of variation in the model we carry out a bayesian hierarchical analysis for the desired finite population quantities first for each cluster county a separate logistic regression relates the individual s probability of a doctor visit to his or her characteristics second a multivariate linear regression links cluster regression parameters to covariates measured at the cluster level we describe the numerical methods needed to obtain the desired posterior moments then we compare estimates produced using the exact numerical method with approximations finally we compare the hierarchical bayes estimates to empirical bayes estimates and to stand

small area estimation based on natural exponential family quadratic variance function models and survey weights summary we propose pseudo empirical best linear unbiased estimators of small area means based on natural exponential family quadratic variance function models when the basic data consist of survey weighted estimators of these means area specific covariates and certain summary measures involving the weights we also provide explicit approximate mean squared errors of these estimators in the spirit of prasad rao 1990 and these estimators can be readily evaluated a simulation study is undertaken to evaluate the performance of the proposed inferential procedure we estimate also the proportion of poor children in the 5 17 years age group for the different counties in one of the states in the united states

approximating extreme points of infinite dimensional convex sets the property that an optimal solution to the problem of minimizing a continuous concave function over a compact convex set in rn is attained at an extreme point is generalized by the bauer minimum principle to the infinite dimensional context the problem of approximating and characterizing infinite dimensional extreme points thus becomes an important problem consider now an infinite dimensional compact convex set in the nonnegative orthant of the product space rinfinity we show that the sets of extreme points en of its corresponding finite dimensional projections onto rn converge in the product topology to the closure of the set of extreme points e of the infinite dimensional set as an application we extend the concept of total unimodularity to infinite systems of linear equalities in nonnegative variables where we show when extreme points inherit integrality from approximating finite systems an application to infinite horizon production planning is considered

parametric empirical bayes inference theory and applications abstract this article reviews the state of multiparameter shrinkage estimators with emphasis on the empirical bayes viewpoint particularly in the case of parametric prior distributions some successful applications of major importance are considered recent results concerning estimates of error and confidence intervals are described and illustrated with data

asymptotic properties of maximum likelihood estimators and likelihood ratio tests under nonstandard conditions abstract large sample properties of the likelihood function when the true parameter value may be on the boundary of the parameter space are described specifically the asymptotic distribution of maximum likelihood estimators and likelihood ratio statistics are derived these results generalize the work of moran 1971 chant 1974 and chernoff 1954 some of chant s results are shown to be incorrect the approach used in deriving these results follows from comments made by moran and chant the problem is shown to be asymptotically equivalent to the problem of estimating the restricted mean of a multivariate gaussian distribution from a sample of size 1 in this representation the gaussian random variable corresponds to the limit of the normalized score statistic and the estimate of the mean corresponds to the limit of the normalized maximum likelihood estimator thus the limiting distribution of the maximum likelihood estimator is the same as the distribution of the projection of the gaussian random v

small area estimation under informative probability sampling of areas and within the selected areas in this article we show how to predict small area means and obtain valid mean squared error estimators and confidence intervals when the areas represented in the sample are sampled with unequal probabilities possibly related to the true unknown area means and the sampling of units within the selected areas is with probabilities possibly related to the outcome values ignoring the effects of the sampling process on the distribution of the observed outcomes in such cases may bias the inference very severely classical design based inference that uses the randomization distribution of probability weighted estimators cannot be applied for predicting the means of nonsampled areas we propose simple test statistics for testing the informativeness of the selection of areas and sampling of units within the selected areas we illustrate the proposed procedures by a simulation study and a real application of estimating mean body mass index in u s counties using data from the third national health and nutrition

composite cyclotomic fourier transforms with reduced complexities discrete fourier transforms dfts over finite fields have widespread applications in digital communication and storage systems hence reducing the computational complexities of dfts is of great significance recently proposed cyclotomic fast fourier transforms cffts are promising due to their low multiplicative complexities unfortunately there are two issues with cffts 1 they rely on efficient short cyclic convolution algorithms which have not been sufficiently investigated in the literature and 2 they have very high additive complexities when directly implemented to address both issues we make three main contributions in this paper first for any odd prime p we reformulate a p point cyclic convolution as the product of a p 1 p 1 toeplitz matrix and a vector which has well known efficient algorithms leading to efficient bilinear algorithms for p point cyclic convolutions second to address the high additive complexities of cffts we propose composite cyclotomic fourier transforms ccfts in comparison to previously proposed fast fourier transforms our ccfts achieve lower overall complexities for moderate to long lengths and the improvement significantly increases as the length grows third our efficient algorithms for p point cyclic convolutions and ccfts allow us to obtain longer dfts over larger fields e g the 2047 point dft over gf 211 and 4095 point dft over gf 212 which are first efficient dfts of such lengths to the best of our knowledge finally our ccfts are also advantageous for hardware implementations due to their modular structure

