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.     

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.     

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.     

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 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.     

Error bounds for the perturbation solution of the transition density under a multi-factor CIR term structure model with weak mean-reversion effect

Abstract

We consider a multi-factor Cox-Ingersoll-Ross (CIR) model of the term structure of interest rates with weak mean-reversion effect. We use perturbation theory to analyze its conditional characteristic function illustrated by a system of Riccati equations and derive the error bounds for the perturbation approximations. Using the Fourier inversion theorem, we clarify that the perturbation approximation of the conditional characteristic function can be applied to estimate the transition density and likelihood function. We provide their error bounds and accuracy orders. Finally, we discuss the performance of the perturbation approximation in estimating the transition density via simulation.     

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.     

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     

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.     

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.     

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.     

VIF-based adaptive matrix perturbation method for heteroskedasticity-robust covariance estimators in the presence of multicollinearity

In this study, we investigate linear regression having both heteroskedasticity and collinearity problems. We discuss the properties related to the perturbation method. Important observations are summarized as theorems. We then prove the main result that states the heteroskedasticity-robust variances can be improved and that the resulting bias is minimized by using the matrix perturbation method. We analyze a practical example for validation of the method.     

A performance counterexample of Billor–Chatterjee–Hadi procedure and an improvement proposal for robust regression

The BCH procedure introduced by Billor, Chatterjee, and Hadi for fitting linear models was found to be inefficient for y-outliers in the presence of a high perturbation level. We propose to modify the first step of the BCH procedure, so that the robust distances are computed on the matrix Z = (y, X) of the basic subset. The performance of the present note procedure (PNP), as compared to the BCH procedure and the ordinary least-square (OLS) method, was studied by processing several datasets used in the literature for robust regression and by performing a Monte Carlo experiment. PNP performs better particularly with datasets having high perturbation.     

More on local influence in pllincipal components analysis

Local influence on the eigenvalues of sample covariance matrices in

principal components analysis is examined for a reasonable modification of Shi's (1997) perturbation scheme, The modification is suggested for samples from populations with both unknown mean vector and covariance matrix. While Shi's detection indexes (1997) consist of only quadratic terms, the modified perturbation scheme leads to detection indexes constituted by both linear and quadratic terms associated with centralized observations. These linear and quadratic terms reflect local influences on the first two sample moments. Examples are investigated based on the two detection indexes.     

Bayesian latent variable model for mixed continuous and ordinal responses with possibility of missing responses

A general framework is proposed for joint modelling of mixed correlated ordinal and continuous responses with missing values for responses, where the missing mechanism for both kinds of responses is also considered. Considering the posterior distribution of unknowns given all available information, a Markov Chain Monte Carlo sampling algorithm via winBUGS is used for estimating the posterior distribution of the parameters. For sensitivity analysis to investigate the perturbation from missing at random to not missing at random, it is shown how one can use some elements of covariance structure. These elements associate responses and their missing mechanisms. Influence of small perturbation of these elements on posterior displacement and posterior estimates is also studied. The model is illustrated using data from a foreign language achievement study.     

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.     

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.     

Corrected Phase-Type Approximations of Heavy-Tailed Queueing Models in a Markovian Environment

We develop accurate approximations for the delay distribution of the MArP/G/1 queue that capture the exact tail behavior and provide bounded relative errors. Motivated by statistical analysis, we consider the service times as a mixture of a phase-type and a heavy-tailed distribution. With the aid of perturbation analysis, we derive corrected phase-type approximations as a sum of the delay in a MArP/PH/1 queue and a heavy-tailed component depending on the perturbation parameter. We exhibit their performance with numerical examples.     

On the validity of the perturbation method in asymptotic theory

The perturbation methods and the Taylor expansions are very often used to obtain test statistics approximations in multivariate analysis (Specially in Principal Component and Canonical Analyses). These approximations are then used to obtain formal Edgeworth expransions of the distribution functions of the statistics. BHATTACHARYA and GHOSH 1978 have justified these practices under suitable assumptions. In this paper a non classical perturbation problem is solved in order to obtain almost surely expansions of test statistics