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 masking breakdown point of multivariate outlier identification rules abstract in this article we consider simultaneous outlier identification rules for multivariate data generalizing the concept of so called outlier identifiers as presented by davies and gather for the case of univariate samples such multivariate outlier identifiers are based on estimators of location and covariance therefore it seems reasonable that characteristics of the estimators influence the behavior of outlier identifiers several authors mentioned that using estimators with low finite sample breakdown point is not recommended for identifying outliers to give a formal explanation we investigate how the finite sample breakdown points of estimators used in these identification rules influence the masking behavior of the rules

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

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

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

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

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

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

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

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

unified treatment for dealing with auxiliary conditions in blade dynamics use of lagrange multipliers is shown here to lead to a convenient method for the numerical analysis of blade dynamics in which a variety of auxiliary conditions may be imposed general equations are derived that can deal with blades having virtually all of the possible combinations of boundary conditions the method is combined with a generalized coordinates approach to obtain a unified and efficient formulation that is applicable to both linear and nonlinear situations

fourier transform and valuations abstract it is proved that the classical fourier transform is a continuous positively gl n covariant and logarithmic translation covariant valuation moreover the fourier transform is shown to be the only such valuation on complex valued functions

a class of locally and globally robust regression estimates abstract we present a new class of regression estimates called generalized estimates these estimates are defined by minimizing the scale of the weighted residuals with weights that penalize high leverage observations like the estimates the generalized estimates utilize for their definition two loss functions 1 and 2 which together with the weights can be chosen to achieve simultaneously high breakdown point finite gross error sensitivity and high efficiency we recommend however choosing these functions so as to control the bias behavior of the estimate for a large range of possible contaminations and then boosting the efficiency by a simple least squares reweighting step the generalized estimate with loss functions 1 and 2 is related to the hill ryan gm estimate with a loss function which is a linear combination of 1 and r in fact both estimates have the same influence function and asymptotic distribution under the central model we show that a certain generalized estima

k clustering as a detection tool for influential subsets in regression this article describes a new methodology for the detection of influential subsets in regression the method is based on an adaptation of computational and graphical techniques used in cluster analysis and makes use of some general properties of influential subsets but it is independent of any specific measure of influence for small to moderate data sets the proposed method is computationally efficient compared to existing search methods and it identifies subset candidates that merit attention according to some or all measures of joint influence that have appeared in the literature to date examples are given illustrating the method applied to two data sets previously analyzed in published studies

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

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

robust response transformations based on optimal prediction nonlinear regression problems can often be reduced to linearity by transforming the response variable e g using the box cox family of transformations the classic estimates of the parameter defining the transformation as well as of the regression coefficients are based on the maximum likelihood criterion assuming homoscedastic normal errors for the transformed response these estimates are nonrobust in the presence of outliers and can be inconsistent when the errors are nonnormal or heteroscedastic this article proposes new robust estimates that are consistent and asymptotically normal for any unimodal and homoscedastic error distribution for this purpose a robust version of conditional expectation is introduced for which the prediction mean squared error is replaced with an m scale this concept is then used to develop a nonparametric criterion to estimate the transformation parameter as well as the regression coefficients a finite sample estimate of this criterion based on a robust version of s

second order approach to local influence to examine global and local influence and their relations for regression models we study the perturbation formed surface of a variable such as the maximum likelihood estimate of a parameter by evaluating the second derivative or curvature of the surface the corresponding slope of this surface is related to the curvature of the likelihood displacement surface of cook examples show that this approach with the aid of plots is helpful not only to discover influential cases including those hidden in an individual global sense but also to understand the nature of influence

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

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

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

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

biased and unbiased cross validation in density estimation abstract nonparametric density estimation requires the specification of smoothing parameters the demands of statistical objectivity make it highly desirable to base the choice on properties of the data set in this article we introduce some biased cross validation criteria for selection of smoothing parameters for kernel and histogram density estimators closely related to one investigated in scott and factor 1981 these criteria are obtained by estimating l 2 norms of derivatives of the unknown density and provide slightly biased estimates of the average squared l 2 error or mean integrated squared error these criteria are roughly the analog of wahba s 1981 generalized cross validation procedure for orthogonal series density estimators we present the relationship of the biased cross validation procedure to the least squares cross validation procedure which provides unbiased estimates of the mean integrated squared error both methods are shown to be based on u statistics we compare the two metho