transformations in regression a robust analysis we consider two approaches to robust estimation for the box cox power transformation model one approach maximizes weighted modified likelihoods a second approach bounds a measure of gross error sensitivity among our primary concerns is the performance of these estimators on actual data in examples that we study there seem to be only minor differences between these two robust estimators but they behave rather differently than the maximum likelihood estimator or estimators that bound only the influence of the residuals these examples show that model selection determination of the transformation parameter and outlier identification are fundamentally interconnected

periods and feynman integrals we consider multiloop integrals in dimensional regularization and the corresponding laurent series we study the integral in the euclidean region and where all ratios of invariants and masses have rational values we prove that in this case all coefficients of the laurent series are periods

estimation in generalized mixed models regression models containing fixed and random effects may have a response variable which is not normally distributed the generalized mixed model includes both discrete and continuous response variables and is developed here for problems in which the regression variables enter linearly into the model best linear unbiased predictor methods are extended to maximum likelihood and residual maximum likelihood estimation procedures applications in modelling discrete response variables and in survival analysis are discussed

a robust comparison of biological shapes summary localized differences in the form of two related animal skeletons are more effectively determined when resistant fitting techniques are used rather than least squares the repeated median resistant fitting algorithm is introduced the methods are tested by comparing hominid skulls with those of other primates and the male ostracode with the female in each case the resistant fit is superior to the least squares fit in that the difierences are more readily identified and agl ee more closely with the structural differences perceived on biological grounds le llltroductioll studies of allometric growth and of morphological change often run into difficulties in identifying and measuring the location and extent of relative shape deformations a reference system is needed for measuring these deformations and in the absence of an inherent fixed reference system one must be estimated the problem of estimating shape change in animal morphology provides an elegant setting for the comparison of least squares with recently developed robust and resistant statistical techniques being robust these new methods are protected against departures from the assumptions of analysis such as independent identically and normally distributed errors being resistant they are also protected against the potentially strong influences of atypical or incorrect data values two shapes can rarely be superimposed perfectly different fitting criteria will generally yield different results by allowing regions with large deformations to have a large impact on the fit least squares methods can minimize true shape differences and thereby obscure them a resistant technique however limits the influence of large deformations and the resulting fit is close in similar regions and not close in relatively deformed regions in this way resistant techniques can help to identify similarities and diflerences in form more effectively than least squares methods when sneath 1967 used least squares as a basis for some of the first quantitative estimates of relative deformation and so extended the fundamental descriptive work of thompson 1917 useful robust methods were not generally available sneath did in fact consider using the available robust methods but concluded that alternative methods that are less affected by wild values are not attractive in practice since then however the development of new theory and methods as reported by huber 1972 1981 hampel

a comparison of generalized linear mixed model procedures with estimating equations for variance and covariance parameter estimation in longitudinal studies and group randomized trials by b a evans z feng and a v peterson statistics in medicine 2001 20 3353 3373

controlling the bias of robust small area estimators sinha rao 2009 proposed estimation procedures designed for small area means based on robustified maximum likelihood estimators and robust empirical best linear unbiased predictors their methods are of the plug in type and may be biased bias corrected estimators have been proposed by chambers et al 2013 here we investigate two new approaches one relying on the work of chambers 1986 and the second using the concept of conditional bias to measure the influence of units in the population these two classes of estimators also include correction terms for the bias but are both fully bias corrected in the sense that the corrections account for the potential impact of the other domains on the small area of interest monte carlo simulations suggest that the sinha rao method and the bias adjusted estimator of chambers et al 2013 may exhibit a large bias while the new procedures often offer lower bias and mean squared error a parametric bootstrap procedure is considered for constructing confidence intervals copyright 2013 oxford university press

robust autoregressive estimates using quadratic programming abstract the robust estimation of the autoregressive parameters is formulated in terms of the quadratic programming problem this article s main contribution is to present an estimator that down weights both types of outliers in time series and improves the forecasting results new robust estimates are yielded by combining optimally two weight functions suitable for innovation and additive outliers in time series the technique which is developed here is based on an approach of mathematical programming applications to ip approximation the behavior of the estimators are illustrated numerically under the additive outlier generating model monte carlo results show that the proposed estimators compared favorably with respect to m estimators and bounded influence estimators based on these results we conclude that one can improve the robust properties of ar p estimators using quadratic programming

reml estimation for survival models with frailty a method of estimation for generalised mixed models is applied to the estimation of regression parameters in proportional hazards models for failure times when there are repeated observations of failure on each subject the subject effect is incorporated into the model as a random frailty term best linear unbiased predictors are used as an initial step in the computation of maximum likelihood and residual maximum likelihood estimates

robust restricted maximum likelihood in mixed linear models summary definitions of robust maximum likelihood robust ml and robust restricted maximum likelihood robust reml are introduced and the definitions are applied to data from biological and chemical experiments a simulation study is undertaken to investigate the asymptotic properties of robust ml and robust reml in small samples and to examine the advantages of using robust methods 1 introduction and definitions linear models with multiple sources of error are widely used in designed experiments across many scientific fields an example of such an experiment is described by patterson and nabugoomu 1992 from patterson and silvey 1980 six varieties of wheat were grown at ten centres that formed a sample of the main types of growing area for wheat in scotland and the yields in tonnes hectare were recorded the experiment is unbalanced because of 60 possible variety centre combinations only 46 were used at seven centres four varieties were grown and at the remaining three centres all six varieties were grown in this paper we fit the simplest mixed linear model proposed for this data by patterson and nabugoomu namely

nonsmooth analysis and frechet differentiability of m functionals abstract a necessary requirement for existence of the frechet derivative is that the defining psi function is uniformly bounded and this naturally excludes those nonrobust estimators such as the maximum likelihood estimator in normal parametric models in this paper the methods of nonsmooth analysis described in the book by f h clarke 1983 are introduced to the theory of statistical expansions and are used here in the proofs of weak continuity and frechet differentiability of m functionals subsequently the conditions for frechet differentiability given in clarke 1983 can be relaxed to include most popular m functionals additional keywords distribution functions m estimators robustness gross error sensitivity weak continuity asymptotic expansions asymptotic normality selection functional local uniqueness author

the standardized influence matrix and its applications abstract in this article we introduce the standardized influence matrix sim of parameter estimators as a conjugate of the sample covariance matrix of standardized influence functions sifs evaluated at the data points we propose principal component analysis for the sim and its complement sim analysis as a diagnostic tool for regression or other statistical inference and provide theoretical insight to the local influence defined by cook we show that sim analysis reveals the multivariate structure of outlying and or influential points specifically sim analysis uncovers hidden structures of influence such as clustering that cannot be identified by the lengths of the standardized influences finally examples in linear regression show that a diagnostic method using sim is more effective if robust parameter estimates are used in calculating the sample sim

