detecting shocks outliers and breaks in time series abstract a single outlier in a regression model can be detected by the effect of its deletion on the residual sum of squares an equivalent procedure is the simple intervention in which an extra parameter is added for the mean of the observation in question similarly for unobserved components or structural time series models the effect of elaborations of the model on inferences can be investigated by the use of interventions involving a single parameter such as trend or level changes because such time series models contain more than one variance the effect of the intervention is measured by the change in individual variances we examine the effect on the estimated parameters of moving various kinds of intervention along the series the horrendous computational problems involved are overcome by the use of score statistics combined with recent developments in filtering and smoothing interpretation of the resulting time series plots of diagnostics is aided by simulation envelopes our procedures illustrated with four example permit keen insights into the fragility of inferences to specific shocks such as outliers and level breaks although the emphasis is mostly on parameter estimation forecast are also considered possible extensions include seasonal adjustment and detrending of series

a note on box cox transformation diagnostics with the box cox transformation one or a few responses can greatly influence the transformation parameter estimate to detect influential observations several diagnostics have been suggested we highlight the importance of multiple case deletion especially when the masking effect is present and derive an analytic expression for tsai and wu s diagnostic

on prediction and the power transformation family summary the power transformation family is often used for transforming to a normal linear model the variance of the regression parameter estimators can be much larger when the transformation parameter is unknown and must be estimated compared to when the transformation parameter is known we consider prediction of future untransformed observations when the data can be transformed to a linear model when the transformation must be estimated the prediction error is not much larger than when the parameter is known

a transformation weighting model for estimating michaelis menten parameters abstract there has been considerable disagreement about how best to estimate the parameters in michaelis menten models this document points out that many fitting methods are based on different stochastic models being weighted least squares estimates after appropriate transformation the authors propose a flexible model which can be used to help determine the proper transformation and choice of weights the method is illustrated by examples keywords nonlinear regression lineweaver burke transformation

heteroscedastic nonlinear regression several parameter estimation methods for dealing with heteroscedasticity in nonlinear regression are described these include variations on ordinary weighted iteratively reweighted extended and generalized least squares some of these variations are new and one of them in particular modified extended iteratively reweighted least squares meirls allows parameters of an assumed heteroscedastic variance model to be estimated with an adjustment for bias due to estimation of the regression parameters the context of the discussion is primarily that of pharmacokinetic type data although an example is given involving chemical reaction data using simulated data from 21 heteroscedastic pharmacokinetic type models some of the methods are compared in terms of mean absolute error and 95 confidence interval coverage from these comparisons meirls and the variations on generalized least squares emerge as the methods of choice

the breakdown points of the mean combined with some rejection rules in the past methods for rejection of outliers have been investigated mostly without regard to the quantitative consequences for subsequent estimation or testing procedures moreover although rejection of outliers with subsequent application of least squares methods is one of the oldest and most widespread classes of robust procedures until recently no comparison was made with other robust methods in this article the simplest situation namely estimation of a location parameter in the potential presence of outliers is treated by means of a monte carlo study this study yields monte carlo variances of the arithmetic mean after rejection of outliers according to several classical and recent formal rules the results are also compared with those for other robust estimators of location parameters it turns out that a simple summary and theoretical explanation of the monte carlo results is provided by the breakdown points of the combined rejection estimation procedures as a by product the concept of br

a simulation experiment for comparing tests of regression transformation abstract an example is given of the use of graphical methods in the analysis of a simulation experiment comments are made on the design of such studies

two graphical displays for outlying and influential observations in regression summary the paper describes two procedures for detecting observations with outlying values either in the response variable or in the explanatory variables in multiple regression these procedures are presented as half normal plots with envelopes derived from simulation in order to avoid overinterpretation of the data analysis of a well known data set leads to the use of a data transformation a simple test for which is commended and to some comments on the relationship with robust regression the widespread availability of sophisticated computer software has made the fitting of multiple regression equations painless but the very ease of these procedures may cause insufficient care to be given to checking and scrutiny of the data transcription punching and data manipulation errors may lead to bad values both of the observations y and of the independent or explanatory variables x it is the purpose of the present paper to describe and exemplify two plots which provide checks against such bad values it is argued that these plots and a test for transformations should accompany any thorough regression analysis bad values of the response or outliers have long been detected by a variety of plots of residuals bad values of the explanatory variables are less easily detected even when they lead to an extreme point in the design space since the fitted equation will pass close to such an influential observation the residual may not be especially large even after allowance has been made for the small variance of the fitted value at such a point to detect such behaviour a quantity is needed which exhibits the dependence of the fitted model on each point or group of points discussions of such quantities are given by hoaglin welsch 1978 and more recently by belsley kuh welsch 1980 2 1 by cook weisberg 1980 and by pregibon 1981 the quantity used here is a modification of a statistic proposed by cook 1977 which is derived in 2 two half normal plots are presented in 3 as an aid to interpretation envelopes to the plots are generated by simulation examples of the plots are given in 4 both for simulated data and for the stack loss data given by brownlee 1965 p 454 transformations of this data set are discussed in 5 the paper concludes with some comments on the relationship with robust

parameter orthogonality and approximate conditional inference on propose une statistique du rapport de vraisemblance construite a partir de la distribution conditionnelle des observations etant donne les estimateurs du maximum de vraisemblance pour les parametres de nuisance

