Statistical analysis of controlled calibration model with replicates

In this work, we generalize the controlled calibration model by assuming replication on both variables. Likelihood-based methodology is used to estimate the model parameters and the Fisher information matrix is used to construct confidence intervals for the unknown value of the regressor variable. Further, we study the local influence diagnostic method which is based on the conditional expectation of the complete-data log-likelihood function related to the EM algorithm. Some useful perturbation schemes are discussed. A simulation study is carried out to assess the effect of the measurement error on the estimation of the parameter of interest. This new approach is illustrated with a real data set.     

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.     

Local influence assessment in the growth curve model with unstructured covariance

In this paper, a local influence approach is employed to assess adequacy of the growth curve model with an unstructured covariance, based on likelihood displacement. The Hessian matrix of the model is investigated in detail under an abstract perturbation scheme. For illustration, covariance-weighted perturbation is discussed and used to analyze two real-life biological data sets, which show that the criteria presented in this article are useful in practice.     

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 analysis of additive mixed-effects nonlinear regression models via EM algorithm

This paper presents a unified method for influence analysis to deal with random effects appeared in additive nonlinear regression models for repeated measurement data. The basic idea is to apply the Q-function, the conditional expectation of the complete-data log-likelihood function obtained from EM algorithm, instead of the observed-data log-likelihood function as used in standard influence analysis. Diagnostic measures are derived based on the case-deletion approach and the local influence approach. Two real examples and a simulation study are examined to illustrate our methodology.     

On estimation and local influence analysis for measurement errors models under heavy-tailed distributions

Scale mixtures of normal distributions form a class of symmetric thick-tailed distributions that includes the normal one as a special case. In this paper we consider local influence analysis for measurement error models (MEM) when the random error and the unobserved value of the covariates jointly follow scale mixtures of normal distributions, providing an appealing robust alternative to the usual Gaussian process in measurement error models. In order to avoid difficulties in estimating the parameter of the mixing variable, we fixed it previously, as recommended by Lange et al. (J Am Stat Assoc 84:881–896, 1989) and Berkane et al. (Comput Stat Data Anal 18:255–267, 1994). The local influence method is used to assess the robustness aspects of the parameter estimates under some usual perturbation schemes. However, as the observed log-likelihood associated with this model involves some integrals, Cook’s well–known approach may be hard to apply to obtain measures of local influence. Instead, we develop local influence measures following the approach of Zhu and Lee (J R Stat Soc Ser B 63:121–126, 2001), which is based on the EM algorithm. Results obtained from a real data set are reported, illustrating the usefulness of the proposed methodology, its relative simplicity, adaptability and practical usage.     

Local influence analysis in the growth curve model with Rao's simple covariance structure

In this paper we discuss the likelihood-based local influence in a growth curve model with Rao's simple covariance structure. Under an abstract perturbation, the Hessian matrix is provided in which the eigenvector corresponding to the maximum absolute eigenvalue is used to assess the influence of observations. Specifically, we employ covariance-weighted perturbation to demonstrate the use of the proposed approach. A practical example is analysed using the proposed local influence approach.     

Log-Weibull extended regression model: Estimation,sensitivity and residual analysis

The failure rate function commonly has a bathtub shape in practice. In this paper we discuss a regression model considering new Weibull extended distribution developed by Xie et al. (2002) that can be used to model this type of failure rate function. Assuming censored data, we discuss parameter estimation: maximum likelihood method and a Bayesian approach where Gibbs algorithms along with Metropolis steps are used to obtain the posterior summaries of interest. We derive the appropriate matrices for assessing the local influence on the parameter estimates under different perturbation schemes, and we also present some ways to perform global influence. Also, some discussions on case deletion influence diagnostics are developed for the joint posterior distribution based on the Kullback–Leibler divergence. Besides, for different parameter settings, sample sizes and censoring percentages, are performed various simulations and display and compare the empirical distribution of the Martingale-type residual with the standard normal distribution. These studies suggest that the residual analysis usually performed in normal linear regression models can be straightforwardly extended to the martingale-type residual in log-Weibull extended models with censored data. Finally, we analyze a real data set under a log-Weibull extended regression model. We perform diagnostic analysis and model check based on the martingale-type residual to select an appropriate model.     

Influence measure in maximum likelihood estimate for models of lifetime data

We use the local influence approach to develop influence measures for identifying observations that strike a disproportionate effect on the maximum likelihood estimate of parameters in models for lifetime data. The proposed method for developing influence measures can be applied to a wide variety of models and we use the exponential model to illustrate the details. In particular, we show that the proposed measure is equivalent to the martingale residual under the exponential model.     

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.     

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.     

