Influential observations in GARCH models

This paper examines local influence assessment in generalized autoregressive conditional heteroscesdasticity models with Gaussian and Student-t errors, where influence is examined via the likelihood displacement. The analysis of local influence is discussed under three perturbation schemes: data perturbation, innovative model perturbation and additive model perturbation. For each case, expressions for slope and curvature diagnostics are derived. Monte Carlo experiments are presented to determine the threshold values for locating influential observations. The empirical study of daily returns of the New York Stock Exchange composite index shows that local influence analysis is a useful technique for detecting influential observations; most of the observations detected as influential are associated with historical shocks in the market. Finally, based on this empirical study and the analysis of simulated data, some advice is given on how to use the discussed methodology.     

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 local influence for survival models

The aim of this paper is to develop a Bayesian local influence method (Zhu et al. 2009, submitted) for assessing minor perturbations to the prior, the sampling distribution, and individual observations in survival analysis. We introduce a perturbation model to characterize simultaneous (or individual) perturbations to the data, the prior distribution, and the sampling distribution. We construct a Bayesian perturbation manifold to the perturbation model and calculate its associated geometric quantities including the metric tensor to characterize the intrinsic structure of the perturbation model (or perturbation scheme). We develop local influence measures based on several objective functions to quantify the degree of various perturbations to statistical models. We carry out several simulation studies and analyze two real data sets to illustrate our Bayesian local influence method in detecting influential observations, and for characterizing the sensitivity to the prior distribution and hazard function.     

Local influence analysis for regression models with scale mixtures of skew-normal distributions

The robust estimation and the local influence analysis for linear regression models with scale mixtures of multivariate skew-normal distributions have been developed in this article. The main virtue of considering the linear regression model under the class of scale mixtures of skew-normal distributions is that they have a nice hierarchical representation which allows an easy implementation of inference. Inspired by the expectation maximization algorithm, we have developed a local influence analysis based on the conditional expectation of the complete-data log-likelihood function, which is a measurement invariant under reparametrizations. This is because the observed data log-likelihood function associated with the proposed model is somewhat complex and with Cook's well-known approach it can be very difficult to obtain measures of the local influence. Some useful perturbation schemes are discussed. In order to examine the robust aspect of this flexible class against outlying and influential observations, some simulation studies have also been presented. Finally, a real data set has been analyzed, illustrating the usefulness of the proposed methodology.     

Influence diagnostics for censored regression models with autoregressive errors

Observations collected over time are often autocorrelated rather than independent, and sometimes include observations below or above detection limits (i.e. censored values reported as less or more than a level of detection) and/or missing data. Practitioners commonly disregard censored data cases or replace these observations with some function of the limit of detection, which often results in biased estimates. Moreover, parameter estimation can be greatly affected by the presence of influential observations in the data. In this paper we derive local influence diagnostic measures for censored regression models with autoregressive errors of order p (hereafter, AR(p)‐CR models) on the basis of the Q‐function under three useful perturbation schemes. In order to account for censoring in a likelihood‐based estimation procedure for AR(p)‐CR models, we used a stochastic approximation version of the expectation‐maximisation algorithm. The accuracy of the local influence diagnostic measure in detecting influential observations is explored through the analysis of empirical studies. The proposed methods are illustrated using data, from a study of total phosphorus concentration, that contain left‐censored observations. These methods are implemented in the R package ARCensReg.     

Diagnostic tools in generalized Weibull linear regression models

We propose some statistical tools for diagnosing the class of generalized Weibull linear regression models [A.A. Prudente and G.M. Cordeiro, Generalized Weibull linear models, Comm. Statist. Theory Methods 39 (2010), pp. 3739–3755]. This class of models is an alternative means of analysing positive, continuous and skewed data and, due to its statistical properties, is very competitive with gamma regression models. First, we show that the Weibull model induces ma-ximum likelihood estimators asymptotically more efficient than the gamma model. Standardized residuals are defined, and their statistical properties are examined empirically. Some measures are derived based on the case-deletion model, including the generalized Cook's distance and measures for identifying influential observations on partial F-tests. The results of a simulation study conducted to assess behaviour of the global influence approach are also presented. Further, we perform a local influence analysis under the case-weights, response and explanatory variables perturbation schemes. The Weibull, gamma and other Weibull-type regression models are fitted into three data sets to illustrate the proposed diagnostic tools. Statistical analyses indicate that the Weibull model fitted into these data yields better fits than other common alternative models.     

