positive periodic solutions of nonlinear differential systems with impulses abstract by using a fixed point theorem of strict set contraction sufficient conditions for the existence of positive periodic solutions of nonlinear differential systems with impulses are obtained as applications of our results the existence of positive periodic solutions are established for delay lotka volterra systems with impulses or without impulses

logistic regression analysis of sample survey data summary standard chi squared x2 or likelihood ratio g2 test statistics for logistic regression analysis involving a binary response variable are adjusted to take account of the survey design these adjustments are based on certain generalized design effects logistic regression diagnostics to detect any outlying cell proportions in the table and influential points in the factor space are also developed taking account of the survey design finally the results are used to analyse some data from the october 1980 canadian labour force survey

lipschitz quasistability of impulsive differential difference equations with variable impulsive perturbations in the present paper by means of a suitable comparison lemma sufficient conditions for uniform lipschitz stability of an arbitrary solution of an impulsive system of differential difference equations with variable impulsive perturbations are obtained

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

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

on measuring the variability of small area estimators under a basic area level model summary in this paper based on a basic area level model we obtain second order accurate approxi mations to the mean squared error of model based small area estimators using the fay herriot 1979 iterative method of estimating the model variance based on weighted residual sum of squares we also obtain mean squared error estimators unbiased to second order based on simulations we compare the finite sample performance of our mean squared error estimators with those based on method of moments maximum likelihood and residual maximum likelihood estimators of the model variance our results suggest that the fay herriot method performs better in terms of relative bias of mean squared error estimators than the other methods across different combinations of number of areas pattern of sampling variances and distribution of small area effects we also derive a noninformative prior on the model parameters for which the posterior variance of a small area mean is second order unbiased for the mean squared error the posterior variance based on such a prior possesses both bayesian and frequentist interpretations

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

an h likelihood method for spatial mixed linear models based on intrinsic auto regressions summary we consider sparse spatial mixed linear models particularly those described by besag and higdon and develop an h likelihood method for their statistical inference the method proposed allows for singular precision matrices as it produces estimates that coincide with those from the residual maximum likelihood based on appropriate differencing of the data and has a novel approach to estimating precision parameters by a gamma linear model furthermore we generalize the h likelihood method to include continuum spatial variations by making explicit use of scaling limit connections between gaussian intrinsic markov random fields on regular arrays and the de wijs process keeping various applications of spatial mixed linear models in mind we devise a novel sparse conjugate gradient algorithm that allows us to achieve fast matrix free statistical computations we provide two applications the first is an extensive analysis of an agricultural variety trial that brings forward various new aspects of nearest neighbour adjustment such as effects on statistical analyses to changes of scale and use of implicit continuum spatial formulation the second application concerns an analysis of a large cotton field which gives a focus to matrix free computations the paper closes with some further considerations such as applications to irregularly spaced data use of the parametric bootstrap and some generalizations to the gaussian matern mixed effect models

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

nonparametric estimation of mean squared prediction error in nested error regression models nested error regression models are widely used for analyzing clustered data for example they are often applied to two stage sample surveys and in biology and econometrics prediction is usually the main goal of such analyses and mean squared prediction error is the main way in which prediction performance is measured in this paper we suggest a new approach to estimating mean squared prediction error we introduce a matched moment double bootstrap algorithm enabling the notorious underestimation of the naive mean squared error estimator to be substantially reduced our approach does not require specific assumptions about the distributions of errors additionally it is simple and easy to apply this is achieved through using monte carlo simulation to implicitly develop formulae which in a more conventional approach would be derived laboriously by mathematical arguments

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

a quasi empirical bayes method for small area estimation abstract this article develops an empirical bayes type approach for small area estimation based only on the specification of a set of conditionally independent hierarchical mean and variance functions describing the first two moments of the process generating the data the objective of the analysis is to draw inference about the intermediate or the random means we combine the quasi likelihood functions to construct the quasi posterior density of the random means conditional on the data and the marginal parameters we describe a method for estimating the marginal parameters that are then substituted in the quasi posterior density of the random means the empirical quasi posterior density so derived is used to obtain the quasi empirical bayes estimates of the random parameters we apply the methodology to estimate the utilization rates of cancer chemotherapy for the selected counties in the state of washington

