On local influence for elliptical linear models

The local influence method plays an important role in regression diagnostics and sensitivity analysis. To implement it, we need the Delta matrix for the underlying scheme of perturbations, in addition to the observed information matrix under the postulated model. Galea, Paula and Bolfarine (1997) has recently given the observed information matrix and the Delta matrix for a scheme of scale perturbations and has assessed of local influence for elliptical linear regression models. In the present paper, we consider the same elliptical linear regression models. We study the schemes of scale, predictor and response perturbations, and obtain their corresponding Delta matrices, respectively. To illustrate the methodology for assessment of local influence for these schemes and the implementation of the obtained results, we give an example.     

Model influence functions based on mixtures

Influence functions are considered as diagnostics for model departures in parametric Bayesian inference. A baseline model density is expressed as a mixture; then the mixing distribution is perturbed. This is designed to engender perturbations which are plausible a priori. The influence of perturbations is measured for both Bayes estimates and their associated posterior expected losses. To assess the plausibility of perturbations a posteriori, an additional influence function is constructed for the Bayes factor comparing the perturbed and baseline models. The effect of perturbation on various estimands is incorporated in the analysis.     

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.     

Local influence in ridge semiparametric models

ABSTRACT

Ridge penalized least-squares estimators has been suggested as an alternative to the minimum penalized sum of squares estimates in the presence of collinearity among the explanatory variables in semiparametric regression models (SPRMs). This paper studies the local influence of minor perturbations on the ridge estimates in the SPRM. The diagnostics under the perturbation of ridge penalized sum of squares, response variable, explanatory variables and ridge parameter are considered. Some local influence diagnostics are given. A Monte Carlo simulation study and a real example are used to illustrate the proposed perturbations.     

Sensitivity analysis of partially linear models with response missing at random

This article investigates case-deletion influence analysis via Cook’s distance and local influence analysis via conformal normal curvature for partially linear models with response missing at random. Local influence approach is developed to assess the sensitivity of parameter and nonparametric estimators to various perturbations such as case-weight, response variable, explanatory variable, and parameter perturbations on the basis of semiparametric estimating equations, which are constructed using the inverse probability weighted approach, rather than likelihood function. Residual and generalized leverage are also defined. Simulation studies and a dataset taken from the AIDS Clinical Trials are used to illustrate the proposed methods.     

Influence Diagnostics for Simultaneous Equations Models

This paper presents influence diagnostics for simultaneous equations models. It proposes residuals, leverage and other influence measures. A missing data method is adopted to minimize the masking effect due to case deletions. The assessment of local influence is also considered. The paper shows how to evaluate the effects that perturbations to the endogenous variables, predetermined variables and case weights may have on the parameter estimates. The diagnostics are illustrated with two examples.     

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 IN NONLINEAR REPRODUCTIVE DISPERSION MODELS

Nonlinear reproductive dispersion models (NRDM, Jorgensen 1997) include a wider range of distributions and nonlinear models such as the possibility of correlated errors and nonlinear hypotheses dropping the exponential family assumption. Based on the generalized Cook distance and the conformal normal curvature of Poon & Poon (1999), local influence of minor perturbations on the data set is investigated for NRDM. Two examples are used to illustrate our results.     

Diagnostics in elliptical regression models with stochastic restrictions applied to econometrics

We propose an influence diagnostic methodology for linear regression models with stochastic restrictions and errors following elliptically contoured distributions. We study how a perturbation may impact on the mixed estimation procedure of parameters in the model. Normal curvatures and slopes for assessing influence under usual schemes are derived, including perturbations of case-weight, response variable, and explanatory variable. Simulations are conducted to evaluate the performance of the proposed methodology. An example with real-world economy data is presented as an illustration.     

Influence Assessment in an Heteroscedastic Errors-in-Variables Model

The main goal of this article is to consider influence assessment in models with error-prone observations and variances of the measurement errors changing across observations. The techniques enable to identify potential influential elements and also to quantify the effects of perturbations in these elements on some results of interest. The approach is illustrated with data from the WHO MONICA Project on cardiovascular disease.     

Local Influence Analysis for The Ridge Regression Under Stochastic Linear Restrictions

In this article, we assess the local influence for the ridge regression of linear models with stochastic linear restrictions in the spirit of Cook by using the log-likelihood of the stochastic restricted ridge regression estimator. The diagnostics under the perturbations of constant variance, responses and individual explanatory variables are derived. We also assess the local influence of the stochastic restricted ridge regression estimator under the approach suggested by Billor and Loynes. At the end, a numerical example on the Longley data is given to illustrate the theoretic results.     