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

impact of imputation of missing values on classification error for discrete data numerous industrial and research databases include missing values it is not uncommon to encounter databases that have up to a half of the entries missing making it very difficult to mine them using data analysis methods that can work only with complete data a common way of dealing with this problem is to impute fill in the missing values this paper evaluates how the choice of different imputation methods affects the performance of classifiers that are subsequently used with the imputed data the experiments here focus on discrete data this paper studies the effect of missing data imputation using five single imputation methods a mean method a hot deck method a nai ve bayes method and the latter two methods with a recently proposed imputation framework and one multiple imputation method a polytomous regression based method on classification accuracy for six popular classifiers ripper c4 5 k nearest neighbor support vector machine with polynomial and rbf kernels and nai ve bayes on 15 datasets this experimental study shows that imputation with the tested methods on average improves classification accuracy when compared to classification without imputation although the results show that there is no universally best imputation method nai ve bayes imputation is shown to give the best results for the ripper classifier for datasets with high amount i e 40 and 50 of missing data polytomous regression imputation is shown to be the best for support vector machine classifier with polynomial kernel and the application of the imputation framework is shown to be superior for the support vector machine with rbf kernel and k nearest neighbor the analysis of the quality of the imputation with respect to varying amounts of missing data i e between 5 and 50 shows that all imputation methods except for the mean imputation improve classification error for data with more than 10 of missing data finally some classifiers such as c4 5 and nai ve bayes were found to be missing data resistant i e they can produce accurate classification in the presence of missing data while other classifiers such as k nearest neighbor svms and ripper benefit from the imputation

identifying variables responsible for data not missing at random when data are not missing at random nmar maximum likelihood ml procedure will not generate consistent parameter estimates unless the missing data mechanism is correctly modeled understanding nmar mechanism in a data set would allow one to better use the ml methodology a survey or questionnaire may contain many items certain items may be responsible for nmar values in other items the paper develops statistical procedures to identify the responsible items by comparing ml estimates mle statistics are developed to test whether the mles are changed when excluding items the items that cause a significant change of the mles are responsible for the nmar mechanism normal distribution is used for obtaining the mles a sandwich type covariance matrix is used to account for distribution violations the class of nonnormal distributions within which the procedure is valid is provided both saturated and structural models are considered effect sizes are also defined and studied the results indicate that more missing data in a sample does not necessarily imply more significant test statistics due to smaller effect sizes knowing the true population means and covariances or the parameter values in structural equation models may not make things easier either

robust estimation in generalized linear mixed models generalized linear mixed models glmms are widely used to analyse non normal response data with extra variation but non robust estimators are still routinely used we propose robust methods for maximum quasi likelihood and residual maximum quasi likelihood estimation to limit the influence of outlying observations in glmms the estimation procedure parallels the development of robust estimation methods in linear mixed models but with adjustments in the dependent variable and the variance component the methods proposed are applied to three data sets and a comparison is made with the nonparametric maximum likelihood approach when applied to a set of epileptic seizure data the methods proposed have the desired effect of limiting the influence of outlying observations on the parameter estimates simulation shows that one of the residual maximum quasi likelihood proposals has a smaller bias than those of the other estimation methods we further discuss the equivalence of two glmm formulations when the response variable follows an exponential family their extensions to robust glmms and their comparative advantages in modelling are described some possible modifications of the robust glmm estimation methods are given to provide further flexibility for applying the method

testing ignorable missingness in estimating equation approaches for longitudinal data summary we address the matter of determining whether or not missing data in longitudinal studies are ignorable with regard to quasilikelihood or estimating equations approaches this involves testing for whether or not the zero mean property of estimating equations holds true chen little 1999 proposed testing for significant differences among parameter estimators calculated from sample subsets with different patterns of missing data whereas we propose a more unified generalised score type test this avoids exhaustive estimation of parameters for each missing data pattern testing instead with a single quadratic score test statistic whether or not there is a common parameter under which the means of all the pattern specific estimating equations are zero comparisons are made for the two approaches with both simulations and real data examples

diagnostics for mixed model analysis of variance we describe a new method for assessment of model inadequacy in maximum likelihood mixed model analysis of variance in particular we discuss its use in diagnosing perturbations from the usual assumption of constant error variance and from the assumption that each realization of a given random factor has been drawn from the same normal population computer implementation of the procedure is described and an example is presented involving the analysis of filter cartridges used with commercial respirators

robust estimation of the process standard deviation for control charts control charts are tools used to detect aberrant behavior in manufacturing processes the x chart plots subsample averages as a function of time and the r chart plots subsample ranges both of these charts rely on an estimate of the standard deviation of the process when it is operating correctly the estimate has traditionally been based on the average range of 2040 subgroups but this will produce an estimate that is biased high when outliers are present one standard solution is to construct a range chart for the original subgroups and estimate the standard deviation only from those subgroups within the control limits repeating the procedure as necessary proposals have also recently been made to use a trimmed mean of the subsample ranges with a fixed percentage of trimming as well as the trimmed mean of the subsample interquartile ranges this article presents a new approach to robust estimation of the process standard deviation the procedure first centers each subsample on its own median and then

