The mean-shift outlier model in general weighted regression and its applications

Consider the general weighted linear regression model y=Xβ+, where E()=0, Cov()=Vσ², σ² is an unknown positive scalar, and V is a symmetric positive-definite matrix not necessary diagonal. Two models, the mean-shift outlier model and the case-deletion model, can be employed to develop multiple case-deletion diagnostics for the linear model. The multiple case-deletion diagnostics are obtained via the mean-shift outlier model in this article and are shown to be equivalent to the deletion diagnostics via the case deletion model obtained by Preisser and Qaqish (1996, Biometrika, 83, 551–562). In addition, computing the multiple case-deletion diagnostics obtained via the mean-shift outlier model is faster than computing the one based on the more commonly used case-deletion model in some situations. Applications of the multiple deletion diagnostics developed from the mean-shift outlier model are also given for regression analysis with the likelihood function available and regression analysis based on generalized estimating equations. These applications include survival models and the generalized estimating equations of Liang and Zeger (1986, Biometrika, 73, 13–22). Several numerical experiments as well as a real example are given as illustrations.     

Assessment of diagnostic procedures in symmetrical nonlinear regression models

The aim of this paper is to derive diagnostic procedures based on case-deletion model for symmetrical nonlinear regression models, which complements Galea et al. (2005) that developed local influence diagnostics under some perturbation schemes. This class of models includes all symmetric continuous distributions for errors covering both light- and heavy-tailed distributions such as Student-t, logistic-I and -II, power exponential, generalized Student-t, generalized logistic and contaminated normal, among others. Thus, these models can be checked for robustness to outliers in the response variable and diagnostic methods may be a useful tool for an appropriate choice. First, an iterative process for the parameter estimation as well as some inferential results are presented. Besides, we present the results of a simulation study in which the characteristics of heavy-tailed models are evaluated in the presence of outliers. Then, we derive some diagnostic measures such as Cook distance, W-K statistic, one-step approach and likelihood displacement, generalizing results obtained for normal nonlinear regression models. Also, we present simulation studies that illustrate the behavior of diagnostic measures proposed. Finally, we consider two real data sets previously analyzed under normal nonlinear regression models. The diagnostic analysis indicates that a Student-t nonlinear regression model seems to fit the data better than the normal nonlinear regression model as well as other symmetrical nonlinear models in the sense of robustness against extreme observations.     

Influence diagnostics in log-Birnbaum-Saunders regression models with censored data

In this paper we discuss log-Birnbaum-Saunders regression models with censored observations. This kind of model has been largely applied to study material lifetime subject to failure or stress. The score functions and observed Fisher information matrix are given as well as the process for estimating the regression coefficients and shape parameter is discussed. The normal curvatures of local influence are derived under various perturbation schemes and two deviance-type residuals are proposed to assess departures from the log-Birnbaum-Saunders error assumption as well as to detect outlying observations. Finally, a data set from the medical area is analyzed under log-Birnbaum-Saunders regression models. A diagnostic analysis is performed in order to select an appropriate model.     

A variance shift model for detection of outliers in the linear mixed model

A variance shift outlier model (VSOM), previously used for detecting outliers in the linear model, is extended to the variance components model. This VSOM accommodates outliers as observations with inflated variance, with the status of the ith observation as an outlier indicated by the size of the associated shift in the variance. Likelihood ratio and score test statistics are assessed as objective measures for determining whether the ith observation has inflated variance and is therefore an outlier. It is shown that standard asymptotic distributions do not apply to these tests for a VSOM, and a modified distribution is proposed. A parametric bootstrap procedure is proposed to account for multiple testing. The VSOM framework is extended to account for outliers in random effects and is shown to have an advantage over case-deletion approaches. A simulation study is presented to verify the performance of the proposed tests. Challenges associated with computation and extensions of the VSOM to the general linear mixed model with correlated errors are discussed.     

Deletion, replacement and mean-shift for diagnostics in linear mixed models

Deletion, replacement and mean-shift model are three approaches frequently used to detect influential observations and outliers. For general linear model with known covariance matrix, it is known that these three approaches lead to the same update formulae for the estimates of the regression coefficients. However if the covariance matrix is indexed by some unknown parameters which also need to be estimated, the situation is unclear. In this paper, we show under a common subclass of linear mixed models that the three approaches are no longer equivalent. For maximum likelihood estimation, replacement is equivalent to mean-shift model but both are not equivalent to case deletion. For restricted maximum likelihood estimation, mean-shift model is equivalent to case deletion but both are not equivalent to replacement. We also demonstrate with real data that misuse of replacement and mean-shift model in place of case deletion can lead to incorrect results.     

