Pena's statistic for the Liu regression

In fitting regression model, one or more observations may have substantial effects on estimators. These unusual observations are precisely detected by a new diagnostic measure, Pena's statistic. In this article, we introduce a type of Pena's statistic for each point in Liu regression. Using the forecast change property, we simplify the Pena's statistic in a numerical sense. It is found that the simplified Pena's statistic behaves quite well as far as detection of influential observations is concerned. We express Pena's statistic in terms of the Liu leverages and residuals. The normality of this statistic is also discussed and it is demonstrated that it can identify a subset of high Liu leverage outliers. For numerical evaluation, simulated studies are given and a real data set has been analysed for illustration.     

Mean shift and influence measures in linear measurement error models with stochastic linear restrictions

We present influence diagnostics for linear measurement error models with stochastic linear restrictions using the corrected likelihood of Nakamura in 1990. The case deletion and mean shift outlier models are developed to identify outlying and influential observations. We derive a corrected score test statistic for outlier detection based on mean shift outlier models. The analogs of Cook's distance and likelihood distance are proposed to determine influential observations based on case deletion models. A parametric bootstrap procedure is used to obtain empirical distributions of the test statistics and a simulation study has been used to evaluate the performance of the proposed estimators based on the mean squares error criterion and the score test statistic. Finally, a numerical example is given to illustrate the theoretical results.     

Outlier detection in linear models: a comparative study in simple linear regression

Five widely used test statistics for detecting outliers and influential observations were studied using Monte Carlo method . The test statistic based on Studentized residuals, with critical values given by Tietjen, Moore and Beckman (1973), appears to be the best procedure for detecting a single outlier in simple linear regression.     

The Prediction Sum of Squares as a General Measure for Regression Diagnostics

Statistics that usually accompany the regression model do not provide insight into the quality of the data or the potential influence of the individual observations on the estimates. In this study, the Q² statistic is used as a criterion for detecting influential observations or outliers. The statistic is derived from the jackknifed residuals, the squared sum of which is generally known as the prediction sum of squares or PRESS. This article compares R ² with Q² and suggests that the latter be used as part of the data-quality check. It is shown, for two separate data sets obtained from regional cost of living and U.S. food industry studies, that in the presence of outliers the Q² statistic can be negative, because it is sensitive to the choice of regressors and the inclusion of influential observations. Once the outliers are dropped from the sample, the discrepancy between Q² and R ² values is negligible.     

Influence measures in affine combination type regression

The detection of outliers and influential observations has received a great deal of attention in the statistical literature in the context of least-squares (LS) regression. However, the explanatory variables can be correlated with each other and alternatives to LS come out to address outliers/influential observations and multicollinearity, simultaneously. This paper proposes new influence measures based on the affine combination type regression for the detection of influential observations in the linear regression model when multicollinearity exists. Approximate influence measures are also proposed for the affine combination type regression. Since the affine combination type regression includes the ridge, the Liu and the shrunken regressions as special cases, influence measures under the ridge, the Liu and the shrunken regressions are also examined to see the possible effect that multicollinearity can have on the influence of an observation. The Longley data set is given illustrating the influence measures in affine combination type regression and also in ridge, Liu and shrunken regressions so that the performance of different biased regressions on detecting and assessing the influential observations is examined.     

Influential Observations in Time Series

This article studies how to identify influential observations in univariate autoregressive integrated moving average time series models and how to measure their effects on the estimated parameters of the model. The sensitivity of the parameters to the presence of either additive or innovational outliers is analyzed, and influence statistics based on the Mahalanobis distance are presented. The statistic linked to additive outliers is shown to be very useful for indicating the robustness of the fitted model to the given data set. Its application is illustrated using a relevant set of historical data.     

Rejoinder: "Comment on 'a new Statistic for Testing Suspected Outliers'"

The values of the power of Tiku's (1975) T statistic for testing outliers in normal samples are evaluated. The statistic T is shown to be more powerful than other comparable statistics under Tiku's outlier model, although slightly less powerful under Dixon's (1950) contamination model.     

On detection of outliers in symmetric normal models

Extensions of recent results for detection of mean slippage type outliers from i.i.d. multivariate normal and elliptically symmetric distributions are made to symmetric case, that is, when the observations are equicorrelated. The main tool used is Wijsman's (1967) representation theorem. The results obtained can be viewed as a robustness property of the use of Mardia's multivariate kurtosis as a locally optimal test statistic to detect outliers against equicorrelated distributions.     

The Relationship Between the T2 Statistic and the Influence Function

Hotelling's T² statistic has many applications in multivariate analysis. In particular, it can be used to measure the influence that a particular observation vector has on parameter estimation. For example, in the bivariate case, there exists a direct relationship between the ellipse generated using a T² statistic for individual observations and the hyperbolae generated using Hampel's influence function for the corresponding correlation coefficient. In this paper, we jointly use the components of an orthogonal decomposition of the T² statistic and some influence functions to identify outliers or influential observations. Since the conditional components in the T² statistic are related to the possible changes in the correlation between a variable and a group of other variables, we consider the theoretical influence functions of the correlations and multiple correlation coefficients. Finite-sample versions of these influence functions are used to find the estimated influence function values.     

Stepwise local influence in generalized autoregressive conditional heteroskedasticity models

Detection of outliers or influential observations is an important work in statistical modeling, especially for the correlated time series data. In this paper we propose a new procedure to detect patch of influential observations in the generalized autoregressive conditional heteroskedasticity (GARCH) model. Firstly we compare the performance of innovative perturbation scheme, additive perturbation scheme and data perturbation scheme in local influence analysis. We find that the innovative perturbation scheme give better result than other two schemes although this perturbation scheme may suffer from masking effects. Then we use the stepwise local influence method under innovative perturbation scheme to detect patch of influential observations and uncover the masking effects. The simulated studies show that the new technique can successfully detect a patch of influential observations or outliers under innovative perturbation scheme. The analysis based on simulation studies and two real data sets show that the stepwise local influence method under innovative perturbation scheme is efficient for detecting multiple influential observations and dealing with masking effects in the GARCH model.     