uniform asymptotics for robust location estimates when the scale is unknown most asymptotic results for robust estimates rely on regularity conditions that are dicult to verify and that real data sets rarely satisfy moreover these results apply to fixed distribution functions in the robustness context the distribution of the data remains largely unspecified and hence results that hold uniformly over a set of possible distribution functions are of theoretical and practical interest in this paper we study the problem of obtaining verifiable and realistic conditions that suce to obtain uniform consistency and uniform asymptotic normality for location robust estimates when the scale of the errors is unknown we study m location estimates calculated with an s scale and we obtain uniform asymptotic results over contamination neighbourhoods there is a trade o between the size of these neighbourhoods and the breakdown point of the scale estimate we also show how to calculate the maximum size of the contamination neighbourhoods where these uniform results hold

bounded influence estimation in the mixed linear model abstract bounded influence estimation also known as generalized m or gm estimation in the regression model is reviewed the definitions of bounded influence estimation proposed by mallows and schweppe are then extended to the mixed linear model this is achieved by applying appropriate weight functions to maximum likelihood and restricted maximum likelihood estimating equations the asymptotic properties of the new estimators are obtained and the estimators are applied to an artificial dataset the article concludes with an extension of the example into a small simulation study designed to test some properties of the estimators in samples of moderate size

mallows type bounded influence regression trimmed means abstract the influence functions of the regression trimmed mean estimators proposed by koenker and bassett 1978 and welsh 1987 are bounded in the dependent variable space but not in the independent variable space this article follows the approach of mallows 1973 1975 and modifies these estimators so that the resulting estimators have bounded influence functions the large sample behavior of these estimators is studied and it is shown that they have the same asymptotic distribution the small sample behaviors of the ordinary regression and bounded influence regression trimmed means are then investigated by means of a monte carlo study and by applying the estimators to water salinity data see ruppert and carroll 1980 based on these results we conclude that one can potentially gain much by using bounded influence regression trimmed means over ordinary regression trimmed means however there does not seem to be a clear choice between the koenker bassett and welsh versions

estimation in generalized linear models with random effects summary a conceptually very simple but general algorithm for the estimation of the fixed effects random effects and components of dispersion in generalized linear models with random effects is proposed conditions are described under which the algorithm yields approximate maximum likelihood or quasi maximum likelihood estimates of the fixed effects and dispersion components and approximate empirical bayes estimates of the random effects the algorithm is applied to two data sets to illustrate the estimation of components of dispersion and the modelling of overdispersion

frechet differentiability p variation and uniform donsker classes classically a stastistical functional is defined on a space of distribution f on the real line with the supremum norn the values may be real or themselves functions such as the quantile function f 1 nonlinear functionals are studied via their derivatives this paper and a related one dudley 1991a will show that the differentiability properties originally proved by reeds 1976 also treated in fernholz 1983 can be improved substantially

bounded influence regression via local minimax mean squared error abstract the class of general m estimators for the stochastic linear regression model is considered and influence and change of variance functions are used to approximate the main term of the mean squared error mse the estimator is then found in the class that minimizes the maximum of that approximate mse over the contamination neighborhood this approach allows for additional flexibility of designing estimators optimal for prediction in specific regions of the factor space and it provides a practical way of choosing estimators tuning constants as functions of a fraction of contamination and sample size

efficient bounded influence regression estimation abstract the least squares estimator for in the classical linear regression model is strongly efficient under certain conditions however in the presence of heavy tailed errors and or anomalous data the least squares efficiency can be markedly reduced in this article we propose an estimator that limits the influence of any small subset of the data and show that it satisfies a first order condition for strong efficiency subject to the constraint we then show that the estimator is asymptotically normal the article concludes with an outline of an algorithm for computing a bounded influence regression estimator and with an example comparing least squares robust regression as developed by huber and the estimator proposed in this article

outlier induced collinearities when an observation in a regression analysis has very large values on two or more predictor variables artificial collinearities can be induced the effects of such collinearities on a regression analysis are not well documented although they can be shown to be similar in many respects to those resulting from approximate linear dependencies among the columns of predictor variable values the purpose of this article is to explore the effects of outlier induced collinearities on the estimation of regression coefficients

laurent series expansion of a class of massive scalar one loop integrals up to o e2 in terms of multiple polylogarithms in a recent paper we have presented results for a set of massive scalar one loop master integrals needed in the nnlo parton model description of the hadroproduction of heavy flavors the one loop integrals were evaluated in n 4 2e dimension and the results were presented in terms of a laurent series expansion up to o e2 we found that some of the e2 coefficients contain a new class of functions which we termed the l functions the l functions are defined in terms of one dimensional integrals involving products of logarithm and dilogarithm functions in this paper we derive a complete set of algebraic relations that allow one to convert the l functions of our previous approach to a sum of classical and multiple polylogarithms using these results we are now able to present the e2 coefficients of the one loop master integrals in terms of classical and multiple polylogarithms

robust bounded influence tests in linear models abstract a robust test that we call an aligned generalized m test for testing subhypotheses in general linear models is developed and its asymptotic properties are studied the test is a robustification of the well known f test and it is an elegant alternative to ronchetti s 1982 class of tests p values associated with it can be approximated readily using existing chi square tables the test is based on an appropriately constructed quadratic form and uses the generalized m estimators of the parameters in the reduced model under the null hypothesis the asymptotic distribution is a central chi square and under contiguous alternatives it is a noncentral chi square with the same degrees of freedom the test can be viewed as a generalization of sen s 1982 m test for linear models the influence function of the test is bounded the bound not only applies to the influence of residuals but to the influence of position in the factor space as well on the other hand sen s test has bounded influence only

a test of missing completely at random for generalised estimating equations with missing data we consider inference from generalised estimating equations when data are incomplete a test for missing completely at random is proposed to help decide whether or not we should adjust estimating equations to correct the possible bias introduced by a missing data mechanism that is not missing completely at random likelihood ratio tests have been introduced to test the missing completely at random hypothesis fuchs 1982 little 1988 for the estimating equation setting following the basic idea of little 1988 we propose a wald type test based on an information decomposition and recombination procedure which also provides an alternative method for estimating parameters one application of the test is to assess the adequacy of the marginal generalised estimating equation for longitudinal data with missing values simulations are done to evaluate its performance