a comparison of asymptotically equivalent test statistics for regression transformation summary a simulation comparison is made of six asymptotically equivalent test statistics for transformation of the response in regression the tests typically have observed sizes up to twice the nominal size a likelihood ratio statistic a score statistic and a wald statistic behave very similarly both at the null hypothesis and in terms of power values of all the statistics are highly positively correlated the greater part of any differences between the statistics is concentrated in their standard deviations to test hypotheses about parametric families of transformation box cox 1964 use a likelihood ratio test more recently several other asymptotically equivalent tests have been suggested in the present paper we present empirical comparisons of six tests in terms of size power and diagnostic efficacy the interesting relevance of these results to small sample comparisons of score wald and likelihood ratio tests is briefly discussed the notation and background of the paper follow those of box cox 1964 a vector of n response values y is linearly related with an initial constant term to p 1 explanatory variables whose values and an initial column of ones are contained in an n xp matrix x the box cox family of transformations indexed by a scalar parameter a is employed to produce the transformed response variable y the aim is to choose a by maximum likelihood so that y follows the linear model y a x 3 e 1 1 where e is a vector of errors which are mutually independent with normal distributions of zero mean and constant variance a2 and 3 is a vector of coefficients the estimate a satisfies the equation

weyl spectra and weyl s theorem two variants of the weyl spectrum are discussed we find for example that if one of them coincides with the browder spectrum then weyl s theorem holds and conversely for isoloid operators

new tools for residual analysis techniques are presented for stepwise selection of observations in order of discrepancy from a linear model and for assigning a p value to each selected observation a general class of criteria for discrepancy is defined and leading members of the class are discussed two ways to define the sequence of p values for a general selection criterion are discussed and their relative advantages are compared the techniques are illustrated on several data sets using only the computationally simpler of the two approaches to p values

adding a variable in generalized linear models the likelihood ratio statistic can be used to determine the significance of an explanatory variable in a generalized linear model in order to obtain such a statistic however we need two sets of iterations for two maximum likelihoods moreover the statistic is not directed to detect influential or outlying observations that affect the importance of the variable considered therefore we develop relatively simple procedures to help the analyst select an appropriate model and detect the effects of such observations on adding a variable into any model two examples are given for illustration

high breakdown point and high efficiency robust estimates for regression a class of robust estimates for the linear model is introduced these estimates called mm estimates have simultaneously the following properties i they are highly efficient when the errors have a normal distribution and ii their breakdown point is 0 5 the mm estimates are defined by a three stage procedure in the first stage an initial regression estimate is computed which is consistent robust and with high breakdown point but not necessarily efficient in the second stage an m estimate of the errors scale is computed using residuals based on the initial estimate finally in the third stage an m estimate of the regression parameters based on a proper redescending psi function is computed consistency and asymptotical normality of the mm estimates assuming random carriers are proved a convergent iterative numerical algorithm is given finally the asymptotic biases under contamination of optimal bounded influence estimates and mm estimates are compared

property ω1 and weyl type theorem abstract in this note we define the property 1 a variant of weyl s theorem and establish for a bounded linear operator defined on a banach space the sufficient and necessary conditions for which property 1 holds by means of the variant of the essential approximate point spectrum 1 in addition the relation between property 1 and hypercyclicity or supercyclicity is discussed

an analysis of transformations revisited rebutted abstract in a paper written in 1964 box and cox described a method for estimating transformations and showed how in suitable cases valuable increases in simplicity and efficiency were possible since that time this technique has enjoyed wide practical use and considerable success however a recent theoretical paper by bickel and doksum 1981 seems to suggest that serious dangers are associated with the employment of this method and speaks of instability and cost of estimation of the transformation these difficulties seem to be associated with 1 examples which common sense would rule out namely situations where the effect of transformation on the data is almost linear so that it is a matter of indifference which transformation is used 2 the idea that it makes sense to state conclusions in terms of a number measured on an arbitrary scale 3 failure to take proper account of the jacobian of the transformation

a note on visualizing response transformations in regression a new graphical method for assessing parametric transformations of the response in linear regression is given we simply regress the response variable y on the predictors find the fitted values and then dynamically plot the transformed response y against those fitted values by varying the transformation parameter until the plot is linear the method can be used also to assess the success of numerical response transformation methods and to discover influential observations modifications using robust estimators can be used as well

critical scaling for monodromy fields the large scale asymptotics of the correlations for a family of two dimensional lattice field theories is calculated at the critical temperature

finite size corrections in random matrix theory and odlyzko s dataset for the riemann zeros odlyzko has computed a dataset listing more than 109 successive riemann zeros starting from a zero number to beyond 1023 this dataset relates to random matrix theory as according to the montgomery odlyzko law the statistical properties of the large riemann zeros agree with the statistical properties of the eigenvalues of large random hermitian matrices moreover keating and snaith and then bogomolny and co workers have used n n random unitary matrices to analyse deviations from this law we contribute to this line of study in two ways first we point out that a natural process to apply to the dataset is to minimize it by deleting each member independently with some specified probability and we proceed to compute empirical two point correlation functions and nearest neighbour spacings in this setting second we show how to characterize the order 1 n 2 correction term to the spacing distribution for random unitary matrices in terms of a second order differential equation with coefficients that are painleve transcendents and where the thinning parameter appears only in the boundary condition this equation can be solved numerically using a power series method in comparison to the riemann zero data accurate agreement is exhibited

