Some contributions on the multivariate Poisson–Skellam probability distribution

In this article, we introduce a new form of distribution whose components have the Poisson or Skellam marginal distributions. This new specification allows the incorporation of relevant information on the nature of the correlations between every component. In addition, we present some properties of this distribution. Unlike the multivariate Poisson distribution, it can handle variables with positive and negative correlations. It should be noted that we are only interested in modeling covariances of order 2, which means between all pairs of variables. Some simulations are presented to illustrate the estimation methods. Finally, an application of soccer teams data will highlight the relationship between number of points per season and the goal differential by some covariates.     

ß-expectation tolerance region for the heteroscedastic multiple regression model with multivariate Student-t error

A ß-expectation tolerance region has been constructed for the multivariate regression model with heteroscedastic errors which follow a multivariate Student-t distribution with an unknown number of degrees of freedom. The ß-expectaion tolerance region obtained in this paper is optimal in the sense of having minimum enclosure among all such tolerance regions that guarantees that it would cover any preassigned proportions, namely, ß×100 percent of the future responses from the model.     

Flexible mixture modeling via the multivariate t distribution with?the?Box-Cox transformation: an?alternative to?the?skew-t distribution

Cluster analysis is the automated search for groups of homogeneous observations in a data set. A popular modeling approach for clustering is based on finite normal mixture models, which assume that each cluster is modeled as a multivariate normal distribution. However, the normality assumption that each component is symmetric is often unrealistic. Furthermore, normal mixture models are not robust against outliers; they often require extra components for modeling outliers and/or give a poor representation of the data. To address these issues, we propose a new class of distributions, multivariate t distributions with the Box-Cox transformation, for mixture modeling. This class of distributions generalizes the normal distribution with the more heavy-tailed t distribution, and introduces skewness via the Box-Cox transformation. As a result, this provides a unified framework to simultaneously handle outlier identification and data transformation, two interrelated issues. We describe an Expectation-Maximization algorithm for parameter estimation along with transformation selection. We demonstrate the proposed methodology with three real data sets and simulation studies. Compared with a wealth of approaches including the skew-t mixture model, the proposed t mixture model with the Box-Cox transformation performs favorably in terms of accuracy in the assignment of observations, robustness against model misspecification, and selection of the number of components.     

Optimality of quasi-score in the multivariate mean–variance model with an application to the zero-inflated Poisson model with measurement errors

In a multivariate mean–variance model, the class of linear score (LS) estimators based on an unbiased linear estimating function is introduced. A special member of this class is the (extended) quasi-score (QS) estimator. It is ‘extended’ in the sense that it comprises the parameters describing the distribution of the regressor variables. It is shown that QS is (asymptotically) most efficient within the class of LS estimators. An application is the multivariate measurement error model, where the parameters describing the regressor distribution are nuisance parameters. A special case is the zero-inflated Poisson model with measurement errors, which can be treated within this framework.     

The wild bootstrap for multivariate Nelson–Aalen estimators

Lifetime Data Analysis - We rigorously extend the widely used wild bootstrap resampling technique to the multivariate Nelson–Aalen estimator under Aalen’s multiplicative intensity...     

Functional principal component analysis for the explorative analysis of multisite–multivariate air pollution time series with long gaps

The knowledge of the urban air quality represents the first step to face air pollution issues. For the last decades many cities can rely on a network of monitoring stations recording concentration values for the main pollutants. This paper focuses on functional principal component analysis (FPCA) to investigate multiple pollutant datasets measured over time at multiple sites within a given urban area. Our purpose is to extend what has been proposed in the literature to data that are multisite and multivariate at the same time. The approach results to be effective to highlight some relevant statistical features of the time series, giving the opportunity to identify significant pollutants and to know the evolution of their variability along time. The paper also deals with missing value issue. As it is known, very long gap sequences can often occur in air quality datasets, due to long time failures not easily solvable or to data coming from a mobile monitoring station. In the considered dataset, large and continuous gaps are imputed by empirical orthogonal function procedure, after denoising raw data by functional data analysis and before performing FPCA, in order to further improve the reconstruction.     