median regression for longitudinal data we review and compare three estimators of median regression in linear models with longitudinal data the estimators are constructed based on well known ideas of weighting decorrelating and the working assumption of independence both asymptotic efficiency calculations and finite sample monte carlo studies are used to assess the performance of these estimators we find that their relative performances depend on the nature of covariates the estimator under the working assumption of independence is computationally simple and yet has good relative performance when the covariates are invariant over time or when the within subject correlations are small its relative performance in finite samples is also found to be more favourable than suggested by the asymptotic comparisons

a general maximum likelihood analysis of variance components in generalized linear models this paper describes an em algorithm for nonparametric maximum likelihood ml estimation in generalized linear models with variance component structure the algorithm provides an alternative analysis to approximate mql and pql analyses mcgilchrist and aisbett 1991 biometrical journal 33 131 141 breslow and clayton 1993 journal of the american statistical association 88 9 25 mcgilchrist 1994 journal of the royal statistical society series b 56 61 69 goldstein 1995 multilevel statistical models and to gee analyses liang and zeger 1986 biometrika 73 13 22 the algorithm first given by hinde and wood 1987 in longitudinal data analysis 110 126 is a generalization of that for random effect models for overdispersion in generalized linear models described in aitkin 1996 statistics and computing 6 251 262 the algorithm is initially derived as a form of gaussian quadrature assuming a normal mixing distribution but with only slight variation it can be used for a completely unknown mixing distribution giving a straightforward method for the fully nonparametric ml estimation of this distribution this is of value because the ml estimates of the glm parameters can be sensitive to the specification of a parametric form for the mixing distribution the nonparametric analysis can be extended straightforwardly to general random parameter models with full npml estimation of the joint distribution of the random parameters this can produce substantial computational saving compared with full numerical integration over a specified parametric distribution for the random parameters a simple method is described for obtaining correct standard errors for parameter estimates when using the em algorithm several examples are discussed involving simple variance component and longitudinal models and small area estimation

bias correction in generalised linear mixed models with a single component of dispersion summary general expressions are derived for the asymptotic biases in three approximate estimators of regression coefficients and variance component for small values of the variance component in generalised linear mixed models with canonical link function and a single source of extraneous variation the estimators involve first and second order laplace expansions of the integrated likelihood and a related procedure known as penalised quasilikelihood numerical studies of a series of matched pairs of binary outcomes show that the first order estimators of the variance component are seriously biased easily computed correction factors produce satisfactory estimators of small variance components comparable to those obtained with a second order laplace expansion and markedly improve the asymptotic performance for larger values for a series of matched pairs of binomial observations the variance correction factors rapidly approach one as the binomial denominators increase these results greatly extend the range of parameter values for which the approximate estimation procedures have satisfactory asymptotic properties

two way random effects analyses and gauge r r studies an important prerequisite of any sensible data based engineering study is the quantification of the precision of gauges or measuring equipment to be used in data collection it has long been understood that in the event that more than one individual will use a particular gauge measurement variation for that gauge can include not only a kind of pure error component but an operator or technician component as well furthermore it is well known that the two way random effects model provides a natural framework for quantifying the different components of measurement variation some parts of standard practice in the gauge r r studies aimed at quantifying measurement precision however are unfortunately at odds with what makes sense under this model thus the purpose of this primarily expository article is to explain in elementary terms the use of a two way random effects model for gauge r r studies to critique current practice and to point out some simple improvements that can follow from more c

robust regression based on infinitesimal neighbourhoods we study robust estimation in the general normal regression model with random carriers permitting small departures from the model the framework is that of bickel 1981 we obtain solutions of huber 1982 kraskerhampel 1980 and krasker welsch 1982 as special cases as well as some new procedures our calculations indicate that the optimality properties of these estimates are more limited than suggested by krasker and welsch

minimax aspects of bounded influence regression abstract bounded influence regression in the sense of hampel mallows krasker welsch is examined critically from the point of view of finite sample minimax robust estimation theory the main conclusion is that the influence of position of very high leverage points that is of points where the diagonal element of the hat matrix h x xtx 1 xt exceeds a certain bound should be cut down selectively this bound should be chosen roughly between 2 and 5 for the upper of these values the resulting estimate can be approximated very simply by scaling the residuals by their own estimated standard deviation a conflict between decision theoretic estimation and data analytic diagnostic viewpoints is pointed out and briefly discussed

nested sums expansion of transcendental functions and multiscale multiloop integrals expansion of higher transcendental functions in a small parameter are needed in many areas of science for certain classes of functions this can be achieved by algebraic means these algebraic tools are based on nested sums and can be formulated as algorithms suitable for an implementation on a computer examples such as expansions of generalized hypergeometric functions or appell functions are discussed as a further application we give the general solution of a two loop integral the so called c topology in terms of multiple nested sums in addition we discuss some important properties of nested sums in particular we show that they satisfy a hopf algebra

a copula model for repeated measurements with non ignorable non monotone missing outcome a normal copula based selection model is proposed for continuous longitudinal data with a non ignorable non monotone missing data process the normal copula is used to combine the distribution of the outcome of interest and that of the missing data indicators given the covariates parameters in the model are estimated by a pseudo likelihood method we first use the gee with a logistic link to estimate the parameters associated with the marginal distribution of the missing data indicator given the covariates assuming that covariates are always observed then we estimate other parameters by inserting the estimates from the first step into the full likelihood function a simulation study is conducted to assess the robustness of the assumed model under different missing data processes the proposed method is then applied to one example from a community cohort study to demonstrate its capability to reduce bias

variance components models for dependent cell populations cells grown in culture can be tracked for several generations and measurements taken on size or age at division and other cell characteristics the observations for the offspring of each cell form a family tree of dependent data such cell lineage data are here modeled as repeated measurements on different family trees arising from individual ancestor cells selected at random from a population of cultured cells the bifurcating autoregression model is embedded in a process that allows for measurement error and variation from tree to tree robust methods are presented that accommodate outliers in this time dependent and branching environment while allowing the statistician to interactively build a variance components model for the process the methodology is illustrated on a substantial data set of 41 trees of emt6 cells with the surprising conclusion that after removing measurement error sister cell lifetimes are nearly identical