empirical bayes estimation in finite population sampling abstract empirical bayes methods are becoming increasingly popular in statistics robbins 1955 introduced the method in the context of nonparametric estimation of a completely unspecified prior distribution subsequently the method has been explored very successfully in a series of articles by efron and morris 1973 1975 1977 in a parametric framework in the efron morris setup a family of parametric distributions is used as possible priors but only when one or more of the parameters of the family of prior distributions is estimated from the data morris 1983 listed a number of areas where empirical bayes methods are used one of the main features of empirical bayes analysis is to borrow strength from the ensemble that is use information from similar sources in constructing estimators and predictors in addition to the most directly available source of information there are some situations in finite population sampling where such methods might be suitable for instance in many repetitive surve

mean squared error estimation in transformed fay herriot models summary the problem of accurately estimating the mean squared error of small area estimators within a fay herriot normal error model is studied theoretically in the common setting where the model is fitted to a logarithmically transformed response variable for bias corrected empirical best linear unbiased predictor small area point estimators mean squared error formulae and estimators are provided with biases of smaller order than the reciprocal of the number of small areas the performance of these mean squared error estimators is illustrated by a simulation study and a real data example relating to the county level estimation of child poverty rates in the us census bureau s on going small area income and poverty estimation project

on the design consistency property of hierarchical bayes estimators in finite population sampling we obtain a limit of a hierarchical bayes estimator of a finite population mean when the sample size is large the limit is in the sense of ordinary calculus where the sample observations are treated as fixed quantities our result suggests a simple way to correct the hierarchical bayes estimator to achieve design consistency a well known property in the traditional randomization approach to finite population sampling we also suggest three different measures of uncertainty of our proposed estimator

standard errors of prediction in generalized linear mixed models abstract the unconditional mean squared error of prediction umsep is widely used as a measure of prediction variance for inferences concerning linear combinations of fixed and random effects in the classical normal theory mixed model but the umsep is inappropriate for generalized linear mixed models where the conditional variance of the random effects depends on the data when the random effects describe variation between independent small domains and domain specific prediction is of interest we propose a conditional mean squared error of prediction cmsep as a general measure of prediction variance the cmsep is shown to be the sum of the conditional variance and a positive correction that accounts for the sampling variability of parameter estimates we derive a second order correct estimate of the cmsep that consists of three components a a plug in estimate of the conditional variance b a plug in estimate of a taylor series approximation to the correction term and c a bootstrap estimate of

shadow prices in infinite dimensional linear programming we consider the class of linear programs that can be formulated with infinitely many variables and constraints but where each constraint has only finitely many variables this class includes virtually all infinite horizon planning problems modeled as infinite stage linear programs examples include infinite horizon production planning under time varying demands and equipment replacement under technological change we provide under a regularity condition conditions that are both necessary and sufficient for strong duality to hold moreover we show that under these conditions the lagrangean function corresponding to any pair of primal and dual optimal solutions forms a linear support to the optimal value function thus extending the shadow price interpretation of an optimal dual solution to the infinite dimensional case we illustrate the theory through an application to production planning under time varying demands and costs where strong duality is established

linear score tests for variance components in linear mixed models and applications to genetic association studies summary following the rapid development of genome scale genotyping technologies genetic association mapping has become a popular tool to detect genomic regions responsible for certain disease phenotypes especially in early phase pharmacogenomic studies with limited sample size in response to such applications a good association test needs to be 1 applicable to a wide range of possible genetic models including but not limited to the presence of gene by environment or gene by gene interactions and non linearity of a group of marker effects 2 accurate in small samples fast to compute on the genomic scale and amenable to large scale multiple testing corrections and 3 reasonably powerful to locate causal genomic regions the kernel machine method represented in linear mixed models provides a viable solution by transforming the problem into testing the nullity of variance components in this study we consider score based tests by choosing a statistic linear in the score function when the model under the null hypothesis has only one error variance parameter our test is exact in finite samples when the null model has more than one variance parameter we develop a new moment based approximation that performs well in simulations through simulations and analysis of real data we demonstrate that the new test possesses most of the aforementioned characteristics especially when compared to existing quadratic score tests or restricted likelihood ratio tests