Improved point and interval estimation for a beta regression model

In this paper we consider the beta regression model recently proposed by Ferrari and Cribari-Neto [2004. Beta regression for modeling rates and proportions. J. Appl. Statist. 31, 799-815], which is tailored to situations where the response is restricted to the standard unit interval and the regression structure involves regressors and unknown parameters. We derive the second order biases of the maximum likelihood estimators and use them to define bias-adjusted estimators. As an alternative to the two analytically bias-corrected estimators discussed, we consider a bias correction mechanism based on the parametric bootstrap. The numerical evidence favors the bootstrap-based estimator and also one of the analytically corrected estimators. Several different strategies for interval estimation are also proposed. We present an empirical application.     

Explaining extreme waves by a theory of stochastic wave groups

It is well known that in a Gaussian sea an extreme wave event is a particular realization of the space–time evolution of a well defined linear wave group, in agreement with the theory of quasi-determinism of Boccotti [Boccotti P. On mechanics of irregular gravity waves. Atti Acc Naz Lincei, Memorie 1989;19:11–170] and the Slepian model of Lindgren [Kac M, Slepian D. Large excursions of Gaussian processes. Ann Math Statist 1959;30:1215–28; Lindgren G. Some properties of a normal process near a local maximum. Ann Math Statist 1970;4(6):1870–83]. In this paper, the concept of stochastic wave groups is proposed to explain the occurrence of extreme waves in nonlinear random seas, according to the dynamics imposed by the Zakharov equation [Zakharov VE. Statistical theory of gravity and capillary waves on the surface of a finite-depth fluid. J Eur Mech B—Fluids 1999;18(3):327–44]. As a corollary, a new analytical solution for the probability of exceedance of the crest-to-trough height is derived for the prediction of extreme wave events in nonlinearly modulated long-crested narrow-band seas. Furthermore, a generalization of the Tayfun distribution [Tayfun MA. On narrow-band representation of ocean waves. Part I: Theory. J Geophys Res 1986;91(C6):7743–52] for the wave crest height is also provided. The new analytical distributions explain qualitatively well recent experimental results of Onorato et al. [Onorato M, Osborne AR, Cavaleri L, Brandini C, Stansberg CT. Observation of strongly non-Gaussian statistics for random sea surface gravity waves in wave flume experiments. Phys Rev E 2004;70:067302] and the numerical simulations of Socquet-Juglard et al. [Socquet-Juglard H, Dysthe K, Trulsen K, Krogstad HE, Liu J. Probability distributions of surface gravity waves during spectral changes. J Fluid Mech 2005;542:195–216].     

Estimating joinpoints in continuous time scale for multiple change-point models

Joinpoint models have been applied to the cancer incidence and mortality data with continuous change points. The current estimation method [Lerman, P.M., 1980. Fitting segmented regression models by grid search. Appl. Statist. 29, 77-84] assumes that the joinpoints only occur at discrete grid points. However, it is more realistic that the joinpoints take any value within the observed data range. Hudson [1966. Fitting segmented curves whose join points have to be estimated. J. Amer. Statist. Soc. 61, 1097-1129] provides an algorithm to find the weighted least square estimates of the joinpoint on the continuous scale. Hudson described the estimation procedure in detail for a model with only one joinpoint, but its extension to a multiple joinpoint model is not straightforward. In this article, we describe in detail Hudson's method for the multiple joinpoint model and discuss issues in the implementation. We compare the computational efficiencies of the LGS method and Hudson's method. The comparisons between the proposed estimation method and several alternative approaches, especially the Bayesian joinpoint models, are discussed. Hudson's method is implemented by C++ and applied to the colorectal cancer incidence data for men under age 65 from SEER nine registries.     

A novel method for testing normality in a mixed model of a nested classification

Normality is one of the most common assumptions made in the development of statistical models such as the fixed effect model and the random effect model. White and MacDonald [1980. Some large-sample tests for normality in the linear regression model. JASA 75, 16-18] and Bonett and Woodward [1990. Testing residual normality in the ANOVA model. J. Appl. Statist. 17, 383-387] showed that many tests of normality perform well when applied to the residuals of a fixed effect model. The elements of the error vector are not independent in random effects models and standard tests of normality are not expected to perform properly when applied to the residuals of a random effects model.In this paper, we propose a transformation method to convert the correlated error vector into an uncorrelated vector. Moreover, under the normality assumption, the uncorrelated vector becomes an independent vector. Thus, all the existing methods can then be implemented. Monte-Carlo simulations are used to evaluate the feasibility of the transformation. Results show that this transformation method can preserve the Type I error and provide greater powers under most alternatives.     