estimation of finite population domain means a model assisted empirical best prediction approach in this article we introduce a general methodology for producing a model assisted empirical best predictor ebp of a finite population domain mean using data from a complex survey our method improves on the commonly used design consistent survey estimator by using a suitable mixed model such a model combines information from related sources such as census and administrative data unlike a purely model based ebp the proposed model assisted ebp converges in probability to the customary design consistent estimator as the domain and sample sizes increase the convergence in probability is shown to hold with respect to the sampling design irrespective of the assumed mixed model a property commonly known as design consistency this property ensures robustness of the proposed predictor against possible model failures in addition the convergence in probability is shown to be valid with respect to the assumed mixed model model consistency a new mean squared prediction error mspe estimator is proposed

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

impulsive state feedback control of a predator prey model the dynamics of a predator prey model with impulsive state feedback control which is described by an autonomous system with impulses is studied the sufficient conditions of existence and stability of semi trivial solution and positive period 1 solution are obtained by using the poincare map and analogue of the poincare criterion the qualitative analysis shows that the positive period 1 solution bifurcates from the semi trivial solution through a fold bifurcation the bifurcation diagrams of periodic solutions are obtained by using the poincare map and it is shown that a chaotic solution is generated via a cascade of period doubling bifurcations

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

jackknife estimation of mean squared error of small area predictors in nonlinear mixed models empirical bayes predictors of small area parameters of interest are often obtained under a linear mixed model for continuous response data or a generalized linear mixed model for binary responses or count data however estimation of the unconditional mean squared error of prediction is complicated particularly for a nonlinear mixed model jiang et al 2002 proposed a jackknife method for estimating the unconditional mean squared error and showed that the resulting estimator is nearly unbiased the leading term of this estimator does not depend on the area specific responses in the nonlinear case whereas the posterior variance of the small area parameter given the model parameters is area specific rao 2003 proposed an alternative method that leads to a computationally simpler jackknife estimator with an area specific leading term we show that a modification of rao s method leads to a nearly unbiased area specific jackknife estimator which is also nearly unbiased for the conditional mean squared error given the area specific responses we examine the relative performances of the jackknife estimators conditionally as well as unconditionally in a simulation study and apply the proposed method to estimate small area mean squared errors in disease mapping problems

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

reml estimation asymptotic behavior and related topics the restricted maximum likelihood reml estimates of dispersion parameters variance components in a general non normal mixed model are defined as solutions of the reml equations in this paper we show the reml estimates are consistent if the model is asymptotically identifiable and infinitely informative under the location invariant class and are asymptotically normal a n if in addition the model is asymptotically nondegenerate the result does not require normality or boundedness of the rank p of design matrix of fixed effects moreover we give a necessary and sufficient condition for asymptotic normality of gaussian maximum likelihood estimates mle in non normal cases as an application we show for all unconfounded balanced mixed models of the analysis of variance the reml anova estimates are consistent and are also a n provided the models are nondegenerate the mle are consistent a n if and only if certain constraints on p are satisfied

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

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

empirical bayes estimates of age standardized relative risks for use in disease mapping there have been many attempts in recent years to map incidence and mortality from diseases such as cancer such maps usually display either relative rates in each district as measured by a standardized mortality ratio smr or some similar index or the statistical significance level for a test of the difference between the rates in that district and elsewhere neither of these approaches is fully satisfactory and we propose a new approach using empirical bayes estimation the resulting estimators represent a weighted compromise between the smr the overall mean relative rate and a local mean of the relative rate in nearby areas the compromise solution depends on the reliability of each individual smr and on estimates of the overall amount of dispersion of relative rates over different districts

natural exponential families with quadratic variance functions statistical theory the normal poisson gamma binomial negative binomial and nefghs distributions are the six univariate natural exponential families nef with quadratic variance functions qvf this sequel to morris 1982 treats certain statistical topics that can be handled within this unified nef qvf formulation including unbiased estimation bhattacharyya and cramer rao lower bounds conditional distributions and moments quadratic regression conjugate prior distributions moments of conjugate priors and posterior distributions empirical bayes and g 2 minimax marginal distributions and their moments parametric empirical bayes and characterizations