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

diagnostics and robust estimation in multivariate data transformations abstract this article presents a method for detecting multivariate outliers that might be distorting the estimation of a transformation to normality it also proposes a robust estimator of the transformation parameter

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

detection of influential observation in linear regression a new measure based on confidence ellipsoids is developed for judging the contribution of each data point to the determination of the least squares estimate of the parameter vector in full rank linear regression models it is shown that the measure combines information from the studentized residuals and the variances of the residuals and predicted values two examples are presented

on the estimation of slope and the identification of outliers in linear regression summary the paper presents a large sample method for estimating the slope parameter in a linear model by minimizing a loss function related to the empirical cumulant generating function of the error distribution a family of estimators indexed by a real parameter is obtained and consistency and asymptotic normality established the optimum member of the family is that which has minimum variance with respect to the parameter this minimization together with a characterization result for the normal distribution leads to a procedure for the identification of outliers with respect to least squares

information from the maximized likelihood function suppose x x1 xj is a random sample of either scalar or vector observations from a density f x c where w e q is partitioned into a set 0 01 or of parameters of direct interest and 4 4 1 4 q of nuisance parameters the log likelihood function l x w i logf xi w is assumed to have continuous first and second derivatives on the parameter space q and to have a global maximum at an interior point a i of q at which the likelihood equations are satisfied the information in the sample is defined as minus the matrix of second partial derivatives of the log likelihood that is

entanglement entropy and berezin toeplitz operators we consider berezin toeplitz operators on compact kahler manifolds whose symbols are characteristic functions when the support of the characteristic function has a smooth boundary we prove a two term weyl law the second term being proportional to the riemannian volume of the boundary as a consequence we deduce the area law for the entanglement entropy of integer quantum hall states another application is for the determinantal processes with correlation kernel the bergman kernels of a positive line bundle we prove that the number of points in a smooth domain is asymptotically normal

eigenvalue distributions of random unitary matrices let u be an n n random matrix chosen from haar measure on the unitary group for a fixed arc of the unit circle let x be the number of eigenvalues of m which lie in the specified arc we study this random variable as the dimension n grows using the connection between toeplitz matrices and random unitary matrices and show that x e x var x 1 2 is asymptotically normally distributed in addition we show that for several fixed arcs i1 im the corresponding random variables are jointly normal in the large n limit

diagnostics and robust estimation when tranforming the regression model and the response in regression analysis the response is often transformed to remove heteroscedasticity and or skewness when a model already exists for the untransformed response then it can be preserved by applying the same transform to both the model and the response this methodology which we call transform both sides has been applied in several recent papers and appears highly useful in practice when a parametric transformation family such as the power transformations is used then the transformation can be estimated by maximum likelihood the maximum likelihood estimator however is very sensitive to outliers in this article we propose diagnostics to indicate cases influential for the transformation or regression parameters we also propose a robust bounded influence estimator similar to the krasker welsch regression estimator

transformation model diagnostics we apply the local influence method of lawrance to assess the effect of the case weights perturbation on the transformation power estimator in the box cox regression model we show that this method is the same as the local influence method proposed by cook furthermore the difference in local influence diagnostics between the case weights perturbation and the constant variances perturbation is examined and the relationship between the localinfluence diagnostic and the deletion diagnostic is studied an example is presented to illustrate local influence diagnostics

transformations and influential observations in minimum sum of absolute errors regression the box cox power transformation model is considered for minimum sum of absolute errors msae regression a long tailed error distribution specifically the laplace distribution is included and could accommodate observations that in the normal case are outlying or unduly influencing a choice of transformation the log likelihood procedure of box and cox 1964 for obtaining the optimal transformation parameter is adapted for laplace errors and msae regression graphical methods for detecting the influence of individual observations on the choice of transformation are described application to examples illustrates that this approach can provide valuable additional information to the data analyst

trace formulas for toeplitz matrices with piecewise continuous symbols asymptotically for certain functions f and for 4 a piecewise continuous function the more general problem has been studied for smooth 4 in 9 and also for many other classes of operators 7 8 lo it of course yields information about determinants by choosing f to be the logarithm and gives information about the behavior of the eigenvalues of t closely related to the matrix t q5 is the toeplitz matrix t hz hz where hp felp s if o k o

robustness aspects of score tests for generalized linear and partially linear regression models robustness aspects of score tests for the null hypothesis of no z effect on the mean response for some given subset of explanatory variables z are explored in the context of regression analysis the model considered here is something else f 1z where is a given transformation of via some link function and f representing the form of the z effect on is monotonic and twice differentiable it is shown that the score test for h 0 1 0 remains unchanged whatever the form chosen for f under the alternative h 1 1 0 this type of robustness is desirable in testing how z might affect the mean response when precise knowledge of the form for f under h 1 is lacking further robust score tests against extreme observations either in the rank test sense or in the hampe huber sense can be achieved for the location family with additive errors as is pointed out in one of the examples by replacing in the tests the score vector based on the observations by that based on the ranks or s