assessing the validity of weighted generalized estimating equations the inverse probability weighted generalized estimating equations approach robins et al 1994 robins et al 1995 effectively removes bias and provides valid statistical inference for regression parameter estimation in marginal models when longitudinal data contain missing values the validity of the weighted generalized estimating equations regarding consistent estimation depends on whether the underlying missing data process is properly modelled however there is little work available to examine whether or not this condition holds in this paper we propose a test constructed from two sets of estimating equations one set is known to be unbiased but the other set is not known we utilize the quadratic inference function qu et al 2000 method to assess their compatibility which is equivalent to testing for the validity of the weighted generalized estimating equations approach we conduct simulation studies to assess the performance of the proposed method the test procedure is illustrated through a real data example

tests of homogeneity of means and covariance matrices for multivariate incomplete data existing test statistics for assessing whether incomplete data represent a missing completely at random sample from a single population are based on a normal likelihood rationale and effectively test for homogeneity of means and covariances across missing data patterns the likelihood approach cannot be implemented adequately if a pattern of missing data contains very few subjects a generalized least squares rationale is used to develop parallel tests that are expected to be more stable in small samples three factors were varied for a simulation number of variables percent missing completely at random and sample size one thousand data sets were simulated for each condition the generalized least squares test of homogeneity of means performed close to an ideal type i error rate for most of the conditions the generalized least squares test of homogeneity of covariance matrices and a combined test performed quite well also

bootstrapping robust estimates for clustered data in mixed models the use of robust estimates raises several interesting inferential challenges one of these challenges arises from the realization that the effect of contamination is to increase the variability in the data but robust estimates of variance components are usually smaller than their nonrobust counterparts the robust estimates reflect the variability of the bulk of the data which is not the same as the variability in the data generating process this means that the naive implementation of bootstrap procedures might not work in this article we consider several bootstrap procedures including random effect transformation and weighted bootstraps we give conditions for the asymptotic validity of the bootstraps and assess their performance via a small simulation study both the transformation and generalized cluster bootstrap perform well and are asymptotically valid under reasonable conditions

a robust approach for high resolution frequency estimation a robust estimation method for frequencies of received signals is considered the influence function of a robust estimate is derived for the sinusoidal signals the variance of the robust estimate is also derived using the influence function it is found that the robust estimate attains the cramer rao lower bound crlb for the contaminated gaussian distribution it is of order o n sup 3 and is close to the crlb for the perfect gaussian distribution the authors introduce some basic definitions for the high resolution frequency estimation method and prove that the robust estimate has the high resolution property this property is also confirmed by numerical simulations

estimation in a semiparametric model for longitudinal data with unspecified dependence structure this paper considers an extension of m estimators in semiparametric models for independent observations to the case of longitudinal data we approximate the nonparametric function by a regression spline and any m estimation algorithm for the usual linear models can then be used to obtain consistent estimators of the model and valid largesample inferences about the regression parameters without any specification of the error distribution and the covariance structure included as special cases are the analysis of the conditional mean and median functions for longitudinal data

robust asymptotic statistics contents preface notation chapter 1 von mises functionals chapter 2 log likelihoods chapter 3 asymptotic statistics chapter 4 nonparametric statistics chapter 5 optimal influence curves chapter 6 stable constructions chapter 7 robust regression appendix a weak convergence of measures appendix b some functional analysis appendix c complements bibliography index

high breakdown inference for mixed linear models mixed linear models are used to analyze data in many settings these models have a multivariate normal formulation in most cases the maximum likelihood estimator mle or the residual mle reml is usually chosen to estimate the parameters however the latter are based on the strong assumption of exact multivariate normality welsh and richardson have shown that these estimators are not robust to small deviations from multivariate normality this means that in practice a small proportion of data even only one can drive the value of the estimates on their own because the model is multivariate we propose a high breakdown robust estimator for very general mixed linear models that include for example covariates this robust estimator belongs to the class of s estimators from which we can derive asymptotic properties for inference we also use it as a diagnostic tool to detect outlying subjects we discuss the advantages of this estimator compared with other robust estimators proposed previously and i

approximate inference in generalized linear mixed models statistical approaches to overdispersion correlated errors shrinkage estimation and smoothing of regression relationships may be encompassed within the framework of the generalized linear mixed model glmm given an unobserved vector of random effects observations are assumed to be conditionally independent with means that depend on the linear predictor through a specified link function and conditional variances that are specified by a variance function known prior weights and a scale factor the random effects are assumed to be normally distributed with mean zero and dispersion matrix depending on unknown variance components for problems involving time series spatial aggregation and smoothing the dispersion may be specified in terms of a rank deficient inverse covariance matrix approximation of the marginal quasi likelihood using laplace s method leads eventually to estimating equations based on penalized quasilikelihood or pql for the mean parameters and pseudo likelihood for the variances im

a test of missing completely at random for multivariate data with missing values abstract a common concern when faced with multivariate data with missing values is whether the missing data are missing completely at random mcar that is whether missingness depends on the variables in the data set one way of assessing this is to compare the means of recorded values of each variable between groups defined by whether other variables in the data set are missing or not although informative this procedure yields potentially many correlated statistics for testing mcar resulting in multiple comparison problems this article proposes a single global test statistic for mcar that uses all of the available data the asymptotic null distribution is given and the small sample null distribution is derived for multivariate normal data with a monotone pattern of missing data the test reduces to a standard t test when the data are bivariate with missing data confined to a single variable a limited simulation study of empirical sizes for the test applied to normal and nonnormal data suggests th

uniqueness and frechet differentiability of functional solutions to maximum likelihood type equations solutions of simultaneous equations of the maximum likelihood type or m estimators can be represented as functionals existence and uniqueness of a root in a local region of the parameter space are proved under conditions that are easy to check if only one root of the equation exists the resulting statistical functional is frechet differentiable and robust when several solutions exists conditions on the loss criterion used to select the root for the statistic ensure frechet differentiability an interesting example of a frechet differentiable functional is the solution of the maximum likelihood equations for location and scale parameters in a cauchy distribution the estimator is robust and asymptotically efficient