sumca simple unified monte carlo assisted approach to second order unbiased mean squared prediction error estimation we propose a simple unified monte carlo assisted approach called sumca to second order unbiased estimation of the mean squared prediction error mspe of a small area predictor the mspe estimator proposed is easy to derive has a simple expression and applies to a broad range of predictors that include the traditional empirical best linear unbiased predictor empirical best predictor and post model selection empirical best linear unbiased predictor and empirical best predictor as special cases furthermore the leading term of the mspe estimator proposed is guaranteed positive the lower order term corresponds to a bias correction which can be evaluated via a monte carlo method the computational burden for the monte carlo evaluation is much less compared with other monte carlo based methods that have been used for producing second order unbiased mspe estimators such as the double bootstrap and monte carlo jackknife the sumca estimator also has a nice stability feature theoretical and empirical results demonstrate properties and advantages of the sumca estimator

mean squared error of empirical predictor the term empirical predictor refers to a two stage predictor of a linear combination of fixed and random effects in the first stage a predictor is obtained but it involves unknown parameters thus in the second stage the unknown parameters are replaced by their estimators in this paper we consider mean squared errors mse of empirical predictors under a general setup where ml or reml estimators are used for the second stage we obtain second order approximation to the mse as well as an estimator of the mse correct to the same order the general results are applied to mixed linear models to obtain a second order approximation to the mse of the empirical best linear unbiased predictor eblup of a linear mixed effect and an estimator of the mse of eblup whose bias is correct to second order the general mixed linear model includes the mixed anova model and the longitudinal model as special cases

small area estimates for cross classifications we develop a class of log linear structural models that is suited to estimation of small area cross classified counts based on survey data this allows us to account for various association structures within the data and includes as a special case the restricted log linear model underlying structure preserving estimation the effect of survey design can be incorporated into estimation through the specification of an unbiased direct estimator and its associated covariance structure we illustrate our approach by applying it to estimation of small area labour force characteristics in norway

delta method and bootstrap in linear mixed models to estimate a proportion when no event is observed application to intralesional resection in bone tumor surgery resecting bone tumors requires good cutting accuracy to reduce the occurrence of local recurrence this issue is considerably reduced with a navigated technology the estimation of extreme proportions is challenging especially with small or moderate sample sizes when no success is observed the commonly used binomial proportion confidence interval is not suitable while the rule of three provides a simple solution unfortunately these approaches are unable to differentiate between different unobserved events different delta methods and bootstrap procedures are compared in univariate and linear mixed models with simulations and real data by assuming the normality the delta method on the z score and parametric bootstrap provide similar results but the delta method requires the estimation of the covariance matrix of the estimates in mixed models the observed fisher information matrix with unbounded variance components should be preferred the parametric bootstrap easier to apply outperforms the delta method for larger sample sizes but it may be time costly copyright 2016 john wiley sons ltd

circumventing the slater conundrum in countably infinite linear programs abstract duality results on countably infinite linear programs are scarce subspaces that admit an interior point which is a sufficient condition for a zero duality gap yield a dual where the constraints cannot be expressed using the ordinary transpose of the primal constraint matrix subspaces that permit a dual with this transpose do not admit an interior point this difficulty has stumped researchers for a few decades it has recently been called the slater conundrum we find a way around this hurdle we propose a pair of primal dual spaces with three properties the series in the primal and dual objective functions converge the series defined by the rows and columns of the primal constraint matrix converge and the order of sums in a particular iterated series of a double sequence defined by the primal constraint matrix can be interchanged so that the dual is defined by the ordinary transpose weak duality and complementary slackness are then immediate instead of using interior point conditions to establish a zero duality gap we call upon the planning horizon method when the series in the primal and dual constraints are continuous we prove that strong duality holds if a sequence of optimal solutions to finite dimensional truncations of the primal and dual cilps has an accumulation point we show by counterexample that the requirement that such an accumulation point exist cannot be relaxed our results are illustrated using several examples and are applied to countable state markov decision processes and to a problem in robust optimization