logistic regression diagnostics a maximum likelihood fit of a logistic regression model and other similar models is extremely sensitive to outlying responses and extreme points in the design space we develop diagnostic measures to aid the analyst in detecting such observations and in quantifying their effect on various aspects of the maximum likelihood fit the elements of the fitting process which constitute the usual output parameter estimates standard errors residuals etc will be used for this purpose with a properly designed computing package for fitting the usual maximum likelihood model the diagnostics are essentially free for the asking in particular good data analysis for logistic regression models need not be expensive or time consuming

toeplitz matrices and determinants with fisher hartwig symbols abstract this paper is concerned with toeplitz matrices generated by symbols of the form a t r 1 r t t r 2 r t t r r b t t 1 where t 1 t r are pairwise distinct points on the unit circle b is sufficiently smooth b t 0 t 1 and ind b 0 t t r r is defined as exp i r arg t t r with arg t t r and r r are complex numbers satisfying re r 1 2 re r 1 2 weighted hardy spaces l 2 1 and l 2 2 are defined such that the toeplitz operator t a l 2 1 l 2 2 is bounded and invertible and that the finite section method is applicable to t a considered as acting from l 2 1 onto l 2 2 these results are then applied to prove a conjecture of fisher and hartwig adv in chem phys 15 1968 333 353 which asserts that the determinants of the n n sections of t a are asymptotically equal to g b n n q e a n where q r 2 r 2 and g b e a are certain nonzero constants under quite general conditions on the smoothness of b these constants are completely identified

assessing partial influence in generalized linear models summary a general technique for assessing partial influence in generalized linear models is presented the method consists of a graphical diagnostic display to identify observations that are either influential on a subset of parameters or substantially contributing to a particular model departure application of the method is illustrated with an example in logistic regression this paper is concerned with assessing partial influence for the class of generalized linear models glms described by nelder and wedderburn 1972 we present a unified approach for the detection and assessment of observations that are influential only on one parameter or a subset of parameters throughout the emphasis is on supplementing formal significance tests with effective graphical methods in linear regression the added variable plot and the partial residual plot are two wellknown diagnostic displays that have been suggested for studying the influence of an observation on a single regression parameter explanatory variable see chatterjee and hadi 1986 for a review for glms the corresponding generalizations are given by wang 1985 and landwehr pregibon and shoemaker 1984 the latter proposed a partial residual plot just for binary logistic regression but its glm version can be constructed analogously both types of plots should appear linear if the parameter of interest is significant furthermore it is expected that the plots will give an overall impression of the scatter of the data points individual points that are separated from the rest of the data provide heuristic information about potentially influential observations and may suggest cases for further study however as shown in cook and wang 1983 these plots can fail if important information about the variable arises from observations at leverage points i e points remote in the design space this is because leverage points usually produce relatively small residuals and such points may make only a small contribution to the plot even if they are influential for that variable another difficulty occurs in the situation where a subset of regression parameters are of interest simultaneously since the above plots can handle only one variable parameter at a time the aim of this paper is therefore to derive a reliable diagnostic display for revealing individual observations that are influential on one or more parameters within the framework of glms in section 2 a new diagnostic display called a partial influence plot is described the plot provides additional insight in identifying leverage and influential observations and moreover has general applicability throughout the model checking procedure an example illustrating the usefulness of the partial influence plot is given in section 3 the last section contains further comments on related work on partial influence

from gap probabilities in random matrix theory to eigenvalue expansions we present a method to derive asymptotics of eigenvalues for trace class integral operators acting on a single interval which belongs to the ring of integrable operators its et al 1990 int j mod phys b 4 1003 37 our emphasis lies on the behavior of the spectrum of k as and i is fixed we show that this behavior is intimately linked to the analysis of the fredholm determinant as and in a stokes type scaling regime concrete asymptotic formulae are obtained for the eigenvalues of airy and bessel kernels in random matrix theory

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

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

the analysis of transformed data abstract recently it was suggested bickel and doksum 1981 that when data are used to select a transformation the post transformation analysis of those data may need to be modified considerably from standard form so as to allow for the selection we argue that common sense and the work of box and cox 1964 point to a contrary conclusion our argument is based on considerations of parameter interpretation and subsequent bayesian analysis within the context of fitting normal error linear models numerical examples are used to illustrate the main points

assessing influence in variable selection problems abstract variable selection techniques are often used in combination with multiple linear regression to produce a parsimonious model that fits the data well it is clearly undesirable for the final model to depend strongly on the inclusion of a few influential cases in the data set this article discusses a measure of influence of single cases on the final model based on a similar measure used in ordinary multiple regression when variables are selected objectively deletion of individual cases can strongly affect the choice of model the influence of individual cases on the parameters of the selected model are often assessed as part of the model building process however such conditional measures fail to evaluate the influence of the cases on the variable selection process modern computing environments make it feasible to use an unconditional criterion to determine the influence of each case on the selection procedure a number of examples are discussed to illustrate the differences between these appr