variance component estimation with model based diagnostics the problem of estimating variance components in mixed linear models has no satisfactory solution for the unbalanced data case further there has been little work on diagnostic methods for assessing the data and the model in this article we propose a new class of unbiased estimators these estimators have simple closed form expressions allowing for smallsample analysis and easy computations the structure of the estimators reveals natural diagnostics for examining the data and the model assumptions the source of negative estimates of variance components is revealed limited efficiency studies indicate that these estimators are comparable to existing estimators the estimators are illustrated by two numerical examples that reveal the importance of the diagnostic analysis

robust statistical analysis of interlaboratory studies summary a common procedure in testing analytical methods is to send a portion of each of a number of samples to each of several laboratories the results of such a study are submitted to statistical analysis to determine the two important variance components in the problem replication error and laboratory bias outliers are relatively common in these data both among laboratory effects and among the residuals this paper presents a method of analysis for interlaboratory studies that is robust to the existence of outliers and long tailed distributions of random effects theoretical considerations as well as a monte carlo study are adduced as support for this new technique

influence diagnostics for linear measurement error models summary influence functions are used to show that extreme observations affect linear measurement error model estimators in directions orthogonal to and along the fitted plane rather than vertically and horizontally as with least squares estimators influence diagnostics patterned after those for least squares and derived for generalized linear models by pregibon 1981 are adapted to assess the influence of extreme observations on linear measurement error model estimators perturbation arguments and asymptotic theory are used to support the use of these diagnostics it is well known that least squares estimators are influenced by extreme observations in the response and predictor variables geometrically these influences are in directions defined by the axis of the response variable and the plane defined by the predictors as shown in 3 measurement error model estimators are affected in directions that are perpendicular to and along the fitted plane thus least squares influence diagnostics that measure the effects of observations in the vertical and the horizontal directions may fail to accurately indicate the true effects of extreme observations on measurement error model estimators kelly 1984 derives an influence function for a structural measurement error model in which the error distribution has finite fourth moments and the error covariance matrix is known up to a multiple with the aid of this influence function diagnostics such as cook s d are defined by analogy to the corresponding least squares equivalents fuller 1987 defines a measurement error model estimator of the hat matrix using estimated predictor variable values he also proposes using diagnostics and plotting techniques analogous to those of least squares with estimated model errors and estimated leverage values from the hat matrix miller 1989 provides partial theoretical justification for the use of these least squares based techniques by showing that the estimated model errors asymptotically are a gaussian process apart from these efforts little research has been conducted on influence diagnostics for measurement error models the derivation of influence diagnostics for measurement error models in this paper proceeds as follows measurement error model estimators are briefly described in 2 a more comprehensive discussion of these estimators is given by fuller 1987 ch 2 in 3 influence function arguments are used to establish the need for measurement error model diagnostics measurement error model estimators are expressed as iteratively reweighted least squares estimators in 4 leading to the adaptation of diagnostics for

on the bounded influence regression estimator of krasker and welsch abstract recently krasker and welsch 1982 considered a class of bounded influence regression estimators they showed that within this class the so called krasker welsch estimator is the only solution to a first order necessary condition for strong optimality that is for minimizing in the sense of positive definiteness the asymptotic covariance matrix however whether any strongly optimal estimator in fact exists remained an open question in this article an example is given where no strongly optimal estimator exists moreover the practical significance of the lack of a strongly optimal estimator is discussed

estimation of variance components with high breakdown point and high efficiency summary for the one way random effects model three noniterative robust variance component estimators are investigated for these estimators block breakdown points measurement breakdown points and efficiency are derived it is shown that the block breakdown points are approximating 50 and that the measurement breakdown points are not smaller than 25 by deriving the asymptotic normal distribution of these estimators we show that two of the new estimators have an asymptotic efficiency greater than 82 and under certain circumstances up to 96 competing estimators have lower breakdown points and lower efficiencies a simulation study shows that the high efficiency also appears for finite samples the use of the estimators for interlaboratory tests is demonstrated in a practical example

some covariance models for longitudinal count data with overdispersion a family of covariance models for longitudinal counts with predictive covariates is presented these models account for overdispersion heteroscedasticity and dependence among repeated observations the approach is a quasi likelihood regression similar to the formulation given by liang and zeger 1986 biometrika 73 13 22 generalized estimating equations for both the covariate parameters and the variance covariance parameters are presented large sample properties of the parameter estimates are derived the proposed methods are illustrated by an analysis of epileptic seizure count data arising from a study of progabide as an adjuvant therapy for partial seizures

robust estimation of multivariate covariance components in many settings such as interlaboratory testing small area estimation in sample surveys and heritability studies investigators are interested in estimating covariance components for multivariate measurements however the presence of outliers can seriously distort estimates obtained using standard procedures such as maximum likelihood we propose a procedure based on m estimation for robustly estimating multivariate covariance components in the presence of outliers the procedure applies to balanced and unbalanced data we present an algorithm for computing the robust estimates and examine the performance of the estimator through a simulation study the estimator is used to find covariance components and identify outliers in a study of variability of egg length and breadth measurements of american coots

an autoregressive linear mixed effects model for the analysis of longitudinal data which include dropouts and show profiles approaching asymptotes we are interested in longitudinal data of a continuous response that show profiles with an initial sharp change and approaching asymptotes for each patient and many patients drop out with a reason related to the response in this paper we focus on a model that assumes a dropout process is missing at random mar in this dropout process we can obtain consistent maximum likelihood estimators as long as both the mean and covariance structures are correctly specified however parsimonious covariance structures for the profiles approaching asymptotes are unclear an autoregressive linear mixed effects model can express the profile with random individual asymptotes we show that this model provides a new parsimonious covariance structure the covariance structure at steady state is compound symmetry and the other elements of the covariance depend on the measurement points in simulation studies the estimate of the asymptote is unbiased in mar dropouts but biased in non ignorable dropouts we also applied this model to actual schizophrenia trial data

