Local dependence in bivariate copulae with beta marginals

The local dependence function (LDF) describes changes in the correlation structure of continuous bivariate random variables along their range. Bivariate density functions with Beta marginals can be used to model jointly a wide variety of data with bounded outcomes in the (0,1) range, e.g. proportions. In this paper we obtain expressions for the LDF of bivariate densities constructed using three different copula models (Frank, Gumbel and Joe) with Beta marginal distributions, present examples for each, and discuss an application of these models to analyse data collected in a study of marks obtained on a statistics exam by postgraduate students

Bivariate beta-generated distributions with applications to well-being data

ABSTRACT: The class of beta-generated distributions (Commun. Stat. Theory Methods 31:497–512, 2002; TEST 13:1–43, 2004) has received a lot of attention in the last years. In this paper, three new classes of bivariate beta-generated distributions are proposed. These classes are constructed using three different definitions of bivariate distributions with classical beta marginals and different covariance structures. We work with the bivariate beta distributions proposed in (J. Educ. Stat. 7:271–294, 1982; Metrika 54:215–231, 2001; Stat. Probability Lett. 62:407–412, 2003) for the first proposal, in (Stat. Methods Appl. 18: 465–481, 2009) for the second proposal and (J. Multivariate Anal. 102:1194–1202, 2011) for the third one. In each of these three classes, the main properties are studied. Some specific bivariate beta-generated distributions are studied. Finally, some empirical applications with well-being data are presented

Copula models for epidemiological research and practice

Investigating associations between random variables (rvs) is one of many topics in the heart of statistical science. Graphical displays show emerging patterns between rvs, and the strength of their association is conventionally quantified via correlation coefficients. When two or more of these rvs are thought of as outcomes, their association is governed by a joint probability distribution function (pdf). When the joint pdf is bivariate normal, scalar correlation coefficients will produce a satisfactory summary of the association, otherwise alternative measures are needed. Local dependence functions, together with their corresponding graphical displays, quantify and show how the strength of the association varies across the span of the data. Additionally, the multivariate distribution function can be explicitly formulated and explored. Copulas model joint distributions of varying shapes by combining the separate (univariate) marginal cumulative distribution functions of each rv under a specified correlation structure. Copula models can be used to analyse complex relationships and incorporate covariates into their parameters. Therefore, they offer increased flexibility in modelling dependence between rvs. Copula models may also be used to construct bivariate analogues of centiles, an application for which few references are available in the literature though it is of particular interest for many paediatric applications. Population centiles are widely used to highlight children or adults who have unusual univariate outcomes. Whilst the methodology for the construction of univariate centiles is well established there has been very little work in the area of bivariate analogues of centiles where two outcomes are jointly considered. Conditional models can increase the efficiency of centile analogues in detection of individuals who require some form of intervention. Such adjustments can be readily incorporated into the modelling of the marginal distributions and of the dependence parameter within the copula model

Closed-Form Bayesian Inferences for the Logit Model via Polynomial Expansions

Articles in Marketing and choice literatures have demonstrated the need for incorporating person-level heterogeneity into behavioral models (e.g., logit models for multiple binary outcomes as studied here). However, the logit likelihood extended with a population distribution of heterogeneity doesn't yield closed-form inferences, and therefore numerical integration techniques are relied upon (e.g., MCMC methods). We present here an alternative, closed-form Bayesian inferences for the logit model, which we obtain by approximating the logit likelihood via a polynomial expansion, and then positing a distribution of heterogeneity from a flexible family that is now conjugate and integrable. For problems where the response coefficients are independent, choosing the Gamma distribution leads to rapidly convergent closed-form expansions; if there are correlations among the coefficients one can still obtain rapidly convergent closed-form expansions by positing a distribution of heterogeneity from a Multivariate Gamma distribution. The solution then comes from the moment generating function of the Multivariate Gamma distribution or in general from the multivariate heterogeneity distribution assumed. Closed-form Bayesian inferences, derivatives (useful for elasticity calculations), population distribution parameter estimates (useful for summarization) and starting values (useful for complicated algorithms) are hence directly available. Two simulation studies demonstrate the efficacy of our approach.Comment: 30 pages, 2 figures, corrected some typos. Appears in Quantitative Marketing and Economics vol 4 (2006), no. 2, 173--20