transformations to symmetry and homoscedasticity abstract this article considers transformations in regression to eliminate skewness and heteroscedasticity of the response we work with the transform both sides model where the relationship between the median response and the independent variables has been identified at least tentatively to preserve this relationship the response and the regression model are transformed in the same way extending the work of others for the location parameter case we propose an estimator that eliminates skewness we also develop an estimator to eliminate heteroscedasticity and an estimator that attempts to induce both symmetry and homoscedasticity both and appear new by comparing and we develop a test of the null hypothesis that there exists a transformation to both symmetry and homoscedasticity we study the question when does the estimator of behave in terms of asymptotic variance as if the regression parameter were known and vice versa the results are of use for telling when the optimal estimator of do

on a class of integral operators on a half space with discontinuous symbol this paper is concerned with the spectral asymptotics of the integral operator a on l2 with kernel 2 n ei x y x d where and are regions in rn and is a large parameter more precisely we consider the asymptotics as of tr a for suitable functions f the first order asymptotics are classical and there is a conjectured second order result the conjecture is proved here if one of the regions or is a half space

distribution functions for random variables for ensembles of positive hermitian matrices distribution functions for random variables that depend on a parameter are computed asymptotically for ensembles of positive hermitian matrices the inverse fourier transform of the distribution is shown to be a fredholm determinant of a certain operator that is an analogue of a wiener hopf operator the asymptotic formula shows that up to the terms of order o 1 the distributions are gaussian

counting free fermions on a line a fisher hartwig asymptotic expansion for the toeplitz determinant in the double scaling limit we derive an asymptotic expansion for a wiener hopf determinant arising in the problem of counting one dimensional free fermions on a line segment at zero temperature this expansion is an extension of the result in the theory of toeplitz and wiener hopf determinants known as the generalized fisher hartwig conjecture the coefficients of this expansion are conjectured to obey certain periodicity relations which renders the expansion explicitly periodic in the counting parameter we present two methods to calculate these coefficients and verify the periodicity relations order by order the matrix riemann hilbert problem and the painleve v equation we show that the expansion coefficients are polynomials in the counting parameter and list explicitly first several coefficients

assessment of local influence summary statistical models usually involve some degree of approximation and therefore are nearly always wrong because of this inexactness an assessment of the influence of minor perturbations of the model is important we discuss a method for carrying out such an assessment the method is not restricted to linear regression models and it seems to provide a relatively simple unified approach for handling a variety of problems

the score statistic for regression transformation summary the score statistic for regression transformation using a parametric family is developedfurther a simple formula for the approximate variance matrix of score statistics basedon asymptotic theory is applied to yield the score statistic t s for regression transforma tion this is compared to atkinson s t d statistic for the same problem and yields animprovement to its standardization simulations indicate that the significance levels ofthe proposed score statistic t s are very close to their nominal values some key words power transformation regression score statistic simulation tests for normality 1 introduction the work of box cox 1964 has been the basis of many investigations into theuse of transformations in regression analysis and the current state of the topic is wellsummarized in the monographs of cook weisberg 1982 and atkinson 1985 theprincipal aim of the original work was to transform a response or dependent variable sothat it would satisfy the normality and other standard assumptions of regression analysis the authors used a parametric family of transformations to achieve this and placedemphasis on estimating its parameters in subsequent work andrews 1971 was moreconcerned with testing transformation parameters while atkinson 1973 introducedanother statistic for this purpose from consideration of both andrews s statistic and scoreor c a tests the present paper develops a standardized score statistic in the case of ascalar transformation parameter in effect it is found to provide the standardization ofatkinson s statistic an aspect which atkinson 1973 believed to be theoretically intract able and avoided by using an approximate regression argument

the central limit theorem for local linear statistics in classical compact groups and related combinatorial identities author s soshnikov alexander abstract we discuss clt for the global and local linear statistics of random matrices from classical compact groups the main part of our proofs are certain combinatorial identities much in the spirit of works by kac and spohn

influential observations and outliers in regression statistics offered by cook 1977 and andrews and pregibon 1978 purport to reveal influential observations in a regression analysis detailed examination of these statistics shows that two different types of influence are being measured and this is illustrated with examples derived from a set of data given by mickey dunn and clark 1967 recommendations are given for obtaining the best use of the statistics available

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

least median of squares regression abstract classical least squares regression consists of minimizing the sum of the squared residuals many authors have produced more robust versions of this estimator by replacing the square by something else such as the absolute value in this article a different approach is introduced in which the sum is replaced by the median of the squared residuals the resulting estimator can resist the effect of nearly 50 of contamination in the data in the special case of simple regression it corresponds to finding the narrowest strip covering half of the observations generalizations are possible to multivariate location orthogonal regression and hypothesis testing in linear models

augmented partial residuals a simple way of encouraging a nonlinear effect in a regression model to show itself in a partial residual parres plot component plus residual plot is to include a single quadratic term in the regression it is computationally cheap to do this for each of the independent variables examples show that the resulting set of augmented partial residual plots gives insights that are not available from standard parres or added variable plots

diagnostic regression analysis and shifted power transformations diagnostic displays for outlying and influential observations are reviewed in some examples apparent outliers vanish after a power transformation of the data interpretation of the score statistic for transformations as regression on a constructed variable makes diagnostic methods available for detection of the influence of individual observations on the transformation methods for the power transformation are exemplified and extended to the power transformation after a shift in location for which there are two constructed variables the emphasis is on plots as diagnostic tools