Outlier Detection in Time Series Models Using Local Influence Method

We propose a new procedure for detecting a patch of outliers or influential observations for autoregressive integrated moving average (ARIMA) model using local influence analysis. It is shown that the dependency aspects of time series data gives rise to masking or smearing effects when the local influence analysis is performed using current perturbation schemes. We suggest a new perturbation scheme to take into account the dependent structure of time series data, and employ the stepwise local influence method to give a diagnostic procedure. We show that the new perturbation scheme can avoid the smearing effects, and the stepwise technique of local influence can successfully deal with masking effects. Various simulation studies are performed to show the efficiency of proposed methodology and a real example is used for illustrations.     

Influence diagnostic analysis in the possibly heteroskedastic linear model with exact restrictions

The local influence method has proven to be a useful and powerful tool for detecting influential observations on the estimation of model parameters. This method has been widely applied in different studies related to econometric and statistical modelling. We propose a methodology based on the Lagrange multiplier method with a linear penalty function to assess local influence in the possibly heteroskedastic linear regression model with exact restrictions. The restricted maximum likelihood estimators and information matrices are presented for the postulated model. Several perturbation schemes for the local influence method are investigated to identify potentially influential observations. Three real-world examples are included to illustrate and validate our methodology.     

Influence diagnostics in nonlinear reproductive dispersion mixed models

The class of nonlinear reproductive dispersion mixed models (NRDMMs) is an extension of nonlinear reproductive dispersion models and generalized linear mixed models. This paper discusses the influence analysis of the model based on Laplace approximation. The equivalence of case-deletion models and mean-shift outlier models in NRDMMs is investigated, and some diagnostic measures are proposed via the case-deletion method. We also investigate the assessment of local influence of various perturbation schemes. The proposed method is illustrated with an example.     

Assessing local cluster influence in generalized linear mixed models

This paper investigates local influence measures for assessing cluster influence in generalized linear mixed models. Several cluster-specific perturbation schemes are considered. The proposed local influence diagnostics are applied to analyse maternity length of inpatient stay data where individual observations are nested within hospitals and the relative performance of hospitals is of interest.     

Local influence on bandwidth estimation for kernel smoothing

This paper examines the problem of assessing local influence on the optimal bandwidth estimation in kernel smoothing based on cross validation. The bandwidth for kernel smoothing plays an important role in the model fitting and is often estimated using the cross-validation criterion. Following the argument of the second-order approach to local influence suggested by Wu and Luo (1993), we develop a new diagnostic statistic to examine the local influence of the observations on the estimation of the optimal bandwidth, where the perturbation may belong to one of three schemes. These are the response perturbation, the perturbation in the explanatory variable, and the case-weight

perturbation. The proposed diagnostic is nonparametric and is capable of identifying influential observations with strong influence on the bandwidth estimation. An example is presented to illustrate the application of the proposed diagnostic, and the usefulness of the nonparametric approach is illustrated in comparison with some other approaches to the assessment of local influence     

Assessment of influence in non-linear measurement error models

We consider the assessment of outliers and influential observation in non-linear measurement error models. Residuals, leverage measures and case-deletiondiagonostics are examined. The method of local influence is also applied to the models. In particular, the perturbation of measurement error variances has been found useful in assessing the adequancy of the model assumptions. A numerical example is given to illustrate the application of the diagonostics.     