Local influence for functional comparative calibration models with replicated data

We investigate local influence analysis in functional comparative calibration models with replicated data. A method for selecting appropriate perturbation schemes based on the expected Fisher information matrix with respect to the perturbation vector is proposed. It is shown that arbitrarily perturbing these models may result in misleading inference about the influential subjects. First-order influence measures for identifying the correct influential subjects and replicates on corrected score estimators are defined. We introduce different perturbation schemes including perturbation of subjects and replicates on the corrected likelihood function and obtain the density of the perturbed model from which the methodology is based. Particularly, three perturbation of variances schemes could be a better way to handle badly modeled subjects or replicates. Two real data sets are analyzed to illustrate the use of our local influence measures.     

Influence analysis in skew-Birnbaum–Saunders regression models and applications

In this paper, we propose a method to assess influence in skew-Birnbaum–Saunders regression models, which are an extension based on the skew-normal distribution of the usual Birnbaum–Saunders (BS) regression model. An interesting characteristic that the new regression model has is the capacity of predicting extreme percentiles, which is not possible with the BS model. In addition, since the observed likelihood function associated with the new regression model is more complex than that from the usual model, we facilitate the parameter estimation using a type-EM algorithm. Moreover, we employ influence diagnostic tools that considers this algorithm. Finally, a numerical illustration includes a brief simulation study and an analysis of real data in order to show the proposed methodology.     

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.     

Influence diagnostics for the structural sharpe model under normal/independent distributions

A method is proposed in this paper to assess the local influence of minor perturbations for the Sharpe model when the normal distribution is replaced by normal/independent (NI) distributions. The family of NI distributions is an attractive class of symmetric heavy-tailed densities that includes as special cases the normal, t-Student, slash, and the contaminated normal distributions. Since the returns of the market are not observable, the statistical analysis is carried out in the context of an errors-in-variables model. An influence analysis for detecting influential observations (atypical returns) is developed to investigate the sensitivity of the maximum likelihood estimators. Diagnostic measures are obtained based on the conditional expectation of the complete-data log-likelihood function. The results are illustrated by using a set of shares of companies traded in the Chilean stock market.     

Local influence in binary regression models,and its correspondence with global influence

We develop local influence diagnostics for a general binary regression model,and apply these methods to case-weight perturbations in four examples. In addition, we illustrate the correspondence between case-deletion diagnostics and local case-weight perturbation slopes and curvatures. We demonstrate that local influence diagnostics can provide a more computationally efficient means for obtaining analogous information to that yielded by case-deletion diagnostics, which can be thought of as global influence perturbations. We also assess the global consistency of patterns of local influence using these data examples.     

Influence diagnostics for the structural errors-in-variables model under the Student-t distribution

The influence of observations on the parameter estimates for the simple structural errors-in-variables model with no equation error, under the Student-t distribution, is investigated using the local influence approach. The main conclusion is that the Student-t model with small degrees of freedom is able to incorporate possible outliers and influential observations in the data. The likelihood displacement approach is useful for outlier detection, especially when a masking phenomenon is present and the degrees of freedom parameter is large. The diagnostics are illustrated with two examples.     

Charpy Impact Energy Data: a Markov Chain Monte Carlo Analysis

To assess radiation damage in steel for reactor pressure vessels in the nuclear industry, specimens are subjected to the Charpy test, which measures how much energy a specimen can absorb at a given test temperature before cracking. The resulting Charpy impact energy data are well represented by a three-parameter Burr curve as a function of test temperature, in which the parameters of the Burr curve are themselves dependent on irradiation dose. The resulting non-linear model function, combined with heteroscedastic random errors, gives rise to complicated likelihood surfaces that make conventional statistical techniques difficult to implement. To compute estimates of parameters of practical interest, Markov chain Monte Carlo sampling-based techniques are implemented. The approach is applied to 40 data sets from specimens subjected to no irradiation or one or two doses of irradiation. The influence of irradiation dose on the amount of energy absorbed is investigated.     

Inference with a contrast-based posterior distribution and application in spatial statistics

The likelihood function is often used for parameter estimation. Its use, however, may cause difficulties in specific situations. In order to circumvent these difficulties, we propose a parameter estimation method based on the replacement of the likelihood in the formula of the Bayesian posterior distribution by a function which depends on a contrast measuring the discrepancy between observed data and a parametric model. The properties of the contrast-based (CB) posterior distribution are studied to understand what the consequences of incorporating a contrast in the Bayes formula are. We show that the CB-posterior distribution can be used to make frequentist inference and to assess the asymptotic variance matrix of the estimator with limited analytical calculations compared to the classical contrast approach. Even if the primary focus of this paper is on frequentist estimation, it is shown that for specific contrasts the CB-posterior distribution can be used to make inference in the Bayesian way.The method was used to estimate the parameters of a variogram (simulated data), a Markovian model (simulated data) and a cylinder-based autosimilar model describing soil roughness (real data). Even if the method is presented in the spatial statistics perspective, it can be applied to non-spatial data.