deviance residuals and normal scores plots summary we discuss the use of normal order statistics plots based on deviance residuals to check distributional assumptions in regression models continuous and discrete error distributions are considered as are censored data misspecified error distributions and discrimination between competing models are discussed with an example residual plots to detect inadequacies in normal linear regression models have a long history an example is the normal scores or rankit plot a plot of ordered residuals against normal order statistics which is used to detect outliers and to check distributional assumptions under or overdispersion in such a plot may also indicate a misspecified systematic component of the model the purpose of this paper is to discuss such plots for regressions with nonnormal errors such as generalized linear models for which deviance residuals are commonly used deviance residuals are known to be approximately normal in many cases in this paper we briefly describe the properties of rankit plots based on them for continuous distributions outline analogous results for discrete and censored data and describe a method for discriminating between competing models with different error distributions most roads lead to rome for the normal distribution in the sense that many definitions of residuals are functions of y jt o not so for other distributions for which various

semiparametric box cox power transformation models for censored survival observations the accelerated failure time model specifies that the logarithm of the failure time is linearly related to the covariate vector without assuming a parametric error distribution in this paper we consider the semiparametric box cox transformation model which includes the above regression model as a special case to analyse possibly censored failure time observations inference procedures for the transformation and regression parameters are proposed via a resampling technique prediction of the survival function of future subjects with a specific covariate vector is also provided via pointwise and simultaneous interval estimates all the proposals are illustrated with datasets from two clinical studies

a discriminant rule under transformation we present a new rule for discriminating among continuous populations that are not multivariate normal this new procedure has been motivated by a discriminant problem that appears in particle physics

large deformations of the tracy widom distribution i non oscillatory asymptotics we analyze the left tail asymptotics of deformed tracy widom distribution functions describing the fluctuations of the largest eigenvalue in invariant random matrix ensembles after removing each soft edge eigenvalue independently with probability 1 gamma in 0 1 as gamma varies a transition from tracy widom statistics gamma 1 to classical weibull statistics gamma 0 was observed in the physics literature by bohigas et al phys rev e 79 031117 2009 we provide a description of this transition by rigorously computing the leading order left tail asymptotics of the thinned goe gue and gse tracy widom distributions in this paper we obtain the asymptotic behavior in the non oscillatory region with gamma in 0 1 fixed for the goe gue and gse distributions and gamma uparrow 1 at a controlled rate for the gue distribution this is the first step in an ongoing program to completely describe the transition between tracy widom and weibull statistics as a corollary to our results we obtain a new total integral formula involving the ablowitz segur solution to the second painleve equation

asymptotics for the fredholm determinant of the sine kernel on a union of intervals in the bulk scaling limit of the gaussian unitary ensemble of hermitian matrices the probability that an interval of lengths contains no eigenvalues is the fredholm determinant of the sine kernel frac sin x y pi x y over this interval a formal asymptotic expansion for the determinant ass tends to infinity was obtained by dyson in this paper we replace a single interval of lengths bysj wherej is a union ofm intervals and present a proof of the asymptotics up to second order the logarithmic derivative with respect tos of the determinant equals a constant expressible in terms of hyperelliptic integrals timess plus a bounded oscillatory function ofs zero ifm 1 periodic ifm 2 and in general expressible in terms of the solution of a jacobi inversion problem pluso 1 also determined are the asymptotics of the trace of the resolvent operator which is the ratio in the same model of the probability that the set contains exactly one eigenvalue to the probability that it contains none the proofs use ideas from orthogonal polynomial theory

the x ray problem revisited in this letter we revisit the x ray problem assuming point interaction between the conduction electrons and the instantaneously created core hole the latter s green s function can be represented as a fredholm determinant of certain wiener hopf operators acting on l2 0 t with discontinuous symbols here the symbols are the local conduction electron green s function in the frequency domain and t is the time the core hole spends in the system before removal in this situation the classical theory of singular integral equations usually employed in the literature to compute the large t asymptotics of the fredholm determinant ceased to be applicable a rigorous theory first put forward in the context of operator theory comes into play and universal constants are found in the asymptotics

power transformations and reparameterizations in nonlinear regression models a two stage procedure is proposed to achieve normality and homoscedasticity of the classical error assumptions and to remove nonlinearity of the regression function this systematic approach involves power transformations and reparameterization test statistics are obtained to assess the necessity of power transformations and the validity of homoscedasticity of the errors in addition nonlinearity measures are provided to diagnose the accuracy of linearization based approximate confidence regions for parameters numerical illustrations of the two stage procedure arc presented

more about transformations and influential cases in regression we extend the diagnostic method of cook and wang 1983 for assessing case influence to regression models in which both response and linear predictor are transformed we also suggest a possible improvement of cook and wang s method

powers of large random unitary matrices and toeplitz determinants we study the limiting behavior of tr u k n where u is an n x n random unitary matrix and k n is a natural number that may vary with n in an arbitrary way our analysis is based on the connection with toeplitz determinants the central observation of this paper is a strong szego limit theorem for toeplitz determinants associated to symbols depending on n in a particular way as a consequence of this result we find that for each fixed m n the random variables tru k j n min k j n n j 1 m converge to independent standard complex normals