A two-stage estimation in the Clayton–Oakes model with marginal linear transformation models for multivariate failure time data

This paper considers the analysis of multivariate survival data where the marginal distributions are specified by semiparametric transformation models, a general class including the Cox model and the proportional odds model as special cases. First, consideration is given to the situation where the joint distribution of all failure times within the same cluster is specified by the Clayton–Oakes model (Clayton, Biometrika 65:141–151, l978; Oakes, J R Stat Soc B 44:412–422, 1982). A two-stage estimation procedure is adopted by first estimating the marginal parameters under the independence working assumption, and then the association parameter is estimated from the maximization of the full likelihood function with the estimators of the marginal parameters plugged in. The asymptotic properties of all estimators in the semiparametric model are derived. For the second situation, the third and higher order dependency structures are left unspecified, and interest focuses on the pairwise correlation between any two failure times. Thus, the pairwise association estimate can be obtained in the second stage by maximizing the pairwise likelihood function. Large sample properties for the pairwise association are also derived. Simulation studies show that the proposed approach is appropriate for practical use. To illustrate, a subset of the data from the Diabetic Retinopathy Study is used.     

Robust multivariate control charts based on Birnbaum–Saunders distributions

Multivariate control charts are powerful and simple visual tools for monitoring the quality of a process. This multivariate monitoring is carried out by considering simultaneously several correlated quality characteristics and by determining whether these characteristics are in control or out of control. In this paper, we propose a robust methodology using multivariate quality control charts for subgroups based on generalized Birnbaum–Saunders distributions and an adapted Hotelling statistic. This methodology is constructed for Phases I and II of control charts. We estimate the corresponding parameters with the maximum likelihood method and use parametric bootstrapping to obtain the distribution of the adapted Hotelling statistic. In addition, we consider the Mahalanobis distance to detect multivariate outliers and use it to assess the adequacy of the distributional assumption. A Monte Carlo simulation study is conducted to evaluate the proposed methodology and to compare it with a standard methodology. This study reports the good performance of our methodology. An illustration with real-world air quality data of Santiago, Chile, is provided. This illustration shows that the methodology is useful for alerting early episodes of extreme air pollution, thus preventing adverse effects on human health.     

Spatial modelling for binary data using␣a␣hidden conditional autoregressive Gaussian process: a multivariate extension of the probit model

A Bayesian approach to modelling binary data on a regular lattice is introduced. The method uses a hierarchical model where the observed data is the sign of a hidden conditional autoregressive Gaussian process. This approach essentially extends the familiar probit model to dependent data. Markov chain Monte Carlo simulations are used on real and simulated data to estimate the posterior distribution of the spatial dependency parameters and the method is shown to work well. The method can be straightforwardly extended to regression models.     

EM-based maximum likelihood parameter estimation for multivariate generalized hyperbolic distributions with fixed λ

Generalized Hyperbolic distribution (Barndorff-Nielsen 1977) is a variance-mean mixture of a normal distribution with the Generalized Inverse Gaussian distribution. Recently subclasses of these distributions (e.g., the hyperbolic distribution and the Normal Inverse Gaussian distribution) have been applied to construct stochastic processes in turbulence and particularly in finance, where multidimensional problems are of special interest. Parameter estimation for these distributions based on an i.i.d. sample is a difficult task even for a specified one-dimensional subclass (subclass being uniquely defined by ) and relies on numerical methods. For the hyperbolic subclass ( = 1), computer program hyp (Blæsild and Sørensen 1992) estimates parameters via ML when the dimensionality is less than or equal to three. To the best of the author's knowledge, no successful attempts have been made to fit any given subclass when the dimensionality is greater than three. This article proposes a simple EM-based (Dempster, Laird and Rubin 1977) ML estimation procedure to estimate parameters of the distribution when the subclass is known regardless of the dimensionality. Our method relies on the ability to numerically evaluate modified Bessel functions of the third kind and their logarithms, which is made possible by currently available software. The method is applied to fit the five dimensional Normal Inverse Gaussian distribution to a series of returns on foreign exchange rates.     

The flood algorithm—a multivariate, self-organizing-map-based, robust location and covariance estimator

Self-organizing maps (SOMs) introduced by Kohonen (Biol. Cybern. 43(1):59–69, 1982) are well-known in the field of artificial neural networks. The way SOMs are performing is very intuitive, leading to great popularity and numerous applications (related to statistics: classification, clustering). The result of the unsupervised learning process performed by SOMs is a non-linear, low-dimensional projection of the high-dimensional input data, that preserves certain features of the underlying data, e.g. the topology and probability distribution (Lee and Verleysen in Nonlinear Dimensionality Reduction, Springer, 2007; Kohonen in Self-organizing Maps, 3rd edn., Springer, 2001).     