non parametric inference for clustered binary and count data when only summary information is available summary data in the form of pairs x y where the response y is a count arise in many applications including problems involving stratified or two stage sampling such data are often analysed by using random effects models where the distribution of y conditional on x and on an unobserved random parameter is taken to be either binomial or poisson and the distribution of is connected through a link function to a random effect the latter is sometimes supposed to be normally distributed but that assumption can lead to serious biases if it is not satisfied this difficulty has motivated an extensive literature on non parametric techniques for cases where the random effect distribution is unknown but that methodology requires detailed cluster level data no non parametric techniques are available for instances where only summary data are accessible we show that the random effect distribution is actually non parametrically identifiable from crude summary data and we suggest a sieve method for inference in this case non parametric approaches to both parameter estimation and prediction are introduced and empirical techniques are developed for choosing tuning parameters these methods are shown to outperform their parametric counterparts when the model is misspecified and to perform almost as well as parametric methods when the model is correct

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 posteriori error bounds and global error control for approximation of ordinary differential equations the author analyzes a finite element method for the integration of initial value problems in ordinary differential equations general and contractive problems are treated and quasi optimal a priori and a posteriori error bounds obtained in each case in particular good results are obtained for a class of stiff dissipative problems these results are used to construct a rigorous and robust theory of global error control the author also derives an asymptotic error estimate that is used in a discussion of the behavior of the error in conclusion the properties of the error control are exhibited in a series of numerical experiments

a new class of average moment matching priors summary we derive a new class of priors for the variance component in the fay herriot model a mixed regression model widely used in small area estimation this class includes the well known uniform or superhannonic prior through simulation we illustrate the use of our class of priors

regression analysis of data from a cluster sample abstract for the case of different regression relationships in different subgroups of a finite population only part of which are sampled the extended least squares estimator of any weighted average of the distinct coefficients is derived under assumptions relating only to the first two moments of the distribution of the coefficients under these assumptions the estimator is shown to be the best linear unbiased estimator while under further distributional assumptions it is also the bayesian estimator for a quadratic loss function for the case of unknown variances a method for estimating them from the sample is proposed the empirical estimator thus obtained is shown to perform well by a simulation comparison with the optimal estimator and with other proposed empirical estimators

best predictive small area estimation we derive the best predictive estimator bpe of the fixed parameters under two well known small area models the fay herriot model and the nested error regression model this leads to a new prediction procedure called observed best prediction obp which is different from the empirical best linear unbiased prediction eblup we show that bpe is more reasonable than the traditional estimators derived from estimation considerations such as maximum likelihood ml and restricted maximum likelihood reml if the main interest is estimation of small area means which is a mixed model prediction problem we use both theoretical derivations and empirical studies to demonstrate that the obp can significantly outperform eblup in terms of the mean squared prediction error mspe if the underlying model is misspecified on the other hand when the underlying model is correctly specified the overall predictive performance of the obp is very similar to that of the eblup if the number of small areas is large a

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

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

fast and reliable methods for determining the evolution of uncertain parameters in differential equations a very common problem in science and engineering is the determination of the effects of uncertainty or variation in parameters and data on the output of a deterministic nonlinear operator for example such variations may describe the effect of experimental error or may arise as part of a sensitivity analysis of the model the monte carlo method is a justifiably popular tool for analyzing such effects it does however require many evaluations of the operator and it is difficult to extract precise information about the accuracy of any particular result in this paper we borrow techniques from a posteriori error analysis for finite element methods to compute information about the derivative of an operator with respect to its parameters these techniques employ the generalized green s function to describe how variation propagates into the solution around localized points in the parameter space we show how this derivative information can be used either to create a higher order method or produce an error estimate for information computed from a given representation in the latter case this provides the basis for adaptive sampling according to the variation in the output values both the higher order method and the adaptive sampling method are generally orders of magnitude faster than monte carlo methods in the case that the parameter space is not too high dimensional