diagnostic tests for transformations the evidence for transformation of the response in a regression model may sometimes depend crucially on one or a few observations expressions are derived for estimating the effect of deletion of observations on both the score test and the estimate of the transformation parameter the estimates are presented as index plots which are as easy to calculate as the wellknown added variable plots of constructed variables one advantage of the new procedure is that the importance of individual observations is not masked by the presence of leverage a second is that the plots refer directly to score statistics and parameter estimates both quantities of prime inferential importance examples are given for the power transformation and for the shifted power transformation with unknown origin

a statistic for allocating c p to individual cases a subdivision of the c p statistic into individual components for each case is developed and the properties of this statistic are outlined this subdivision depends on both the change in fitted value or residual from the full model to the subset model and the change in leverage but it does not depend directly on the values of the residuals it is useful both as an aid in understanding c p and as a diagnostic procedure

diagnostics for heteroscedasticity in regression summary for the usual regression model without replication we provide a diagnostic test for heteroscedasticity based on the score statistic a graphical procedure to complement the score test is also presented some key u ords influence linear model residual score test diagnostic methods in linear regression are used to examine the appropriateness of assumptions underlying the modelling process and to locate unusual characteristics of the data that may influence conclusions the recent literature on diagnostics is dominated by studies of methods for the detection of influential observations cook weisberg 1982 provide a review diagnostics for the relevance of specific assumptions however have not received the same degree of attention even though these may be of equal importance our purpose here is to provide appropriate diagnostic techniques to aid in an assessment of the validity of the usual assumption of homoscedasticity when little or no replication is present available methods for studying this assumption include both graphical and nongraphical procedures the usual graphical procedure consists of plotting the ordinary least squares residuals against fitted values or an explanatory variable a megaphone shaped pattern is taken as evidence that the variance depends on the quantity plotted on tlle abscissa weisberg 1980 chapter 6 in 3 we suggest several ways in which this

transformation diagnostics for linear models summary an approximate transformation model is used to present an integrated development of tests for transformable deviations from a normal theory linear model particular emphasis is placed on tukey s test for transformable nonadditivity and its relation to tests for variahce heteroscedasticity and error skewness numerical illustrations of the tests are given

a strong central limit theorem for a class of random surfaces this paper is concerned with d 2 dimensional lattice field models with action v nabla phi cdot where v mathbf r d rightarrow mathbf r is a uniformly convex function the fluctuations of the variable phi 0 phi x are studied for large x via the generating function given by g x mu ln langle e mu phi 0 phi x rangle a in two dimensions g x mu partial 2g x mu partial mu 2 is proportional to ln vert x vert the main result of this paper is a bound on g x mu partial 3 g x mu partial mu 3 which is uniform in vert x vert for a class of convex v the proof uses integration by parts following helffer sjostrand and witten and relies on estimates of singular integral operators on weighted hilbert spaces

transformations and influential cases in regression we describe a diagnostic method for assessing the degree to which individual cases and groups of cases influence the box cox likelihood estimate of the transformation parameter for the response variable in linear regression models we compare the method to a method proposed by atkinson 1982 and sketch the extension to explanatory variables we present two examples

diagnostics in transformation and weighted regression we describe a diagnostic method for assessing the influence of an individual case on the transformation power estimator in the box cox regression model and transform both sides regression model carroll and ruppert 1988 we compare the method to those proposed by cook and wang 1983 and hinkley and wang 1988 the new method takes into account the deletion effect on the jacobian of the variable s transformation it provides a more accurate and reliable transformation power estimator we also extend this method to analyze the case influence on the weighted parameter estimators in both the weighted regression model and the transformed and weighted regression model several examples are used to illustrate the results

regression transformation diagnostics using local influence abstract following box and cox 1964 the use of transformations in regression analysis is now common recently there has been emphasis on diagnostic methods for transformation much of which has involved deletion of data cases summaries were given by cook and weisberg 1982 and atkinson 1985 this article obtains diagnostics for the estimated regression parameter of the box cox transformation of the response variable in the linear model instead of deleting cases the more general notions of perturbing assumptions of the model or components of the data are employed as in the local influence approach of cook 1986 the diagnostics then arise from local changes to the transformation parameter estimate caused by small perturbations the case direction in which small perturbations have the greatest effect is the main diagnostic quantity an appeal of the approach is that it allows simultaneous perturbations affecting all data cases not just one at a time deletions it can thus point to groups of in

another note on weyl s theorem weyl s theorem holds for an operator t on a banach space x when the complement in the spectrum of the weyl spectrum coincides with the isolated points of spectrum which are eigenvalues of finite multiplicity this is close to but not quite the same as equality between the weyl spectrum and the browder spectrum which in turn ought to but does not guarantee the spectral mapping theorem for the weyl spectrum of polynomials in t in this note we try to explore these distinctions recall 2 4 6 that a bounded linear operator t e bl x x on a banach space x is fredholm if t x is closed and both t 1 0 and x cl tx are finite dimensional in this case we define the index of t by index t dim t 1 0 dim x t x an operator t e bl x x is called weyl if it is fredholm of index zero and is called browder if it is fredholm of finite ascent and descent equivalently 6 theorem 7 9 3 if t is fredholm and t ai is invertible for sufficiently small a 0 in c the fredholm essential spectrum uess t the weyl spectrum wess t and the browder spectrum wcmm t of t are defined by 0 1 jess t a e c t ai is not fredholm 0 2 wess t a e c t ai is not weyl and 0 3 wecssmm t e c tai is not browder evidently 0 4 jess t c wess t c wcomm t cess t u acc v t where we write acc k for the accumulation points of k c c and av t for the usual spectrum of t if we write iso k k acc k and 0 5 leftr t a e iso 5 t 0 dim tai 1 0 xo for the isolated eigenvalues of finite multiplicity and 6 9 8 3 4 0 6 troo t a5 t wcomm t for the riesz points of t then 6 theorem 9 8 4 with the help of the punctured neighbourhood theorem 0 7 iso v t jess t iso 5 t wess t troo t c 4left t received by the editors december 18 1995 1991 mathematics subject classification primary 47a10