Commentary on Krishnaswamy et al. - Quantifying and mapping biodiversity and ecosystem services: Utility of a multi-season NDVI based Mahalanobis distance surrogate

Remote sensing is a powerful tool for characterizing, estimating or modelling species diversity. Differences in environmental properties of different habitats should lead to differences of spectral responses, which can be detected by satellite imagery. Hence, spectral distance may be related to species diversity. Based on previous studies, Krishnaswamy et al. [Krishnaswamy, J., Bawa, K. S., Ganeshaiah, K. N., & Kiran, M. C. (2009). Quantifying and mapping biodiversity and ecosystem services: Utility of a multi-season NDVI based Mahalanobis distance surrogate. Remote Sensing of Environment.] used spectral distance to estimate species diversity. Since a noisy scatterplot of species versus spectral diversity is expected, the commonly used Ordinary Least Square regression may fail to detect trends which occur across other quantiles than the mean.Krishnaswamy et al. [Krishnaswamy, J., Bawa, K. S., Ganeshaiah, K. N., & Kiran, M. C. (2009). Quantifying and mapping biodiversity and ecosystem services: Utility of a multi-season NDVI based Mahalanobis distance surrogate. Remote Sensing of Environment.] proposed a quantile-quantile plot method as an alternative to conventional regression based approaches which are inappropriate for dependent pair-wise dissimilarity or similarity data. By this commentary I demonstrate the utility of a quantile regression technique to complement the Krishnaswamy et al. [Krishnaswamy, J., Bawa, K. S., Ganeshaiah, K. N., & Kiran, M. C. (2009). Quantifying and mapping biodiversity and ecosystem services: Utility of a multi-season NDVI based Mahalanobis distance surrogate. Remote Sensing of Environment.] graphical approach in terms of a predictive model.     

A mixed effects log-linear model based on the Birnbaum-Saunders distribution

In lifetime data analysis and particularly in engineering reliability contexts, the Birnbaum-Saunders (BISA) density is often suggested as a suitable model; see Birnbaum and Saunders (1969), Mann et al. (1974), and Desmond (1985). A linear regression model, obtained from a logarithmic transformation of the response variable, is useful in studying the effect of covariates on the response variable; see Rieck and Nedelman (1991), Tsionas (2001) and Galea et al. (2004). In this paper, an extension of the log-linear regression model of Rieck and Nedelman (1991), which considers random effects, is introduced. From a Monte Carlo simulation study, the performance of various estimation and prediction methods are studied. The usefulness of the mixed log-linear model is stressed and compared to the pure fixed effects log-linear regression BISA model. The new model is used to analyze a real data set, for which a fixed effects model is inappropriate.     

Some restriction tests in a new class of regression models for proportions

The main purpose of this work is to study the behaviour of Skovgaard’s [Skovgaard, I.M., 2001. Likelihood asymptotics. Scandinavian Journal of Statistics 28, 3–32] adjusted likelihood ratio statistic in testing simple hypothesis in a new class of regression models proposed here. The proposed class of regression models considers Dirichlet distributed observations, and the parameters that index the Dirichlet distributions are related to covariates and unknown regression coefficients. This class is useful for modelling data consisting of multivariate positive observations summing to one and generalizes the beta regression model described in Vasconcellos and Cribari-Neto [Vasconcellos, K.L.P., Cribari-Neto, F., 2005. Improved maximum likelihood estimation in a new class of beta regression models. Brazilian Journal of Probability and Statistics 19, 13–31]. We show that, for our model, Skovgaard’s adjusted likelihood ratio statistics have a simple compact form that can be easily implemented in standard statistical software. The adjusted statistic is approximately chi-squared distributed with a high degree of accuracy. Some numerical simulations show that the modified test is more reliable in finite samples than the usual likelihood ratio procedure. An empirical application is also presented and discussed.     