two sided confidence intervals for the single proportion comparison of seven methods simple interval estimate methods for proportions exhibit poor coverage and can produce evidently inappropriate intervals criteria appropriate to the evaluation of various proposed methods include closeness of the achieved coverage probability to its nominal value whether intervals are located too close to or too distant from the middle of the scale expected interval width avoidance of aberrations such as limits outside 0 1 or zero width intervals and ease of use whether by tables software or formulae seven methods for the single proportion are evaluated on 96 000 parameter space points intervals based on tail areas and the simpler score methods are recommended for use in each case methods are available that aim to align either the minimum or the mean coverage with the nominal 1 alpha

a score test against one sided alternatives abstract a score type statistic t s is introduced for testing h 0 against k 0 and more general one sided hypotheses when nuisance parameters may be present is a vector parameter the main advantages of t s are that it requires estimation of the model only under the null hypothesis and that it is asymptotically equivalent to the likelihood ratio statistic these are precisely the reasons for the popularity of the score tests for testing against two sided alternatives in this sense t s preserves the main attractive features of the classical two sided score test the theoretical results are presented in a general framework where the likelihood based score function is replaced by an estimating function so that the test is applicable even if the exact population distribution is unknown computation of t s is simplified by the fact that it can be computed easily once the corresponding two sided statistic has been computed the relevance and simplicity of t s are illustrated by discussing

comparisons of alternative predictors under the balanced one way random model abstract prediction of an arbitrary linear combination of the random effects of a balanced one way random model is investigated alternative two stage predictors are compared on the basis of their conditional on the random effects and unconditional bias and mean squared errors when the true value of the ratio of expected mean squares is known there exists a best linear unbiased predictor blup when the true value is unknown a two stage predictor obtained from the blup by replacing the true value with an estimated value can be used when the ratio of expected mean squares is estimated by maximum likelihood bayesian methods or various related methods a two stage predictor is obtained whose properties compare favorably with for example those of the least squares predictor and the positive part james stein predictor

the mean squared error of small area predictors constructed with estimated area variances in the small area estimation literature the sampling error variances are customarily assumed to be known or to depend on a finite number of parameters we consider the empirical best linear unbiased predictor eblup obtained by using the individual directly estimated variance for each small area an approximation for the mean squared error mse of the eblup that recognizes the impact on the predictors of estimation of the variance components is derived simulation studies show that the theoretical expressions are good approximations for the mse of the predictors unless the between area variance component is very small relative to the within area variance an improved estimator of the mse is developed that has smaller overestimation than the original estimator when the between area variance component is small the robustness of the mse estimator is studied and predictors for nonnormal sampling errors are proposed an example from the national resources inventory that motivated the development of the t

duality in infinite dimensional linear programming we consider the class of linear programs with infinitely many variables and constraints having the property that every constraint contains at most finitely many variables while every variable appears in at most finitely many constraints examples include production planning and equipment replacement over an infinite horizon we form the natural dual linear programming problem and prove strong duality under a transversality condition that dual prices are asymptotically zero that is we show under this transversality condition that optimal solutions are attained in both primal and dual problems and their optimal values are equal the transversality condition and hence strong duality is established for an infinite horizon production planning problem

variance components testing in the longitudinal mixed effects model this article discusses the asymptotic behavior of likelihood ratio tests for nonzero variance components in the longitudinal mixed effects linear model described by laird and ware 1982 biometrics 38 963 974 our discussion of the large sample behavior of likelihood ratio tests for nonzero variance components is based on the results for nonstandard testing situations by self and liang 1987 journal of the american statistical association 82 605 610

response surface models with random block effects in many experimental situations a response surface design is divided into several blocks to control an extraneous source of variation the traditional approach in most response surface applications is to treat the block effect as fixed in the assumed model there are however situations in which it is more appropriate to consider the block effect as random this article is concerned with inference about a response surface model in the presence of a random block effect since this model also contains fixed polynomial effects it is considered to be a mixed effects model the main emphasis of the proposed analysis is on estimation and testing of the fixed effects a two stage mixed model procedure is developed for this purpose the variance components due to the random block effect and the experimental error are first estimated and then used to obtain the generalized least squares estimator of the fixed effects this procedure produces the so called yates combined intra and inter block estimator by cont