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

model selection by testing for the presence of small area effects and application to area level data the models used in small area inference often involve unobservable random effects while this can significantly improve the adaptivity and flexibility of a model it also increases the variability of both point and interval estimators if we could test for the existence of the random effects and if the test were to show that they were unlikely to be present then we would arguably not need to incorporate them into the model and thus could significantly improve the precision of the methodology in this article we suggest an approach of this type we develop simple bootstrap methods for testing for the presence of random effects applicable well beyond the conventional context of the natural exponential family if the null hypothesis that the effects are not present is not rejected then our general methodology immediately gives us access to estimators of unknown model parameters and estimators of small area means such estimators can be substantially more effective for example because they enjoy much fa

fast methods for determining the evolution of uncertain parameters in reaction diffusion equations abstract an important problem in mathematical modeling in science and engineering is the determination of the effects of uncertainty or variation in parameters and data on the output of a deterministic nonlinear operator for example such variations may describe the effect of experimental error in measured parameter values or may arise as part of a sensitivity analysis of the model the monte carlo method is a widely used tool for determining such effects it employs random sampling of the input space in order to produce a pointwise representation of the output it is a robust and easily implemented tool with relatively low dependence on the number of parameters unfortunately it generally requires sampling the operator very many times at a significant cost especially when the model is expensive to evaluate moreover standard analysis provides only asymptotic or distributional information about the error computed from a particular realization in this paper we present an alternative approach for ascertaining the effects of variations and uncertainty in parameters in a reaction diffusion equation on the solution the approach is based on techniques borrowed from a posteriori error analysis for finite element methods the generalized green s function solving the adjoint problem is used to efficiently compute the gradient of a quantity of interest with respect to parameters at sample points in the parameter space this derivative information is used in turn to produce an error estimate for the information computed from the sample points the error estimate provides the basis for both deterministic and probabilistic adaptive sampling algorithms the deterministic adaptive sampling method can be orders of magnitude faster than monte carlo sampling in case of a moderate number of parameters the gradient can also be used to compute useful information that cannot be obtained easily from a monte carlo sample for example the adaptive algorithm yields a natural dimensional reduction in the parameter space where applicable after describing the new methodology in detail we apply the new method to analyze the parameter sensitivity of a predator prey model with a holling ii functional response that has six parameters

generalized green s functions and the effective domain of influence one well known approach to a posteriori analysis of finite element solutions of elliptic problems estimates the error in a quantity of interest in terms of residuals and a generalized green s function the generalized green s function solves the adjoint problem with data related to a quantity of interest and measures the effects of stability including any decay of influence characteristic of elliptic problems we show that consideration of the generalized green s function can be used to improve the efficiency of the solution process when the goal is to compute multiple quantities of interest and or to compute quantities of interest that involve globally supported information such as average values and norms in the latter case we introduce a solution decomposition in which we solve a set of problems involving localized information and then recover the desired information by combining the local solutions by treating each computation of a quantity of interest independently the maximum number of elements required to achieve the desired accuracy can be decreased significantly

stability under persistent disturbances of impulsive differential difference equations of neutral type abstract effective sufficient conditions for stability under persistent disturbances of the zero solution of impulsive systems of differential difference equations of neutral type are found the main results are stated in a theorem in its proof an analogue of gronwall bellman s inequality for piecewise continuous functions is used as a corollary the existence of a global solution of a system with quadratic nonlinearities is proved for sufficiently small initial data

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

an extended quasi likelihood function summary wedderburn s original definition of quasi likelihood for generalized linear models is extended to allow the comparison of variance functions as well as those of linear predictors and link functions the relationship between generalized linear models and the use of transformations of the response variable is explored and the ideas are illustrated by three examples generalized linear models by allowing the full distributional assumption about the random component in the model to be replaced by a much weaker assumption in which only the first and second moments were defined in making this extension wedderburn widened the scope of generalized linear models in a way very similar to that of gauss when he replaced the assumption of normality in classical linear models by that of equal variance for generalized linear models with distributions in the exponential family likelihood ratio and score tests are used for testing hypotheses concerning nested subsets of covariates in the linear predictor and for assessing hypothesized link functions these methods are also applicable with wedderburn s form of quasi likelihood however neither of these methods is suitable for the comparison of different variance functions in this paper we introduce an extended quasi likelihood function which allows for the comparison of various forms of all the components of a generalized linear model i e the linear predictor the link function and the variance function we then apply the ideas to the analysis of several sets of data

lipschitz stability of nonlinear systems of differential equations in s the authors introduced the notion of lipschitz stability in differential equations this notion lies somewhere between uniform stability on one side and the notions of asymptotic stability in variation 3 and uniform stability in variation 4 on the other side however lipschitz stability is new only as a nonliner phenomenon since it coincides with uniform stability in linear systems s an important feature of lipschitz stability is that unlike uniform stability the linearized system inherits the property of lipschitz stability from the original nonlinear system 5 in this paper we pursue the study of lipschitz stability that started in s using essentially the techniques of liapunov functions then we give sufficient conditions for the lipschitz stability of certain nonlinearly pertur bed nonlinear systems such systems include among other equations certain integrodifferential and functional differential equations then we give an example which can be investigated successfully using our results but cannot be handled by any previous techniques or results 9