cell lineage analysis variance components models for dependent cell populations abstract cells grown in culture can be tracked for several generations and measurements taken on size or age at division and other cell characteristics the observations for the offspring of each cell form a family tree of dependent data such cell lineage data are here modeled as repeated measurements on different family trees arising from individual ancestor cells selected at random from a population of cultured cells the bifurcating autoregression model is embedded in a process that allows for measurement error and variation from tree to tree robust methods are presented that accommodate outliers in this time dependent and branching environment while allowing the statistician to interactively build a variance components model for the process the methodology is illustrated on a substantial data set of 41 trees of emt6 cells with the surprising conclusion that after removing measurement error sister cell lifetimes are nearly identical

projected partial likelihood and its application to longitudinal data summary an estimating equation which we call the projected partial score is introduced for longitudinal data analysis the estimating equation is obtained by projecting the partial likelihood score function onto the vector space spanned by a class of conditionally linear estimating equations we demonstrate that removing certain terms from the projection of the full likelihood score does not alter important inferential properties of the estimating equation and doing so is advantageous in handling missing data and time varying covariates within a prequential frame of reference it is shown that the estimating equation is optimal among the largest collection of estimating equations determined by the conditional moments furthermore the method possesses similar properties to generalised estimating equations in particular the correct conditional variance specification is necessary for efficiency but not for asymptotic consistency and distribution theory

new robust statistical procedures for the polytomous logistic regression models this article derives a new family of estimators namely the minimum density power divergence estimators as a robust generalization of the maximum likelihood estimator for the polytomous logistic regression model based on these estimators a family of wald type test statistics for linear hypotheses is introduced robustness properties of both the proposed estimators and the test statistics are theoretically studied through the classical influence function analysis appropriate real life examples are presented to justify the requirement of suitable robust statistical procedures in place of the likelihood based inference for the polytomous logistic regression model the validity of the theoretical results established in the article are further confirmed empirically through suitable simulation studies finally an approach for the data driven selection of the robustness tuning parameter is proposed with empirical justifications

a nonparametric test of missing completely at random for incomplete multivariate data missing data occur in many real world studies knowing the type of missing mechanisms is important for adopting appropriate statistical analysis procedure many statistical methods assume missing completely at random mcar due to its simplicity therefore it is necessary to test whether this assumption is satisfied before applying those procedures in the literature most of the procedures for testing mcar were developed under normality assumption which is sometimes difficult to justify in practice in this paper we propose a nonparametric test of mcar for incomplete multivariate data which does not require distributional assumptions the proposed test is carried out by comparing the distributions of the observed data across different missing pattern groups we prove that the proposed test is consistent against any distributional differences in the observed data simulation shows that the proposed procedure has the type i error well controlled at the nominal level for testing mcar and also has good power against a variety of non mcar alternatives

multilevel models for survival analysis with random effects a method for modeling survival data with multilevel clustering is described the cox partial likelihood is incorporated into the generalized linear mixed model glmm methodology parameter estimation is achieved by maximizing a log likelihood analogous to the likelihood associated with the best linear unbiased prediction blup at the initial step of estimation and is extended to obtain residual maximum likelihood reml estimators of the variance component estimating equations for a three level hierarchical survival model are developed in detail and such a model is applied to analyze a set of chronic granulomatous disease cgd data on recurrent infections as an illustration with both hospital and patient effects being considered as random only the latter gives a significant contribution a simulation study is carried out to evaluate the performance of the reml estimators further extension of the estimation procedure to models with an arbitrary number of levels is also discussed

case deletion diagnostics for mixed models mixed linear models arise in many areas of application standard estimation methods for mixed models are sensitive to bizarre observations such influential observations can completely distort an analysis and lead to inappropriate actions and conclusions we develop case deletion diagnostics for detecting influential observations in mixed linear models diagnostics for both fixed effects and variance components are proposed computational formulas are given that make the procedures feasible the methods are illustrated using examples

monte carlo em estimation for time series models involving counts abstract the observations in parameter driven models for time series of counts are generated from latent unobservable processes that characterize the correlation structure these models result in very complex likelihoods and even the em algorithm which is usually well suited for problems of this type involves high dimensional integration in this article we discuss a monte carlo em mcem algorithm that uses a markov chain sampling technique in the calculation of the expectation in the e step of the em algorithm we propose a stopping criterion for the algorithm and provide rules for selecting the appropriate monte carlo sample size we show that under suitable regularity conditions an mcem algorithm will with high probability get close to a maximizer of the likelihood of the observed data we also discuss the asymptotic efficiency of the procedure we illustrate our monte carlo estimation method on a time series involving small counts the polio incidence time series previously analyzed by zeger

m estimators and gnostical estimators for indentification of a regression model abstract this paper shows that the gnostical estimators of kovanic 1986 automatica 22 657 674 for identification of a regression model with imprecisely known output and input derived from non statistical considerations are equivalent to the m estimators for a regression model with random carriers and scale parameters estimated separately both the theoretical and practical importance of this fact for identification is established

missing data mechanisms and homogeneity of means and variances covariances unless data are missing completely at random mcar proper methodology is crucial for the analysis of incomplete data consequently methods for effectively testing the mcar mechanism become important and procedures were developed via testing the homogeneity of means and variances covariances across the observed patterns e g kim bentler in psychometrika 67 609 624 2002 little in j am stat assoc 83 1198 1202 1988 the current article shows that the population counterparts of the sample means and covariances of a given pattern of the observed data depend on the underlying structure that generates the data and the normal distribution based maximum likelihood estimates for different patterns of the observed sample can converge to the same values even when data are missing at random or missing not at random although the values may not equal those of the underlying population distribution the results imply that statistics developed for testing the homogeneity of means and covariances cannot be safely used for testing the mcar mechanism even when the population distribution is multivariate normal

robust estimation of variance components outliers may occur with respect to any of the random components in the mixed linear model a procedure for limiting the influence of these outliers on the estimates of the model parameters is described given the variances or estimates of them the model effects are estimated by augmenting the original observations with auxiliary observations that contain the prior information represented by the variances large residuals among either the original or the auxiliary observations are interpreted as outlying random errors or outlying random effects as appropriate and winsorized the robust estimation of the variances is obtained by modifying the defining equations for the restricted maximum likelihood estimates under normality along the lines of huber s proposal 2 a numerical example illustrates the use of the methodology both as a diagnostic and as an estimation tool

