Context-Specific independencies embedded in Chain Graph Models of type I

For a set of variables collected in a contingency table, we focus on a particular kind of relationships such as the context-specific independencies. These are conditional independencies that hold for particular values of the conditioning set. Given the advantages of the graphical models, we use them to represent different relationships among the variables, including the context-specific independencies. In particular, we enrich chain graph models with labelled arcs. Furthermore, we consider the well-known relationships between chain graph models and hierarchical multinomial marginal models and we introduce new constraints on parameters in order to describe the context-specific relationship. Finally, we provide an application to the study of innovation in Italy by comparing two different periods

Context-specific independencies in hierarchical multinomial marginal models

This paper focuses on studying the relationships among a set of categorical (ordinal) variables collected in a contingency table. Besides the marginal and conditional (in)dependencies, thoroughly analyzed in the literature, we consider the context-specific independencies holding only in a subspace of the outcome space of the conditioning variables. To this purpose we consider the hierarchical multinomial marginal models and we provide several original results about the representation of context-specific independencies through these models. The theoretical results are supported by an application concerning the innovation degree of Italian enterprises

Gram–Charlier-Like Expansions of the Convoluted Hyperbolic-Secant Density

Since financial series are usually heavy tailed and skewed, research has formerly considered well-known leptokurtic distributions to model these series and, recently, has focused on the technique of adjusting the moments of a probability law by using its orthogonal polynomials. This paper combines these approaches by modifying the moments of the convoluted hyperbolic secant. The resulting density is a Gram\u2013Charlier-like (GC-like) expansion capable to account for skewness and excess kurtosis. Multivariate extensions of these expansions are obtained on an argument using spherical distributions. Both the univariate and multivariate (GC-like) expansions prove to be effective in modeling heavy-tailed series and computing risk measures

Context-specific independencies in Hierarchical Multinomial Marginal models

This paper focuses on studying the relationships among a set of categorical (ordinal) variables collected in a contingency table. Besides the marginal and conditional (in)dependencies, thoroughly analyzed in the literature, we consider the context-specific independencies holding only in a subspace of the outcome space of the conditioning variables. To this purpose we consider the Hierarchical Multinomial Marginal models and we provide several original results about the representation of context-specific independencies through these models. The theoretical results are supported by an application concerning the innovation degree of Italian enterprises

Efficient unequal probability resampling from finite populations

A resampling technique for probability-proportional-to size sampling designs is proposed. It is essentially based on a special form of variable probability, without replacement sampling applied directly to the sample data, yet according to the pseudo-population approach. From a theoretical point of view, it is asymptotically correct: as both the sample size and the population size increase, under mild regularity conditions the proposed resampling design tends to coincide with the original sampling design under which sample data were collected. From a computational point of view, the proposed methodology is easy to be implemented and efficient, because it neither requires the actual construction of the pseudo-population nor any form of randomization to ensure integer weights and sizes. Empirical evidence based on a simulation study1 indicates that the proposed resampling technique outperforms its two main competitors for confidence interval construction of various population parameters including quantiles. (c) 2021 Published by Elsevier B.V

Improving the power of hypothesis tests in sparse contingency tables

When analyzing data in contingency tables it is frequent to deal with sparse data, particularly when the sample size is small relative to the number of cells. Most analyses of this kind are interpreted in an exploratory manner and even if tests are performed, little attention is paid to statistical power. This paper proposes a method we call redundant procedure, which is based on the union–intersection principle and increases test power by focusing on specific components of the hypothesis. This method is particularly helpful when the hypothesis to be tested can be expressed as the intersections of simpler models, such that at least some of them pertain to smaller table marginals. This situation leads to working on tables that are naturally denser. One advantage of this method is its direct application to (chain) graphical models. We illustrate the proposal through simulations and suggest strategies to increase the power of tests in sparse tables. Finally, we demonstrate an application to the EU-SILC dataset

Trivariate Burr-III copula with applications to income data

In this work, Bivariate Burr-III copula is extended to the trivariate case. This copula seems to be very general and analytically manageable and it provides an alternative to the commonly employed elliptical copulas (such as the Gaussian or the Stutent's t ones) since they have, roughly, the same number of parameters. Several applications to income and wine data are described in the paper. They show that the Trivariate Burr-III copula is, in general, able to capture the dependence structure implicit in observed trivariate data. Moreover, they show that the third-order interaction parameter results, in some cases, significant at 1\% 1 % significance level while, in other cases, it can be removed from the fitted model. The ability of the Trivariate Burr-III copula in representing the dependence structure implicit in the considered data is compared with the ones of other well known copulas: the Clayton copula, the t copula, and the Skew-t copula. It results that the Trivariate Burr-III copula provides a good fitting and turns out to be the best performer in fitting the considered wine data but, on income data, the best performers are the t and Skew-t copulas. The over-performance of the last two copulas on income data is probably due to their ability in representing right-tail dependence (a kind of dependence that is not taken into account by the Trivariate Burr-III copula)

Type II chain graph models for categorical data : a smooth subclass

The Probabilistic Graphical Models use graphs in order to represent the joint distribution of q variables. These models are useful for their ability to capture and represent the system of independence relationships among the variables involved, even when complex. This work concerns categorical variables and the possibility to represent symmetric and asymmetric dependences among categorical variables. For this reason we use the Chain Graphical Models proposed by Andersson, Madigan and Perlman (Scand. J. Stat. 28 (2001) 33-85), also known as Chain Graphical Models of type II (GMs II). The GMs II allow for symmetric relationships typical of log-linear models and, at the same time, asymmetric dependences typical of Graphical Models for Directed Acyclic Graphs. In general, GMs II are not smooth, however this work provides a subclass of smooth GMs II by parametrizing the probability function through marginal log-linear models. Furthermore, the proposed models are applied to a data-set from the European Value Study for the year 2008 (EVS (2010))

Context-specific independence in innovation study

The study of (in)dependence relationships among a set of categorical variables collected in a contingency table is an amply topic. In this work we want to focus on the so called context-specific independence where the conditional independence holds only in a subspace of the outcome space. The main aspects that we introduce concern the definition in the same model of marginal, conditional and context-specific independencies, through the marginal models. Furthermore, we investigate how it is possible to test these context-specific independencies when there are ordinal variables. Finally, we propose a graphical representation of all the considered independencies taking advantages from the chain graph model. We show the results on an application on \u201dThe Italian Innovation Survey\u201d of Istat (2012)