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

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

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

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

a method for solving stochastic equations by reduced order models and local approximations a method is proposed for solving equations with random entries referred to as stochastic equations ses the method is based on two recent developments the first approximates the response surface giving the solution of a stochastic equation as a function of its random parameters by a finite set of hyperplanes tangent to it at expansion points selected by geometrical arguments the second approximates the vector of random parameters in the definition of a stochastic equation by a simple random vector referred to as stochastic reduced order model srom and uses it to construct a srom for the solution of this equation the proposed method is a direct extension of these two methods it uses sroms to select expansion points rather than selecting these points by geometrical considerations and represents the solution by linear and or higher order local approximations the implementation and the performance of the method are illustrated by numerical examples involving random eigenvalue problems and stochastic algebraic differential equations the method is conceptually simple non intrusive efficient relative to classical monte carlo simulation accurate and guaranteed to converge to the exact solution

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

on self adjoint boundary conditions for singular sturm liouville operators bounded from below abstract we extend the classical boundary values 0 1 g a w u a 0 g a lim x a g x u a 0 x g 1 a p g a w u a 0 g a lim x a g x g a u a 0 x u a 0 x for regular sturm liouville operators associated with differential expressions of the type r x 1 d d x p x d d x q x for a e x a b r to the case where is singular on a b r and the associated minimal operator t m i n is bounded from below here u a 0 and u a 0 denote suitably normalized principal and nonprincipal solutions of u 0 u for appropriate 0 r respectively our approach to deriving the analog of 0 1 in the singular context employing principal and nonprincipal solutions of u 0 u is closely related to a seminal 1992 paper by niessen and zettl 58 we also recall the well known fact that the analog of the boundary values in 0 1 characterizes all self adjoint extensions of t m i n in the singular case in a manner familiar from the regular case we briefly discuss the singular weyl titchmarsh kodaira m function and finally illustrate the theory in some detail with the examples of the bessel legendre and kummer resp laguerre operators

regression analysis with linked data record linkage or exact matching can be used to join together two files that contain information on the same individuals but lack unique personal identification codes the possibility of errors in linkage causes problems for estimating the relationships between variables on the two files the effect is analogous to the impact of measurement error a model of a linear regression relationship between variables in linked files is proposed assuming the probabilities that pairs of records are links are known an unbiased estimator of the regression coefficients is derived methods for estimating the linkage probabilities by using mixture models are discussed a consistent estimator of the covariance matrix of the proposed estimator is proposed a bootstrap estimator is used to reflect the impact of the uncertainty in record linkage model parameters on the estimators of the regression parameters a simulation study compares the performance of the proposed estimator and alternatives

empirical likelihood for small area estimation abstract small areaestimation has become a topic of growingimportance inrecent years due to increasing demand for small area statistics fromboth the public and private sectors the current methodologies inthis area with possibly a handful of exceptions are all parametric be it bayesian or frequentist even the so called nonparamet ric procedures based on empirical best linear unbiased predictors eblup assume linearity of the estimators of small area means resampling methods like jacknife and bootstrap are used mostlyfor mean squared error estimation in this paper we propose analternative bayesian approach based on empirical likelihood whichdoes not require the assumption of either a parametric likelihoodor linearity of estimators the proposed method can also handleboth discrete and continuous data in a uni ed manner empiricallikelihood for both area and unit level models are introduced theimproved performance of our estimator is illustrated through twoexamples involving both discrete and continuous data key words and phrases area level exponentially tilted t prior unit level

decomposition of prediction error abstract the problem considered is that of predicting the value of an unobservable random variable w from the value of an observable random vector y this problem is considered under each of four states of knowledge about the joint distribution of w and y ranging from complete knowledge to no knowledge a point predictor or predictors are presented for each case prediction error is decomposed so that each component reflects an absence of information under certain conditions these components are uncorrelated and have zero means an exact or approximate expression is given for the variance of each component