On the estimation of the general parameter

In this paper, we first discuss the origin, developments and various thoughts by several researchers on the generalized linear regression estimator (GREG) due to Deville and Särndal [Deville, J.C., Särndal, C.E., 1992. Calibration estimators in survey sampling. J. Amer. Statist. Assoc. 87, 376-382]. Then, the problem of estimation of the general parameter of interest considered by Rao [Rao, J.N.K., 1994. Estimating totals and distribution functions using auxiliary information at the estimation stage. J. Official Statist. 10 (2), 153-165], and Singh [Singh, S., 2001. Generalized calibration approach for estimating the variance in survey sampling. Ann. Inst. Statist. Math. 53 (2), 404-417; Singh, S., 2004. Golden and Silver Jubilee Year-2003 of the linear regression estimators. In: Proceedings of the Joint Statistical Meeting, Toronto (Available on the CD), 4382-4380; Singh, S., 2006. Survey statisticians celebrate Golden Jubilee Year-2003 of the linear regression estimator. Metrika 1-18] is further investigated. In addition to that it is shown that the Farrell and Singh [Farrell, P.J., Singh, S., 2005. Model-assisted higher order calibration of estimators of variance. Australian & New Zealand J. Statist. 47 (3), 375-383] estimators are also a special case of the proposed methodology. Interestingly, it has been noted that the single model assisted calibration constraint studied by Farrell and Singh [Farrell, P.J., Singh, S., 2002. Re-calibration of higher order calibration weights. Presented at Statistical Society of Canada conference, Hamilton (Available on CD); Farrell, P.J., Singh, S., 2005. Model-assisted higher order calibration of estimators of variance. Australian & New Zealand J. Statist. 47 (3), 375-383] and Wu [Wu, C., 2003. Optimal calibration estimators in survey sampling. Biometrika 90, 937-951] is not helpful for calibrating the Sen [Sen, A.R., 1953. On the estimate of the variance in sampling with varying probabilities. J. Indian Soc. Agril. Statist. 5, 119-127] and Yates and Grundy [Yates, F., Grundy, P.M., 1953. Selection without replacement from within strata with probability proportional to size. J. Roy. Statist. Soc. Ser. 15, 253-261] estimator of the variance of the linear regression estimator under the optimal designs of Godambe and Joshi [Godambe, V.P., Joshi, V.M., 1965. Admissibility and Bayes estimation in sampling finite populations—I. Ann. Math. Statist. 36, 1707-1722]. Three new estimators of the variance of the proposed linear regression type estimator of the general parameters of interest are introduced and compared with each other. The newly proposed two-dimensional linear regression models are found to be useful, unlike a simulation based on a couple of thousands of random samples, in comparing the estimators of variance. The use of knowledge of the model parameters in assisting the estimators of variance has been found to be beneficial. The most attractive feature is that it has been shown theoretically that the proposed method of calibration always remains more efficient than the GREG estimator.     

Step-up and step-down procedures controlling the number and proportion of false positives

In multiple hypotheses testing, it is important to control the probability of rejecting “true” null hypotheses. A standard procedure has been to control the family-wise error rate (FWER), the probability of rejecting at least one true null hypothesis.For large numbers of hypotheses, using FWER can result in very low power for testing single hypotheses. Recently, powerful multiple step FDR procedures have been proposed which control the “false discovery rate” (expected proportion of Type I errors). More recently, van der Laan et al. [Augmentation procedures for control of the generalized family-wise error rate and tail probabilities for the proportion of false positives. Statist. Appl. in Genetics and Molecular Biol. 3, 1-25] proposed controlling a generalized family-wise error rate k-FWER (also called gFWER(k)), defined as the probability of at least (k+1) Type I errors (k=0 for the usual FWER).Lehmann and Romano [Generalizations of the familywise error rate. Ann. Statist. 33(3), 1138-1154] suggested both a single-step and a step-down procedure for controlling the generalized FWER. They make no assumptions concerning the p-values of the individual tests. The step-down procedure is simple to apply, and cannot be improved without violation of control of the k-FWER.In this paper, by limiting the number of steps in step-down or step-up procedures, new procedures are developed to control k-FWER (and the proportion of false positives) PFP. Using data from the literature, the procedures are compared with those of Lehmann and Romano [Generalizations of the familywise error rate. Ann. Statist. 33(3), 1138-1154], and, under the assumption of a multivariate normal distribution of the test statistics, show considerable improvement in the reduction of the number and PFP.     

Gaussian estimation and joint modeling of dispersions and correlations in longitudinal data

Analysis of longitudinal, spatial and epidemiological data often requires modelling dispersions and dependence among the measurements. Moreover, data involving counts or proportions usually exhibit greater variation than would be predicted by the Poisson and binomial models. We propose a strategy for the joint modelling of mean, dispersion and correlation matrix of nonnormal multivariate correlated data. The parameter estimation for dispersions and correlations is based on the Whittle's [P. Whittle, Gaussian estimation in stationary time series, Bull Inst. Statist. Inst. 39 (1962) 105-129.] Gaussian likelihood of the partially standardized data which eliminates the mean parameters. The model formulation for the dispersions and correlations relies on a recent unconstrained parameterization of covariance matrices and a graphical method [M. Pourahmadi, Joint mean-covariance models with applications to longitudinal data: unconstrained parameterization, Biometrika 86 (1999) 677-690] similar to the correlogram in time series analysis. We show that the estimating equations for the regression and dependence parameters derived from a modified Gaussian likelihood (involving two distinct covariance matrices) are broad enough to include generalized estimating equations and its many recent extensions and improvements. The results are illustrated using two datasets.     