some aspects of analysis of covariance an account from first principles is given of a number of aspects of analysis of covariance six different meanings of analysis of covariance are outlined and the history of this technique is sketched briefly the development of the key formulae from the method of least squares is described and generalizations to other distributions in the exponential family are mentioned special problems of application in randomized experiments and in observational studies are discussed finally the decomposition of regression relations is considered along with components of covariance

a note on jorgensen s iteratively defined statistics summary the newton raphson and fisher scoring iteratively reweighted least squares methods are two fundamental tools for computing parameter estimators and deletion diagnostics jorgensen 1993 proposed a modified method to compute deletion diagnostics in this paper we study the relationship between jorgensen s newton raphson and fisher scoring one step deletion estimators by using the observed and expected fisher information matrices in addition we extend jorgensen s true leverage and approximate leverage from binary regression models to general model settings

more on score tests for transformation in regression summary lawrance s 1987 results concerning tests for transformation in regression are extended to models where response and mean may both be transformed it is also suggested i that the methods of lawrance and of atkinson 1982 lead to statistics with near perfect correlation and ii that observed and expected informations may have considerably different effects as standardizations of the score statistic

residual plots for detecting nonlinearity in generalized linear models the added variable plot partial residual plot and constructed variable plot are three useful diagnostic procedures in assessing the accuracy of a normal linear model the first two have been extended to the generalized linear model for detecting nonlinearity of variables i extend the last one an example is given for illustration

estimation for the box cox transformation model without assuming parametric error distribution box and cox proposed a power transformation for the response variable that yields a linear regression model with normal error and constant variance inference procedures for the regression coefficients and transformation parameter under this model setting have been studied extensively in this article we propose a simple semiparametric estimation method for the box cox transformation model with no specific parametric assumption on the distribution of the error term the resulting estimators are strongly consistent and asymptotically normal their covariance matrix can be estimated through a novel resampling method without involving nonparametric function estimates of the underlying unknown density function for the error term the new proposal is illustrated with a well known dataset in the literature of transformation and its efficiency and robustness are closely examined through numerical studies

prediction and power transformations when the choice of power is restricted to a finite set abstract we study the family of power transformations proposed by box and cox 1964 when the choice of the power parameter is restricted to a finite set r the behavior of the box cox procedure is as anticipated in two extreme cases when the true parameter is an element of r and when is far from r we study the case in which 0 is close to r finding that the resulting methods can be very different from unrestricted maximum likelihood and that inferences may depend on the design the values of the regression parameters and the distance of to r

on the analysis of incomplete spectra in random matrix theory through an extension of the jimbo miwa ueno differential abstract several distribution functions in the classical unitarily invariant matrix ensembles are prime examples of isomonodromic tau functions as introduced by jimbo miwa and ueno jmu in the early 1980s 45 recent advances in the theory of tau functions 41 based on earlier works of b malgrange and m bertola have allowed to extend the original jimbo miwa ueno differential form to a 1 form closed on the full space of extended monodromy data of the underlying lax pairs this in turn has yielded a novel approach for the asymptotic evaluation of isomonodromic tau functions including the exact computation of all relevant constant factors we use this method to efficiently compute the tail asymptotics of soft edge hard edge and bulk scaled distribution and gap functions in the complex wishart ensemble provided each eigenvalue particle has been removed independently with probability 1 0 1

toeplitz and wiener hopf determinants with piecewise continuous symbols abstract the asymptotics for determinants of toeplitz and wiener hopf operators with piecewise continuous symbols are obtained in this paper if w is the wiener hopf operator defined on l2 0 with piecewise continuous symbol having a finite number of discontinuities at r then under appropriate conditions it is shown that det w g r2k where g exp log 0 r 1 2 log r r and k is a completely determined constant an analogous result is obtained for toeplitz operators the main point of the paper is to obtain a result in the wiener hopf case since the toeplitz case had been treated earlier in the toeplitz case it was discovered that one could obtain asymptotics fairly easily for symbols with several singularities if for each singularity one could find a single example of a symbol with a singularity of that kind whose associated asymptotics were known fortunately in the toeplitz case such asymptotics were known the difficulty in the wiener hopf case is that there was not a single singular case where the determinant was explicitly known this problem was overcome by using the fact that wiener hopf determinants when discretized become toeplitz determinants whose entries depend on the size of the matrix no theorem on toeplitz matrices can be applied directly but these theorems are modified to obtain the desired results