confidence prediction and tolerance in linear mixed models the literature about prediction interval pi and tolerance interval ti in linear mixed models is usually developed for specific designs which is a main limitation to their use this paper proposes to reformulate the two sided pi to be generalizable under a wide variety of designs one random factor nested and crossed designs for multiple random factors and balanced or unbalanced designs this new methodology is based on the hessian matrix namely the inverse of observed fisher information matrix and is built with a cell mean model the degrees of freedom for the total variance are calculated with the generalized satterthwaite method and compared to the kenward roger s degrees of freedom for fixed effects construction of two sided tis are also detailed with one random factor and two nested and two crossed random variables an extensive simulation study is carried out to compare the widths and coverage probabilities of confidence intervals ci pis and tis to their nominal levels it shows excellent coverage whatever the design and the sample size are finally these cis pis and tis are applied to two real data sets one from orthopedic surgery study intralesional resection risk and the other from assay validation study during vaccine development

some bayesian and non bayesian procedures for the analysis of comparative experiments and for small area estimation computational aspects frequentist properties and relationships abstract the estimation of a treatment contrast from experimental data and the estimation of a small area mean are special cases of the prediction of the realization of a linear combination of fixed and random effects in a possibly unbalanced two part mixed linear model in this article a bayesian approach to point and interval prediction is presented and its computational requirements are examined differences between the bayesian approach and the traditional classical approach are discussed in general terms and in addition in terms of two examples taken from the literature 1 the comparison of drug formulations in a bioavailability trial westlake and 2 the estimation of corn crop areas using satellite data battese harter and fuller some deficiencies in the classical approach are pointed out and the bayesian approach is considered from a frequentist perspective it is shown via a monte carlo study that for certain noninformative choices of the prior distribution the frequentist prop

non parametric small area estimation using penalized spline regression the paper proposes a small area estimation approach that combines small area random effects with a smooth non parametrically specified trend by using penalized splines as the representation for the non parametric trend it is possible to express the non parametric small area estimation problem as a mixed effect model regression the resulting model is readily fitted by using existing model fitting approaches such as restricted maximum likelihood we present theoretical results on the prediction mean squared error of the estimator proposed and on likelihood ratio tests for random effects and we propose a simple non parametric bootstrap approach for model inference and estimation of the small area prediction mean squared error the applicability of the method is demonstrated on a survey of lakes in north eastern usa

exact tests for variance components ofversten 1993 presented methods for obtaining exact f tests of variance components in three unbalanced mixed linear models he developed methods for models with one random factor models with nested classifications and models with interaction between two random factors in section 2 we review and extend wald s test which is a generalization of one of ofversten s methods ofversten s other method is a generalization of khuri 1987 1990 and khuri and littell 1987 it does not apply to the one random factor model in section 3 we present a unified treatment of ofversten s second method this treats all of ofversten s examples and gives a formal justification to ofversten s claim that the method extends quite generally section 3 also gives a formal condition for when the method does not require resampling from the residuals the unified treatment is conceptually simpler than ofversten s treatment in that it uses a single orthonormal basis rather than a series of matrix decompositions that change slightly from example to example there do not seem to be any particular computational advantages to the unified treatment the unified treatment also appears to generalize ofversten s method in that it applies in some cases where wald s test also applies section 4 compares the two tests when both apply all of the arguments are given in terms of vector space ideas cf christensen 1987

testing against a high dimensional alternative summary as the dimensionality of the alternative hypothesis increases the power of classical tests tends to diminish quite rapidly this is especially true for high dimensional data in which there are more parameters than observations we discuss a score test on a hyperparameter in an empirical bayesian model as an alternative to classical tests it gives a general test statistic which can be used to test a point null hypothesis against a high dimensional alternative even when the number of parameters exceeds the number of samples this test will be shown to have optimal power on average in a neighbourhood of the null hypothesis which makes it a proper generalization of the locally most powerful test to multiple dimensions to illustrate this new locally most powerful test we investigate the case of testing the global null hypothesis in a linear regression model in more detail the score test is shown to have significantly more power than the f test whenever under the alternative the large variance principal components of the design matrix explain substantially more of the variance of the outcome than do the small variance principal components the score test is also useful for detecting sparse alternatives in truly high dimensional data where its power is comparable with the test based on the maximum absolute t statistic

exact tests for variance components in unbalanced mixed linear models a method for deriving exact tests for variance components in some unbalanced mixed linear models is presented the derivation is based on a new kind of preliminary orthogonal transformation and a subsequent resampling procedure the resulting tests are based on mutually independent sums of squares which under the null hypothesis are distributed as scalar multiples of chi square variates for balanced data the tests are identical to the traditional f tests