Asymptotic characterizations of input–output-to-state stability for discrete-time systems

We establish the equivalence between global detectability and output-to-state stability for difference inclusions with outputs, and we present equivalent asymptotic characterizations of input–output-to-state stability for discrete-time nonlinear systems. These new stability characterizations for discrete-time systems parallel what have been developed for continuous-time systems in Angeli et al. [Uniform global asymptotic stability of differential inclusions, J. Dynamical Control Systems 10 (2004) 391–412] and Angeli et al. [Seperation principles for input–output and integral-input-to-state stability, SIAM J. Control Optim. 43 (2004) 256–276].     

A consistent nonparametric Bayesian procedure for estimating autoregressive conditional densities

This article proposes a Bayesian infinite mixture model for the estimation of the conditional density of an ergodic time series. A nonparametric prior on the conditional density is described through the Dirichlet process. In the mixture model, a kernel is used leading to a dynamic nonlinear autoregressive model. This model can approximate any linear autoregressive model arbitrarily closely while imposing no constraint on parameters to ensure stationarity. We establish sufficient conditions for posterior consistency in two different topologies. The proposed method is compared with the mixture of autoregressive model [Wong and Li, 2000. On a mixture autoregressive model. J. Roy. Statist. Soc. Ser. B 62(1), 91-115] and the double-kernel local linear approach [Fan et al., 1996. Estimation of conditional densities and sensitivity measures in nonlinear dynamical systems. Biometrika 83, 189-206] by simulations and real examples. Our method shows excellent performances in these studies.     

Diagnostics for conditional heteroscedasticity models: some simulation results

In this paper, we study the size and power of various diagnostic statistics for univariate conditional heteroscedasticity models. These test statistics include the residual-based tests recently derived by Tse, Li and Mak, and Wooldridge, respectively. Monte-Carlo experiments with 1000 replications are conducted to generate conditional variances which follow the autoregressive conditional heteroscedasticity (ARCH)/GARCH processes. We use quasi-maximum likelihood estimation (MLE) method to obtain estimates of parameters under different ARCH/ generalized ARCH (GARCH) models. It is found that the Tse and Li–Mak diagnostics are more powerful.     

Hand radiographs preprocessing, image representation in the finger regions and joint space width measurements for image interpretation

In this paper, the first stage of studies concerning the computer analysis of hand X-ray digital images is described. The images are preprocessed and then skeletization of the fingers is carried out. Then, the interphapangeal and metacarpophalangeal joints are detected and contoured. Joint widths are also measured. The obtained results largely concur with those obtained by other authors—see Beier et al. [Segmentation of medical images combining local, regional, global, and hierarchical distances into a bottom-up region merging scheme, Proc. SPIE 5747 (2005) 546-555], Klooster et al. [Automatic quantification of osteoarthritis in hand radiographs: validation of a new method to measure joint space width, Osteoarthritis and Cartilage 16 (1) (2008) 18-25], Ogiela et al. [Image languages in intelligent radiological palm diagnostics, Pattern Recognition 39 (2006) 2157-2165] and Ogiela and Tadeusiewicz [Picture languages in automatic radiological palm interpretation, Int. J. Appl. Math. Comput. Sci. 15 (2) (2005) 305-312].     

Robust bandwidth selection in semiparametric partly linear regression models: Monte Carlo study and influential analysis

In this paper, under a semiparametric partly linear regression model with fixed design, we introduce a family of robust procedures to select the bandwidth parameter. The robust plug-in proposal is based on nonparametric robust estimates of the νth derivatives and under mild conditions, it converges to the optimal bandwidth. A robust cross-validation bandwidth is also considered and the performance of the different proposals is compared through a Monte Carlo study. We define an empirical influence measure for data-driven bandwidth selectors and, through it, we study the sensitivity of the data-driven bandwidth selectors. It appears that the robust selector compares favorably to its classical competitor, despite the need to select a pilot bandwidth when considering plug-in bandwidths. Moreover, the plug-in procedure seems to be less sensitive than the cross-validation in particular, when introducing several outliers. When combined with the three-step procedure proposed by Bianco and Boente [2004. Robust estimators in semiparametric partly linear regression models. J. Statist. Plann. Inference 122, 229-252] the robust selectors lead to robust data-driven estimates of both the regression function and the regression parameter.