Measures of biomarker dependence using a copula-based multivariate epsilon–skew–normal family of distributions

In this note we develop a new multivariate copula model based on epsilon–skew–normal marginal densities for the purpose of examining biomarker dependency structures. We illustrate the flexibility and utility of this model via a variety of graphical tools and a data analysis example pertaining to salivary biomarker. The multivariate normal model is a sub-model of the multivariate epsilon–skew–normal distribution.     

A Monte Carlo comparison of Jarque–Bera type tests and Henze–Zirkler test of multivariate normality

In the paper, tests for multivariate normality (MVN) of Jarque-Bera type, based on skewness and kurtosis, have been considered. Tests proposed by Mardia and Srivastava, and the combined tests based on skewness and kurtosis defined by Jarque and Bera have been taken into account. In the Monte Carlo simulations, for each combination of p = 2, 3, 4, 5 number of traits and n = 10(5)50(10)100 sample sizes 10,000 runs have been done to calculate empirical Type I errors of tests under consideration, and empirical power against different alternative distributions. Simulation results have been compared to the Henze–Zirkler’s test. It should be stressed that no test yet proposed is uniformly better than all the others in every combination of conditions examined.     

A multivariate test for detecting fraud based on Benford’s law,with application to music streaming data

Statistical Methods & Applications - Benford’s law became a prevalent concept for fraud and anomaly detection. It examines the frequencies of the leading digits of numbers in a collection...     

Discussion of “The power of monitoring: how to make the most of a contaminated multivariate sample” by Andrea Cerioli,Marco Riani,Anthony C. Atkinson and Aldo Corbellini

This paper discusses the contribution of Cerioli et al. (Stat Methods Appl, 2018), where robust monitoring based on high breakdown point estimators is proposed for multivariate data. The results follow years of development in robust diagnostic techniques. We discuss the issues of extending data monitoring to other models with complex structure, e.g. factor analysis, mixed linear models for which S and MM-estimators exist or deviating data cells. We emphasise the importance of robust testing that is often overlooked despite robust tests being readily available once S and MM-estimators have been defined. We mention open questions like out-of-sample inference or big data issues that would benefit from monitoring.     

Order statistics and their concomitants from multivariate normal mean–variance mixture distributions with application to Swiss Markets Data

In this article, by considering a multivariate normal mean–variance mixture distribution, we derive the exact joint distribution of linear combinations of order statistics and their concomitants. From this general result, we then deduce the exact marginal and conditional distributions of order statistics and their concomitants arising from this distribution. We finally illustrate the usefulness of these results by using a Swiss markets dataset.     

Robust mixture modeling using multivariate skew <Emphasis Type="Italic">t</Emphasis> distributions

This paper presents a robust mixture modeling framework using the multivariate skew t distributions, an extension of the multivariate Student’s t family with additional shape parameters to regulate skewness. The proposed model results in a very complicated likelihood. Two variants of Monte Carlo EM algorithms are developed to carry out maximum likelihood estimation of mixture parameters. In addition, we offer a general information-based method for obtaining the asymptotic covariance matrix of maximum likelihood estimates. Some practical issues including the selection of starting values as well as the stopping criterion are also discussed. The proposed methodology is applied to a subset of the Australian Institute of Sport data for illustration.     

A note on the equality of the OLSE and the BLUE of the parametric function in the general Gauss–Markov model

In this note we consider the equality of the ordinary least squares estimator (OLSE) and the best linear unbiased estimator (BLUE) of the estimable parametric function in the general Gauss–Markov model. Especially we consider the structures of the covariance matrix V for which the OLSE equals the BLUE. Our results are based on the properties of a particular reparametrized version of the original Gauss–Markov model.      

Response to the “letter to the editor”

First of all, the parameters involved in Chen and Hsu (1995) are unknown. For convenience, we discussed the two possibilities of the unknown parameter μ namely μ =M and μ M, separately. In other words, we do not known wheter μ = M or μ M. Therefore, we must estimate it before drawing any statistical conclusion about the index C_pmk, The conclusion should be applied only after th ehypothesis if μ = M is tested.