cyclotomic ffts with reduced additive complexities based on a novel common subexpression elimination algorithm in this paper we first propose a novel common subexpression elimination cse algorithm for matrix vector multiplications over characteristic 2 fields as opposed to previously proposed cse algorithms which usually focus on complexity savings due to recurrences of subexpressions our cse algorithm achieves two types of complexity reductions differential savings and recurrence savings by taking advantage of the cancellation property of characteristic 2 fields using our cse algorithm we reduce the additive complexities of cyclotomic fast fourier transforms cffts using a weighted sum of the numbers of multiplications and additions as a metric our cffts achieve smaller total complexities than previously proposed cffts and other ffts requiring both fewer multiplications and fewer additions in many cases

benchmarked empirical bayes methods in multiplicative area level models with risk evaluation the paper develops empirical bayes and benchmarked empirical bayes estimators of positive small area means under multiplicative models a simple example will be estimation of per capita income for small areas it is now well understood that small area estimation needs explicit or at least implicit use of models one potential difficulty with model based estimators is that the overall estimator for a larger geographical area based on weighted sum of the model based estimators is not necessarily identical to the corresponding direct estimator such as the overall sample mean one way to fix such a problem is the so called benchmarking approach which modifies the model based estimators to match the aggregate direct estimator benchmarked hierarchical and empirical bayes estimators have proved to be particularly useful in this regard however while estimating positive small area parameters the conventional squared error or weighted squared loss subject to the usual benchmark constraint does not necessarily produce positive estimators hence it is necessary to seek other meaningful losses to alleviate this problem in this paper we consider the transformed fay herriot model as a multiplicative model for estimating positive small area means and suggest a weighted kullbackleibler divergence as a loss function we have found out that the resulting bayes estimator is the posterior mean and that the corresponding benchmarked bayes and empirical bayes estimators retain the positivity constraint the prediction errors of the suggested empirical bayes estimators are investigated asymptotically and their second order unbiased estimators are provided in addition bootstrapped estimators of these prediction errors are also provided the

testing against a high dimensional alternative in the generalized linear model asymptotic type i error control summary testing a low dimensional null hypothesis against a high dimensional alternative in a generalized linear model may lead to a test statistic that is a quadratic form in the residuals under the null model using asymptotic arguments we show that the distribution of such a test statistic can be approximated by a ratio of quadratic forms in normal variables for which algorithms are readily available for generalized linear models the asymptotic distribution shows good control of type i error for moderate to small samples even when the number of covariates in the model far exceeds the sample size

influence functions and robust bayes and empirical bayes small area estimation we introduce new robust small area estimation procedures based on area level models we first find influence functions corresponding to each individual area level observation by measuring the divergence between the posterior density functions of regression coefficients with and without that observation next based on these influence functions properly standardized we propose some new robust bayes and empirical bayes small area estimators the mean squared errors and estimated mean squared errors of these estimators are also found a small simulation study compares the performance of the robust and the regular empirical bayes estimators when the model variance is larger than the sample variance the proposed robust empirical bayes estimators are superior

approximations for standard errors of estimators of fixed and random effects in mixed linear models abstract best linear unbiased estimators of the fixed and random effects of mixed linear models are available when the true values of the variance ratios are known if the true values are replaced by estimated values the mean squared errors of the estimators of the fixed and random effects increase in size the magnitude of this increase is investigated and a general approximation is proposed the performance of this approximation is investigated in the context of a the estimation of the effects of the balanced one way random model and b the estimation of treatment contrasts for balanced incomplete block designs

order restricted score tests for homogeneity in generalised linear and nonlinear mixed models lin 1997 building on work by solomon cox 1992 and lin breslow 1996 introduced a score test for homogeneity in the generalised linear mixed model in this paper we propose an improvement to lin s test that exploits the fact that covariance parameters associated with random effects in the model are constrained so as to form a positive semidefinite covariance matrix therefore an omnibus score test will have power against alternatives that are known never to occur our improvement works by concentrating the score test into directions given by the positive semidefiniteness constraint we apply the same sort of modification to the bias corrected version of her score test in the generalised linear mixed model context and in a nonlinear mixed model context to a score test analogous to hers we show how to obtain the limiting distribution of our proposed test statistic we also argue via simulations and local alternative power calculations that the restricted test has better power than the unrestricted one