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

jackknife bootstrap and other resampling methods in regression analysis motivated by a representation for the least squares estimator we propose a class of weighted jackknife variance estimators for the least squares estimator by deleting any fixed number of observations at a time they are unbiased for homoscedastic errors and a special case the delete one jackknife is almost unbiased for heteroscedastic errors the method is extended to cover nonlinear parameters regression m estimators nonlinear regression and generalized linear models interval estimators can be constructed from the jackknife histogram three bootstrap methods are considered two are shown to give biased variance estimators and one does not have the bias robustness property enjoyed by the weighted delete one jackknife a general method for resampling residuals is proposed it gives variance estimators that are bias robust several bias reducing estimators are proposed some simulation results are reported

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

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

empirical bootstrap bias correction and estimation of prediction mean square error in small area estimation we develop a method for bias correction which models the error of the target estimator as a function of the corresponding estimator obtained from bootstrap samples and the original estimators and bootstrap estimators of the parameters governing the model fitted to the sample data this is achieved by considering a number of plausible parameter values generating a pseudo original sample for each parameter and bootstrap samples for each such sample and then searching for an appropriate functional relationship under certain conditions the procedure also permits estimation of the mean square error of the bias corrected estimator the method is applied for estimating the prediction mean square error in small area estimation of proportions under a generalized mixed model empirical comparisons with jackknife and bootstrap methods are presented

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

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

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

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

a measure theoretic computational method for inverse sensitivity problems i method and analysis we consider the inverse sensitivity analysis problem of quantifying the uncertainty of inputs to a deterministic map given specified uncertainty in a linear functional of the output of the map this is a version of the model calibration or parameter estimation problem for a deterministic map we assume that the uncertainty in the quantity of interest is represented by a random variable with a given distribution and we use the law of total probability to express the inverse problem for the corresponding probability measure on the input space assuming that the map from the input space to the quantity of interest is smooth we solve the generally ill posed inverse problem by using the implicit function theorem to derive a method for approximating the set valued inverse that provides an approximate quotient space representation of the input space we then derive an efficient computational approach to compute a measure theoretic approximation of the probability measure on the input space imparted by the approximate set valued inverse that solves the inverse problem

weighting for unequal selection probabilities in multilevel models when multilevel models are estimated from survey data derived using multistage sampling unequal selection probabilities at any stage of sampling may induce bias in standard estimators unless the sources of the unequal probabilities are fully controlled for in the covariates this paper proposes alternative ways of weighting the estimation of a two level model by using the reciprocals of the selection probabilities at each stage of sampling consistent estimators are obtained when both the sample number of level 2 units and the sample number of level 1 units within sampled level 2 units increase scaling of the weights is proposed to improve the properties of the estimators and to simplify computation variance estimators are also proposed in a limited simulation study the scaled weighted estimators are found to perform well although non negligible bias starts to arise for informative designs when the sample number of level 1 units becomes small the variance estimators perform extremely well the procedures are illustrated using data from the survey of psychiatric morbidity

a unified jackknife theory for empirical best prediction with m estimation the paper presents a unified jackknife theory for a fairly general class of mixed models which includes some of the widely used mixed linear models and generalized linear mixed models as special cases the paper develops jackknife theory for the important but so far neglected prediction problem for the general mixed model for estimation of fixed parameters a jackknife method is considered for a general class of m estimators which includes the maximum likelihood residual maximum likelihood and anova estimators for mixed linear models and the recently developed method of simulated moments estimators for generalized linear mixed models for both the prediction and estimation problems a jackknife method is used to obtain estimators of the mean squared errors mse asymptotic unbiasedness of the mse estimators is shown to hold essentially under certain moment conditions simulation studies undertaken support our theoretical results 1 introduction due to the advent of high speed computers and powerful software computer oriented statistical methods including various resampling methods have received considerable attention in recent years as statisticians are constantly facing complex problems the jackknife method is one such simple resampling method which is very popular among survey samplers primarily due

comparison results for initial value problems of second order impulsive integro differential inclusions abstract in this paper we present new maximum and minimum principles for the initial value problem for a class of second order nonlinear impulsive integro differential inclusions with discontinuous multivalued functions on the right hand side in a real banach space this allows us to lead on to the development of the comparison principle and monotone iterative technique for a nonlinear second order impulsive integro differential inclusion

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