Robust Hotelling's T2 control charts under non-normality: the case of t-Student distribution

Traditional multivariate control charts are based upon the assumption that the observations follow a multivariate normal distribution. In many practical applications, however, this supposition may be difficult to verify. In this paper, we use control charts based on robust estimators of location and scale to improve the capability of detection observations out of control under non-normality in the presence of multiple outliers. Concretely, we use a simulation process to analyse the behaviour of the robust alternatives to Hotelling's T ², which use minimum volume ellipsoidal (MVE) and minimum covariance determinant (MCD) in the presence of observations with a Student's t-distribution. The results show that these robust control charts are good alternatives for small deviations from normality due to the fact that the percentage of out-of-control observations detected for these charts in the Phase II are higher.     

Robust fitting of hidden Markov regression models under a longitudinal setting

We propose a robust estimation procedure for the analysis of longitudinal data including a hidden process to account for unobserved heterogeneity between subjects in a dynamic fashion. We show how to perform estimation by an expectation–maximization-type algorithm in the hidden Markov regression literature. We show that the proposed robust approaches work comparably to the maximum-likelihood estimator when there are no outliers and the error is normal and outperform it when there are outliers or the error is heavy tailed. A real data application is used to illustrate our proposal. We also provide details on a simple criterion to choose the number of hidden states.     

A procedure for approximate negative binomial tolerance intervals

In this article, we present a procedure for approximate negative binomial tolerance intervals. We utilize an approach that has been well-studied to approximate tolerance intervals for the binomial and Poisson settings, which is based on the confidence interval for the parameter in the respective distribution. A simulation study is performed to assess the coverage probabilities and expected widths of the tolerance intervals. The simulation study also compares eight different confidence interval approaches for the negative binomial proportions. We recommend using those in practice that perform the best based on our simulation results. The method is also illustrated using two real data examples.     

Data cloning estimation of GARCH and COGARCH models

GARCH models include most of the stylized facts of financial time series and they have been largely used to analyse discrete financial time series. In the last years, continuous-time models based on discrete GARCH models have been also proposed to deal with non-equally spaced observations, as COGARCH model based on Lévy processes. In this paper, we propose to use the data cloning methodology in order to obtain estimators of GARCH and COGARCH model parameters. Data cloning methodology uses a Bayesian approach to obtain approximate maximum likelihood estimators avoiding numerically maximization of the pseudo-likelihood function. After a simulation study for both GARCH and COGARCH models using data cloning, we apply this technique to model the behaviour of some NASDAQ time series.     

A small-sample correction factor for S-estimators

S-estimators are frequently used as robust estimators of regression and of location and dispersion. Under certain differentiability conditions, S-estimators of multivariate location and dispersion parameters are consistent [Davies PL. Asymtotic behaviour of S-estimators of multivariate location parameters and dispersion matrices. Ann Stat. 1987;15(3):1269–1292]. However, it has been observed that the S-estimators of dispersion parameters give biased results in the case of small-sample data sets. In this work, we constructed formulas based on simulation studies, which allow us to compute small-sample correction factors for all sample sizes and dimensions for S-estimators of dispersion parameters without having to carry out any new simulations. We considered real data to illustrate the effects of the small-sample correction factor.     

A universal robust method for analysing bivariate continuous and proportion data

Traditionally, analysis of Hydrology employs only one hydrological variable. Recently, Nadarajah [A bivariate distribution with gamma and beta marginals with application to drought data. J Appl Stat. 2009;36:277–301] proposed a bivariate model with gamma and beta as marginal distributions to analyse the drought duration and the proportion of drought events. However, the validity of this method hinges on fulfilment of stringent assumptions. We propose a robust likelihood approach which can be used to make inference for general bivariate continuous and proportion data. Unlike the gamma–beta (GB) model which is sensitive to model misspecification, the new method provides legitimate inference without knowing the true underlying distribution of the bivariate data. Simulations and the analysis of the drought data from the State of Nebraska, USA, are provided to make contrasts between this robust approach and the GB model.     

A clustering-based coordinate exchange algorithm for generating G-optimal experimental designs

In the optimal experimental design literature, the G-optimality is defined as minimizing the maximum prediction variance over the entire experimental design space. Although the G-optimality is a highly desirable property in many applications, there are few computer algorithms developed for constructing G-optimal designs. Some existing methods employ an exhaustive search over all candidate designs, which is time-consuming and inefficient. In this paper, a new algorithm for constructing G-optimal experimental designs is developed for both linear and generalized linear models. The new algorithm is made based on the clustering of candidate or evaluation points over the design space and it is a combination of point exchange algorithm and coordinate exchange algorithm. In addition, a robust design algorithm is proposed for generalized linear models with modification of an existing method. The proposed algorithm are compared with the methods proposed by Rodriguez et al. [Generating and assessing exact G-optimal designs. J. Qual. Technol. 2010;42(1):3–20] and Borkowski [Using a genetic algorithm to generate small exact response surface designs. J. Prob. Stat. Sci. 2003;1(1):65–88] for linear models and with the simulated annealing method and the genetic algorithm for generalized linear models through several examples in terms of the G-efficiency and computation time. The result shows that the proposed algorithm can obtain a design with higher G-efficiency in a much shorter time. Moreover, the computation time of the proposed algorithm only increases polynomially when the size of model increases.     

Measure of location-based estimators in simple linear regression

In this note we consider certain measure of location-based estimators (MLBEs) for the slope parameter in a linear regression model with a single stochastic regressor. The median-unbiased MLBEs are interesting as they can be robust to heavy-tailed samples and, hence, preferable to the ordinary least squares estimator (LSE). Two different cases are considered as we investigate the statistical properties of the MLBEs. In the first case, the regressor and error is assumed to follow a symmetric stable distribution. In the second, other types of regressions, with potentially contaminated errors, are considered. For both cases the consistency and exact finite-sample distributions of the MLBEs are established. Some results for the corresponding limiting distributions are also provided. In addition, we illustrate how our results can be extended to include certain heteroskedastic and multiple regressions. Finite-sample properties of the MLBEs in comparison to the LSE are investigated in a simulation study.     

Robust logistic regression modelling via the elastic net-type regularization and tuning parameter selection

The penalized logistic regression is a useful tool for classifying samples and feature selection. Although the methodology has been widely used in various fields of research, their performance takes a sudden turn for the worst in the presence of outlier, since the logistic regression is based on the maximum log-likelihood method which is sensitive to outliers. It implies that we cannot accurately classify samples and find important factors having crucial information for classification. To overcome the problem, we propose a robust penalized logistic regression based on a weighted likelihood methodology. We also derive an information criterion for choosing the tuning parameters, which is a vital matter in robust penalized logistic regression modelling in line with generalized information criteria. We demonstrate through Monte Carlo simulations and real-world example that the proposed robust modelling strategies perform well for sparse logistic regression modelling even in the presence of outliers.     

SiZer inference for generalized varying coefficient models

ABSTRACT

SiZer (significant zero crossings of derivatives) is an effective tool for exploring significant features in curves from the viewpoint of the scale space theory. In this paper, a SiZer approach is developed for generalized varying coefficient models (GVCMs) in order to achieve the task of understanding dynamic characteristics of the regression relationship at multiscales. The proposed SiZer method is based on the local-linear maximum likelihood estimation of GVCMs and the one-step estimation procedure is employed to alleviate the computational cost of estimating the coefficients and their derivatives at different scales. Simulation studies are performed to assess the performance of the SiZer inference and two real-world examples are given to demonstrate its applications.