This paper studies outlier detection and accommodation in general spatial models including spatial autoregressive models and spatial error model as special cases. Using mean-shift and variance-weight models respectively, test statistics for multiple outliers are derived and the detecting procedures are proposed. In addition, several key diagnostic measures such as standardized residuals and leverage measure are defined in general spatial models. Outlier modified models are proposed to accommodate outliers in the data set. The performance of test statistics, including size and power, are examined via simulation studies. Three real examples are analyzed and the results show that the proposed methodology is useful for identifying and accommodating outliers in general spatial models.

The nitrogen dioxide is a primary pollutant, regarded for the estimation of the air quality index, whose excessive presence may cause significant environmental and health problems. In the current work, we suggest characterizing the evolution of $$\hbox {NO}_{2}$$ levels, by using geostatistical approaches that deal with both the space and time coordinates. To develop our proposal, a first exploratory analysis was carried out on daily values of the target variable, daily measured in Portugal from 2004 to 2012, which led to identify three influential covariates (type of site, environment and month of measurement). In a second step, appropriate geostatistical tools were applied to model the trend and the space–time variability, thus enabling us to use the kriging techniques for prediction, without requiring data from a dense monitoring network. This methodology has valuable applications, as it can provide accurate assessment of the nitrogen dioxide concentrations at sites where either data have been lost or there is no monitoring station nearby.

This paper combines optimal spatial sampling designs with geostatistical analysis of functional data. We propose a methodology and design criteria to find the set of spatial locations that minimizes the variance of the spatial functional prediction at unsampled sites for three functional predictors: ordinary kriging, simple kriging and simple cokriging. The last one is a modification of an existing predictor that uses ordinary cokriging based on the basis coefficients. Instead, we propose to use a simple cokriging predictor with the scores resulting from a representation of the functional data with the empirical functional principal components, allowing to remove restrictions and complexity of the covariance models and constraints on the estimation procedure. The methodology is applied to a network of air quality in Bogota city, Colombia.

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.

We treat robust M-estimators for independent and identically distributed Poisson data. We introduce modified Tukey M-estimators with bias correction and compare them to M-estimators based on the Huber function as well as to weighted likelihood and other estimators by simulation in case of clean data and data with outliers. In particular, we investigate the problem of combining robustness and high efficiencies at small Poisson means caused by the strong asymmetry of such Poisson distributions and propose a further estimator based on adaptive trimming. The advantages of the constructed estimators are illustrated by an application to smoothing count data with a time varying mean and level shifts.

Firstly we congratulate the authors on a wonderful paper full of new nice ideas, which can be considered as a major breakthrough in the functional outlier detection using visual procedures. These ideas include a taxonomy of outliers, the definition of bag distance and the centrality-stability plots. Indeed, each of the last two ideas is fundamental in the development of each of the two corresponding procedures to detect multivariate functional outliers introduced in the paper. The first procedure consists in constructing a heat map using the functional bag distance based on the Tukey depth while the second in a scatter-plot based on the skew-adjusted projection depth, SPD, named the centrality-stability plot, CSP. Moreover, both procedures complement themselves because the heat maps are good in detecting all kind of outliers excepting the shape outliers, but those are clearly identified with the CSP’s. Our discussion focus, firstly, on shedding light on the behaviour of the proposed procedures when applied to multivariate functional data whose dimension is entitled to be extremely high. Secondly, a simplification of the CSP is proposed. Furthermore, we encourage the authors to comment on the advantages/disadvantages of applying, what they call, the MFSPD versus, what they call, the 1/(1+ FAO), as the difference between them just lies in the reverse order of the integral and inverse functionals.

The classic susceptible-infectious-recovered (SIR) model, has been used extensively to study the dynamical evolution of an infectious disease in a large population. The SIR-susceptible (SIRS) model is an extension of the SIR model to allow modeling imperfect immunity (those who have recovered might become susceptible again). SIR(S) models assume observed counts are “mass balanced.” Here, mass balance means that total count equals the sum of counts of the individual components of the model. However, since the observed counts have errors, we propose a model that assigns the mass balance to the hidden process of a (Bayesian) hierarchical SIRS (HSIRS) model. Another challenge is to capture the stochastic or random nature of an epidemic process in a SIRS. The HSIRS model accomplishes this through modeling the dynamical evolution on a transformed scale. Through simulation, we compare the HSIRS model to the classic SIRS model, a model where it is assumed that the observed counts are mass balanced and the dynamical evolution is deterministic.

In this paper, the use of orthogonal arrays with strength $$s<p,$$ where $$p$$ is the required strength, for global sensitivity analysis is considered. We first generalize the alias matrix for ANOVA high-dimensional model representation based on matrix image, and then by sequentially minimizing the squared alias degrees, we present a approach for the estimation of sensitivity indices. A two-level orthogonal array with 16 runs and a four-level orthogonal array with 64 runs are studied for estimating both low-order and high-order significant sensitivity indices. Moreover, models containing larger than 10 input factors are also investigated. All cases show that designs with smaller squared alias degree have less bias and variance for the estimations of global sensitivity indices.

In this work we introduce a copula-based method for imputing missing data by using conditional density functions of the missing variables given the observed ones. In theory, such functions can be derived from the multivariate distribution of the variables of interest. In practice, it is very difficult to model joint distributions and derive conditional distributions, especially when the margins are different. We propose a natural solution to the problem by exploiting copulas so that we derive conditional density functions through the corresponding conditional copulas. The approach is appealing since copula functions enable us (1) to fit any combination of marginal distribution functions, (2) to take into account complex multivariate dependence relationships and (3) to model the marginal distributions and the dependence structure separately. We describe the method and perform a Monte Carlo study in order to compare it with two well-known imputation techniques: the nearest neighbour donor imputation and the regression imputation by EM algorithm. Our results indicate that the proposal compares favourably with classical methods in terms of preservation of microdata, margins and dependence structure.