Quantile Regression for Location‐Scale Time Series Models with Conditional Heteroscedasticity

This paper considers quantile regression for a wide class of time series models including autoregressive and moving average (ARMA) models with asymmetric generalized autoregressive conditional heteroscedasticity errors. The classical mean‐variance models are reinterpreted as conditional location‐scale models so that the quantile regression method can be naturally geared into the considered models. The consistency and asymptotic normality of the quantile regression estimator is established in location‐scale time series models under mild conditions. In the application of this result to ARMA‐generalized autoregressive conditional heteroscedasticity models, more primitive conditions are deduced to obtain the asymptotic properties. For illustration, a simulation study and a real data analysis are provided.     

Diagnostics for Autocorrelated Regression Models

This paper discusses the local influence approach to the linear regression model with AR(1) errors. Diagnostics for the autocorrelation models and for the autocorrelation coefficient only are proposed and developed respectively, when simultaneous perturbations of the response vector are allowed. Furthermore, the direction of maximum curvature of local influence analysis is shown to be exactly the same as that in Tsai & Wu (1992) when only the autocorrelation coefficient is of special interest.     

Empirical likelihood for linear regression models under imputation for missing responses

The authors study the empirical likelihood method for linear regression models. They show that when missing responses are imputed using least squares predictors, the empirical log‐likelihood ratio is asymptotically a weighted sum of chi‐square variables with unknown weights. They obtain an adjusted empirical log‐likelihood ratio which is asymptotically standard chi‐square and hence can be used to construct confidence regions. They also obtain a bootstrap empirical log‐likelihood ratio and use its distribution to approximate that of the empirical log‐likelihood ratio. A simulation study indicates that the proposed methods are comparable in terms of coverage probabilities and average lengths of confidence intervals, and perform better than a normal approximation based method.     

Local influence in multilevel models

The authors study the local influence of observations in multilevel regression models. To this end, they perturb simultaneously the variances, responses and design matrix. To measure the local change caused by these perturbations, they use generalized Cook statistics for the fixed and random parameter estimates. Closed form local influence measures also allow them to assess the joint influence of various observations. They suggest a simple computation method and illustrate their results using two examples.     

联合均值与方差模型的统计诊断

研究了联合均值与方差模型,考虑了基于数据删除模型的参数估计和统计诊断,比较删除模型与未删除模型相应统计量之间的差异。首次提出了基于联合均值与方差模型的诊断统计量和局部影响分析。通过模拟研究和实例分析,给出了不同的诊断统计量来判别异常点或强影响点,研究表明提出的理论和方法是有用和有效的。     

Conformal normal curvature and assessment of local influence

In 1986, R. D. Cook proposed differential geometry to assess local influence of minor perturbations of statistical models. We construct a conformally invariant curvature, the conformal normal curvature, for the same purpose. This curvature provides a measure of local influence ranging from 0 to 1, with objective bench-marks to judge largeness. We study various approaches to using the conformal normal curvature and the relationships between these approaches.     

Locally optimal designs for binary dose‐response models

In this article, we consider the problem of seeking locally optimal designs for nonlinear dose‐response models with binary outcomes. Applying the theory of Tchebycheff Systems and other algebraic tools, we show that the locally D‐, A‐, and c‐optimal designs for three binary dose‐response models are minimally supported in finite, closed design intervals. The methods to obtain such designs are presented along with examples. The efficiencies of these designs are also discussed. The Canadian Journal of Statistics 46: 336–354; 2018 © 2018 Statistical Society of Canada     

Functional quasi-likelihood regression models with smooth random effects

Summary. We propose a class of semiparametric functional regression models to describe the influence of vector-valued covariates on a sample of response curves. Each observed curve is viewed as the realization of a random process, composed of an overall mean function and random components. The finite dimensional covariates influence the random components of the eigenfunction expansion through single-index models that include unknown smooth link and variance functions. The parametric components of the single-index models are estimated via quasi-score estimating equations with link and variance functions being estimated nonparametrically. We obtain several basic asymptotic results. The functional regression models proposed are illustrated with the analysis of a data set consisting of egg laying curves for 1000 female Mediterranean fruit-flies (medflies).     

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.