generalized linear models for small area estimation abstract bayesian methods have been used quite extensively in recent years for solving small area estimation problems particularly effective in this regard has been the hierarchical or empirical bayes approach which is especially suitable for a systematic connection of local areas through models however the development to date has mainly concentrated on continuous valued variates often the survey data are discrete or categorical so that hierarchical or empirical bayes techniques designed for continuous variates are inappropriate this article considers hierarchical bayes generalized linear models for a unified analysis of both discrete and continuous data a general theorem is provided that ensures the propriety of posteriors under diffuse priors this result is then extended to the case of spatial generalized linear models the hierarchical bayes procedure is implemented via markov chain monte carlo integration techniques two examples one featuring spatial correlation structure are given to illu

an r2 statistic for fixed effects in the linear mixed model statisticians most often use the linear mixed model to analyze gaussian longitudinal data the value and familiarity of the r2 statistic in the linear univariate model naturally creates great interest in extending it to the linear mixed model we define and describe how to compute a model r2 statistic for the linear mixed model by using only a single model the proposed r2 statistic measures multivariate association between the repeated outcomes and the fixed effects in the linear mixed model the r2 statistic arises as a 1 1 function of an appropriate f statistic for testing all fixed effects except typically the intercept in a full model the statistic compares the full model with a null model with all fixed effects deleted except typically the intercept while retaining exactly the same covariance structure furthermore the r2 statistic leads immediately to a natural definition of a partial r2 statistic a mixed model in which ethnicity gives a very small p value as a longitudinal predictor of blood pressure bp compellingly illustrates the value of the statistic in sharp contrast to the extreme p value a very small r2 a measure of statistical and scientific importance indicates that ethnicity has an almost negligible association with the repeated bp outcomes for the study copyright 2008 john wiley sons ltd

longitudinal data analysis using generalized linear models summary this paper proposes an extension of generalized linear models to the analysis of longitudinal data we introduce a class of estimating equations that give consistent estimates of the regression parameters and of their variance under mild assumptions about the time dependence the estimating equations are derived without specifying the joint distribution of a subject s observations yet they reduce to the score equations for multivariate gaussian outcomes asymptotic theory is presented for the general class of estimators specific cases in which we assume independence m dependence and exchangeable correlation structures from each subject are discussed efficiency of the proposed estimators in two simple situations is considered the approach is closely related to quasi likelih ood some key ironh estimating equation generalized linear model longitudinal data quasi likelihood repeated measures

combining census dual system and evaluation study data to estimate population shares abstract the 1990 census and post enumeration survey produced census and dual system estimates dse of population by domain together with an estimated sampling covariance matrix of the dse estimates of the bias of the dse were derived from various pes evaluation programs of the three sources the unadjusted census is the least variable but is believed to be the most biased the dse is less biased but more variable and the bias estimates may be regarded as unbiased but are the most variable this article addresses methods for combining the census the dse and bias estimates obtained from the evaluation programs to produce accurate estimates of population shares as measured by weighted squared or absolute error loss functions applied to estimated population shares of domains several procedures are reviewed that choose between the census and the dse using the bias evaluation data or that average the two with weights that are constant across domains a multivariate hierarchical bayes model is propose

detection of significant disease risks using a spatial conditional autoregressive model summary the conditional autoregressive car model is widely used to describe the geographical distribution of a specific disease risk in lattice mapping successful developments based on frequentist and bayesian procedures have been extensively applied to obtain two stage disease risk predictions at the subregional level bayesian procedures are preferred for making inferences as the posterior standard errors se of the two stage prediction account for the variability in the variance component estimates however some recent work based on frequentist procedures and the use of bootstrap adjustments for the se has been undertaken in this article we investigate the suitability of an analytical adjustment for disease risk inference that provides accurate interval predictions by using the penalized quasilikelihood pql technique to obtain model parameter estimates the method is a first order approximation of the naive se based on a taylor expansion and is interpreted as a conditional measure of variability providing conditional calibrated prediction intervals given the data we conduct a simulation study to demonstrate how the method can be used to estimate the specific subregion risk by interval we evaluate the proposed methodology by analyzing the commonly used example data set of lip cancer incidence in the 56 counties of scotland for the period 1975 1980 this evaluation reveals a close similarity between the solutions provided by the method proposed here and those of its fully bayesian counterpart