On estimation and diagnostics analysis in log-generalized gamma regression model for interval-censored data

The interval-censored survival data appear very frequently, where the event of interest is not observed exactly but it is only known to occur within some time interval. In this paper, we propose a location-scale regression model based on the log-generalized gamma distribution for modelling interval-censored data. We shall be concerned only with parametric forms. The proposed model for interval-censored data represents a parametric family of models that has, as special submodels, other regression models which are broadly used in lifetime data analysis. Assuming interval-censored data, we consider a frequentist analysis, a Jackknife estimator and a non-parametric bootstrap for the model parameters. We derive the appropriate matrices for assessing local influence on the parameter estimates under different perturbation schemes and present some techniques to perform global influence.     

Local influence diagnostics for incomplete overdispersed longitudinal counts

We develop local influence diagnostics to detect influential subjects when generalized linear mixed models are fitted to incomplete longitudinal overdispersed count data. The focus is on the influence stemming from the dropout model specification. In particular, the effect of small perturbations around an MAR specification are examined. The method is applied to data from a longitudinal clinical trial in epileptic patients. The effect on models allowing for overdispersion is contrasted with that on models that do not.     

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.     

A note on influence diagnostics in AR(1) time series models

The purpose of this paper is to develop influence diagnostics for AR(1) models under the innovative and the data perturbation schemes. There are four main contributions. First, we derive analytical expressions for the slope and curvature statistics. Second, we establish a relationship between the slope and curvature showing that the standardised slope and standardised curvature are equal for the innovative perturbation scheme, and these vectors are nearly identical for several values of the autoregressive parameter, for the data perturbation scheme. Third, we present a connection between the influence statistics and the tests for outlier detection. Fourth, for the innovative perturbation scheme, we derive the asymptotic distribution of a new influence statistic, whereas for the data perturbation scheme, the distribution of the influence statistics is obtained via Monte Carlo simulation. We additionally discuss practical guidelines for the use of local influence statistics, which are illustrated on a chemical process data set.     

Local influence assessment in canonical correlation analysis

The local influence method is adapted to canonical correlation analysis for the purpose of investigating the influence of observations. We consider a perturbation based on the empirical distribution function. An illustrative example is given to show the effectiveness of the local influence method for the identification of influential observations.     

Local Influence Under Parameter Constraints

Calculations of local influence curvatures and leverage have been well developed when the parameters are unrestricted. In this article, we discuss the assessment of local influence and leverage under linear equality parameter constraints with extensions to inequality constraints. Using a penalized quadratic function we express the normal curvature of local influence for arbitrary perturbation schemes and the generalized leverage matrix in interpretable forms, which depend on restricted and unrestricted components. The results are quite general and can be applied in various statistical models. In particular, we derive the normal curvature under three useful perturbation schemes for generalized linear models. Four illustrative examples are analyzed by the methodology developed in the article.     

Influence diagnostics in a vector autoregressive model

In this paper, we use a likelihood approach and the local influence method introduced by Cook [Assessment of local influence (with discussion). J Roy Statist Soc Ser B. 1986;48:133–149] to study a vector autoregressive (VAR) model. We present the maximum likelihood estimators and the information matrix. We establish the normal curvature and slope diagnostics for the VAR model under several perturbation schemes and use the Monte Carlo method to obtain benchmark values for determining the influence of directional diagnostics and possible influential observations. An empirical study using the VAR model to fit real data of monthly returns of IBM and S&P500 index illustrates the effectiveness of our proposed diagnostics.     

Heteroscedastic log-exponentiated Weibull regression model

We introduce a new class of heteroscedastic log-exponentiated Weibull (LEW) regression models. The class of regression models can be applied to censored data and be used more effectively in survival analysis. Maximum likelihood estimation of the model parameters with censored data as well as influence diagnostics for the new regression model is investigated. For different parameter settings, sample sizes and censoring percentages, various simulation studies are performed and compared to the performance of the heteroscedastic LEW regression model. The normal curvatures for studying local influence are derived under various perturbation schemes. An empirical application to a real data set is provided to illustrate the usefulness of the new class of heteroscedastic regression models.