Bayesian influence diagnostics in radiocarbon dating

Linear models constitute the primary statistical technique for any experimental science. A major topic in this area is the detection of influential subsets of data, that is, of observations that are influential in terms of their effect on the estimation of parameters in linear regression or of the total population parameters. Numerous studies exist on radiocarbon dating which propose a value consensus and remove possible outliers after the corresponding testing. An influence analysis for the value consensus from a Bayesian perspective is developed in this article.     

Exploring Cook's Statistic Graphically

An index plot of Cook's statistic is frequently used to highlight influential observations. In this article we illustrate how enhanced higher dimensional plots of Cook's statistic can provide further useful information about sets of influential observations. We provide examples using normal and generalized linear models.     

Distributions of test statistics for multiple outliers in exponential samples

Null as well as alternative distributions of two types of statistics used to test for multiple outliers in exponential samples are obtained. Of these two types one is based on the ratio of sum of the observations suspected to be outliers to sum of the sample observations and the other is Dixon's type. Powers of the tests based on these statistics are compared.

     

Jackknife-After-Bootstrap as Logistic Regression Diagnostic Tool

In this study, we propose using Jackknife-after-Bootstrap (JaB) method to detect influential observations in binary logistic regression model. Performance of the proposed method has been compared with the traditional method for standardized Pearson residuals, Cook's distance, change in the Pearson chi-square and change in the deviance statistics by both real world examples and simulation studies. The results reveal that under the various scenarios considered in this article, JaB performs better than the traditional method and is more robust to masking effect especially for Cook's distance.     

Goodness-of-fit statistics for log-link regression models

The use of log binomial regression, regression on binary outcomes using a log link, is becoming increasingly popular because it provides estimates of relative risk. However, little work has been done on model evaluation. We used simulations to compare the performance of five goodness-of-fit statistics applied to different models in a log binomial setting, namely the Hosmer–Lemeshow, the normalized Pearson chi-square, the normalized unweighted sum of squares, Le Cessie and van Howelingen's statistic based on smoothed residuals and the Hjort–Hosmer test. The normalized Pearson chi-square was unsuitable as the rejection rate depended also on the range of predicted probabilities. The Le Cessie and van Howelingen's test statistic had poor sampling properties when evaluating a correct model and was also considered to be unsuitable in this context. The performance of the remaining three statistics was comparable in most simulations. However, using real data the Hjort–Hosmer outperformed the other two statistics.     

Detection of outliers in familial data

Procedures for detection of outliers in familial data is given for mean-slippage and dispersion-slippage model of outliers for equal and unequal family sizes. The distributions of the test statistics are derived and Bonferroni's bounds for the values of significant probabilities are given.     

Detection of outliers in growth curve models

The growth curve model introduced by Potthoff and Roy (1964) is a general statistical model which includes as special cases regression models and both univariate and multivariate analysis of variance models. In this paper, we discuss procedures for detection of outliers in growth curve models for mean-slippage and dispersion-slippage outlier model. The distributions of the test statistics are discussed and the values of significant probabilities are given using Bonferronl's bounds. Some simulation results are also presented.     

Procedures for the identification of multiple influential observations in linear regression

Since the seminal paper by Cook (1977) in which he introduced Cook's distance, the identification of influential observations has received a great deal of interest and extensive investigation in linear regression. It is well documented that most of the popular diagnostic measures that are based on single-case deletion can mislead the analysis in the presence of multiple influential observations because of the well-known masking and/or swamping phenomena. Atkinson (1981) proposed a modification of Cook's distance. In this paper we propose a further modification of the Cook's distance for the identification of a single influential observation. We then propose new measures for the identification of multiple influential observations, which are not affected by the masking and swamping problems. The efficiency of the new statistics is presented through several well-known data sets and a simulation study.     

An investigation of lag identification tools for vector nonlinear time series

Exploratory methods for determining appropriate lagged vsrlables in a vector nonlinear time series model are investigated. The first is a multivariate extension of the R statistic considered by Granger and Lin (1994), which is based on an estimate of the mutual information criterion. The second method uses Kendall's ρ and partial ρ statistics for lag determination. The methods provide nonlinear analogues of the autocorrelation and partial autocorrelation matrices for a vector time series. Simulation studies indicate that the R statistic reliabiy identifies appropriate lagged nonlinear moving average terms in a vector time series, while Kendall's ρ and partial ρ statistics have some power in identifying appropirate lagged nonlinear moving average and autoregressive terms, respectively, when the nonlinear relationship between lagged variables is monotonic. For illustration, the methods are applied to set of annual temperature and tree ring measurements at Campito Mountain In California.     

Outlier Detection in a Circular Regression Model Using COVRATIO Statistic

In this article, we model the relationship between two circular variables using the circular regression models, to be called JS circular regression model, which was proposed by Jammalamadaka and Sarma (1993). The model has many interesting properties and is sensitive enough to detect the occurrence of outliers. We focus our attention on the problem of identifying outliers in this model. In particular, we extend the use of the COVRATIO statistic, which has been successfully used in the linear case for the same purpose, to the JS circular regression model via a row deletion approach. Through simulation studies, the cut-off points for the new procedure are obtained and its power of performance is investigated. It is found that the performance improves when the resulting residuals have small variance and when the sample size gets larger. An example of the application of the procedure is presented using a real dataset.