bayesian and frequentist predictive inference for the patterns of care studies abstract the patterns of care studies were conducted to determine the quality of care received by cancer patients whose primary treatment modality is radiation therapy in this article we propose and evaluate models which if acceptable permit bayesian and frequentist model based predictive inference for the desired finite population parameters using both hierarchical bayesian and frequentist mixed linear models we describe methodology for making the desired inferences emphasizing the use of transformed random variables finally we compare the frequentist bayes and empirical bayes approaches using data from one of the surveys all three methods produce essentially the same value for the finite population mean the standard empirical bayes and frequentist measures of variability are very much smaller than those derived from the bayesian approach the latter reflecting uncertainty about the values of the scale parameters in the model

residuals for the linear model with general covariance structure a general theory is presented for residuals from the general linear model with correlated errors it is demonstrated that there are two fundamental types of residual associated with this model referred to here as the marginal and the conditional residual these measure respectively the distance to the global aspects of the model as represented by the expected value and the local aspects as represented by the conditional expected value these residuals may be multivariate some important dualities are developed which have simple implications for diagnostics the results are illustrated by reference to model diagnostics in time series and in classical multivariate analysis with independent cases

robust estimation of parameters in a mixed unbalanced model this paper describes a method of robust estimation of shift and scale parameters in a mixed unbalanced interlaboratory model estimators presented result from easily computable fr echet differentiable functionals which enjoy some optimal properties in a small neighborhood of the model a rigorous treatment of their asymptotic behaviour under departures from the model assumptions and a simulation study are given 1 introduction a method of robust analysis of random effects in mixed models is introduced in the paper by rocke 1991 see also iglewicz 1983 and a general treatment of robust estimators of variance components is presented by fellner 1986 rocke proposes estimators of variance components applying robust scale estimators to residuals and laboratory effects in the rst step he uses huber s method to robustly assess location parameters a similar point of view is taken by lischer 1994 who improves breakdown properties of the estimators he also gives a comprehensive practical motivation for the use of robust methods in interlaboratory experiments the above mentioned estimation technique may lead to high dimensional nonlinear equations and it is inconvenient for assessing variability of estimators a possible improvement may be in the simultaneous estimation of xed effects and components of variation where problems of a rigorous asymptotic treatment smoothness of estimators and their optimality become more feasible in a preliminary study by bednarski zmy slony and zontek 1992 it is shown that fr echet differentiability of statistical functionals easily leads to reasonable robust estimators of variance components and treatment fixed effects in a simple interlaboratory model the basic idea there is to construct a smooth fisher consistent functional t for the parameter de 1 a ee t at the model

quasi likelihood regression models with missing covariates summary this paper presents methods to handle missing covariates when the quasi likelihood equations for the complete data are available our suggestion is to replace the functions of the missing data appearing in the quasi likelihood equation with their conditional means given the observed data or with unbiased predictors so that the resulting equation is unbiased we focus on two models one is a random effects model for count data where random effects are treated as missing covariates the other is the overdispersed binomial regression model with partially missing covariates we also investigate the efficiency of the proposed estimates relative to the maximum likelihood estimators

minimum distance methods based on quadratic distances for transforms in simple linear regression models minimum distance estimators and test statistics based on quadratic distances for transforms are introduced in simple linear regression models asymptotic properties of quadratic distance estimators and tests for the error distributions are derived the modified quadratic distance estimator is shown to be efficient provided that the error distribution is symmetric and the derivative of the logarithm of the error density is a linear combination of the transforms

bias correction in generalized linear mixed models with multiple components of dispersion abstract general formulas are derived for the asymptotic bias in regression coefficients and variance components estimated by penalized quasi likelihood pql in generalized linear mixed models with canonical link function and multiple sets of independent random effects easily computed correction matrices result in variance component estimates that have satisfactory asymptotic behavior for small values of the variance components and significantly reduce bias for larger values both first order and second order correction procedures are developed for regression coefficients estimated by pql the methods are illustrated through an analysis of an experiment on salamander matings involving crossed male and female random effects and their properties are evaluated in a simulation study

a regression model for time series of counts summary this paper discusses a model for regression analysis with a time series of counts correlation is assumed to arise from an unobservable process added to the linear predictor in a log linear model an estimating equation approach used for parameter estimation leads to an iterative weighted and filtered least squares algorithm asymptotic properties for the regression coefficients are presented we illustrate the technique with an analysis of trends in u s polio incidence since 1970

asymptotic behavior of general m estimates for regression and scale with random carriers let xini y i be a sequence of independent identically distributed random variables where x i r p and y i r and let r p be an unknown vector such that y i x i u i where u i is independent of x i and has distribution function f u where 0 is an unknown parameter this paper deals with a general class of m estimates of regression and scale defined as solutions of the system sum limits i phi text x i r i x i 0 sum limits i chi r i 0 where r y i xi1 with r p r r and r r this class contains estimators of proposed by huber mallows and krasker and welsch the consistency and asymptotic normality of the general m estimators are proved assuming general regularity conditions on and and assuming the joint distribution of x i y i to fulfill the model only approximately

biometrika centenary theory and general methodology note reprinted in biometrika one hundred years edited by d m titterington and d r cox oxford university press 11 50 reference stat article 2001 002doi 10 1093 biomet 88 1 13view record in web of science record created on 2006 04 21 modified on 2017 05 12

a mean score method for missing and auxiliary covariate data in regression models summary we consider regression analysis when incomplete or auxiliary covariate data are available for all study subjects and in addition for a subset called the validation sample true covariate data of interest have been ascertained the term auxiliary data refers to data not in the regression model but thought to be informative about the true missing covariate data of interest we discuss a method which is nonparametric with respect to the association between available and missing data allows missingness to depend on available response and covariate values and is applicable to both cohort and case control study designs the method previously proposed by flanders greenland 1991 and by zhao lipsitz 1992 is generalised and asymptotic theory is derived our expression for the asymptotic variance of the estimator provides intuition regarding performance of the method optimal sampling strategies for the validation set are also suggested by the asymptotic results