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.     

Adaptive Modeling Procedure Selection by Data Perturbation

Many procedures have been developed to deal with the high-dimensional problem that is emerging in various business and economics areas. To evaluate and compare these procedures, modeling uncertainty caused by model selection and parameter estimation has to be assessed and integrated into a modeling process. To do this, a data perturbation method estimates the modeling uncertainty inherited in a selection process by perturbing the data. Critical to data perturbation is the size of perturbation, as the perturbed data should resemble the original dataset. To account for the modeling uncertainty, we derive the optimal size of perturbation, which adapts to the data, the model space, and other relevant factors in the context of linear regression. On this basis, we develop an adaptive data-perturbation method that, unlike its nonadaptive counterpart, performs well in different situations. This leads to a data-adaptive model selection method. Both theoretical and numerical analysis suggest that the data-adaptive model selection method adapts to distinct situations in that it yields consistent model selection and optimal prediction, without knowing which situation exists a priori. The proposed method is applied to real data from the commodity market and outperforms its competitors in terms of price forecasting accuracy.     

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.     

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 Grubbs's model

In this paper we consider applications of local influence (Cook, 1986) to evaluate small perturbations in the model or in data sets of several measuring devices, assuming Grubbs's model. Different perturbation schemes are investigated and an application is considered to two real data sets.     

Local influence of nonlinear mixed effects model based on M-estimation

Abstract

In this work we mainly study the local influence in nonlinear mixed effects model with M-estimation. A robust method to obtain maximum likelihood estimates for parameters is presented, and the local influence of nonlinear mixed models based on robust estimation (M-estimation) by use of the curvature method is systematically discussed. The counting formulas of curvature for case weights perturbation, response variable perturbation and random error covariance perturbation are derived. Simulation studies are carried to access performance of the methods we proposed. We illustrate the diagnostics by an example presented in Davidian and Giltinan, which was analyzed under the non-robust situation.     

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.     

Local influence in null intercept measurement error regression under a student_t model

In this paper we discuss the application of local influence in a measurement error regression model with null intercepts under a Student_t model with dependent populations. The Student_t distribution is a robust alternative to modelling data sets involving errors with longer than Normal tails. We derive the appropriate matrices for assessing the local influence for different perturbation schemes and use real data as an illustration of the usefulness of the application.     

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     

Sensitivity analysis for the identifiability with application to latent random effect model for the mixed data

In this paper, we study the indentifiability of a latent random effect model for the mixed correlated continuous and ordinal longitudinal responses. We derive conditions for the identifiability of the covariance parameters of the responses. Also, we proposed sensitivity analysis to investigate the perturbation from the non-identifiability of the covariance parameters, it is shown how one can use some elements of covariance structure. These elements associate conditions for identifiability of the covariance parameters of the responses. Influence of small perturbation of these elements on maximal normal curvature is also studied. The model is illustrated using medical data.     

Asymptotics of the Finite-time Ruin Probability for the Sparre Andersen Risk Model Perturbed by an Inflated Stationary Chi-process

In this article, we consider the Sparre Andersen risk model that is perturbed by an inflated chi-process with non-negative random inflator R. Under some conditions on the perturbation and the random inflator, which allow for both small and large fluctuations, exact asymptotic behaviour of the finite-time ruin probability is obtained when initial reserve tends to infinity.     

Answer-perturbation techniques for the protection of statistical databases

This paper presents an answer-perturbation model for the protection of statistical databases. Compared to previous approaches, this model saves both space and time since neither a copy of the original database nor an extra processing of the query set is needed. The modularity of our protection mechanism (perturbation is applied after the query processing, not during it or upon data entry, as in the other approaches) facilitates the implementation. The method presented is user-oriented and since the usability (accuracy) of the statistically altered answers can be fine-tuned, the balance between usability and protection (security) is under the control of the database administrator.     

Influence diagnostics in a multivariate normal regression model with general parameterization

Patriota and Lemonte [24] introduced a quite general multivariate normal regression model. This model considers that the mean vector and the covariance matrix share the same vector of parameters. In this paper we present some influence assessment for this model, such as the local influence, total local influence of an individual and generalized leverage which are discussed. Additionally, the normal curvatures for local influence studies are derived under some perturbation schemes.     

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.     

The Power Series Cure Rate Model: An Application to a Cutaneous Melanoma Data

In this article we propose a new cure rate survival model. In our approach the number of competing causes of the event of interest is assumed to follow an exponential discrete power series distribution. An advantage of our model is that it is very flexible, including several particular cases, such as, Bernoulli, geometric, Poisson, etc. Moreover, we derive the appropriate matrices for assessing local influence on the parameter estimates under different perturbation schemes and present some ways to perform global influence analysis. Distribution fitting can be tested for the best fitting in a straightforward way. Maximum likelihood estimation is discussed. Our proposed model is illustrated through cutaneous melanoma data.     

The Poisson Generalized Linear Failure Rate Model

We formulate a new cure rate survival model by assuming that the number of competing causes of the event of interest has the Poisson distribution, and the time to this event has the generalized linear failure rate distribution. A new distribution to analyze lifetime data is defined from the proposed cure rate model, and its quantile function as well as a general expansion for the moments is derived. We estimate the parameters of the model with cure rate in the presence of covariates for censored observations using maximum likelihood and derive the observed information matrix. We obtain the appropriate matrices for assessing local influence on the parameter estimates under different perturbation schemes and present some ways to perform global influence analysis. The usefulness of the proposed cure rate survival model is illustrated in an application to real data.     

A bivariate regression model with cure fraction

The use of bivariate distributions plays a fundamental role in survival and reliability studies. In this paper, we introduce a location-scale model for bivariate survival times based on the copula to model the dependence of bivariate survival data with cure fraction. We create the correlation structure between the failure times using the Clayton family of copulas, which is assumed to have any distribution. It turns out that the model becomes very flexible with respect to the choice of the marginal distributions. For the proposed model, we consider inferential procedures based on constrained parameters under maximum likelihood. We derive the appropriate matrices for assessing local influence under different perturbation schemes and present some ways to perform global influence analysis. The relevance of the approach is illustrated using a real data set and a diagnostic analysis is performed to select an appropriate model.     

On generalized degrees of freedom with application in linear mixed models selection

The concept of degrees of freedom plays an important role in statistical modeling and is commonly used for measuring model complexity. The number of unknown parameters, which is typically used as the degrees of freedom in linear regression models, may fail to work in some modeling procedures, in particular for linear mixed effects models. In this article, we propose a new definition of generalized degrees of freedom in linear mixed effects models. It is derived from using the sum of the sensitivity of the expected fitted values with respect to their underlying true means. We explore and compare data perturbation and the residual bootstrap to empirically estimate model complexity. We also show that this empirical generalized degrees of freedom measure satisfies some desirable properties and is useful for the selection of linear mixed effects models.     

Likelihood estimation for longitudinal zero-inflated power series regression models

In this paper, a zero-inflated power series regression model for longitudinal count data with excess zeros is presented. We demonstrate how to calculate the likelihood for such data when it is assumed that the increment in the cumulative total follows a discrete distribution with a location parameter that depends on a linear function of explanatory variables. Simulation studies indicate that this method can provide improvements in obtaining standard errors of the estimates. We also calculate the dispersion index for this model. The influence of a small perturbation of the dispersion index of the zero-inflated model on likelihood displacement is also studied. The zero-inflated negative binomial regression model is illustrated on data regarding joint damage in psoriatic arthritis.