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There exist many bivariate parametric copulas to model bivariate data with different dependence features. We propose a new bivariate parametric copula family that cannot only handle various dependence patterns that appear in the existing parametric bivariate copula families, but also provides a more enriched dependence structure. The proposed copula construction exploits finite mixtures of bivariate normal distributions. The mixing operation, the distinct correlation and mean parameters at each mixture component introduce quite a flexible dependence. The new parametric copula is theoretically investigated, compared with a set of classical bivariate parametric copulas and illustrated on two empirical examples from astrophysics and agriculture where some of the variables have peculiar and asymmetric dependence, respectively.

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.

It has been established that the bivariate log-normal distribution is appropriate for modelling certain paired observations. In this paper, we have developed large-sample confidence intervals of the dependence and reliability R=P(X>Y) parameters from a bivariate log-normal distribution with equal log-normal means. The parameter R provides a general measure of difference between the two populations and has applications in many areas. The performance of these confidence intervals has been examined by extensive simulation studies. The results are illustrated with an example dealing with a quantitative assay problem.

The bivariate negative binomial regression (BNBR) and the bivariate Poisson log-normal regression (BPLR) models have been used to describe count data that are over-dispersed. In this paper, a new bivariate generalized Poisson regression (BGPR) model is defined. An advantage of the new regression model over the BNBR and BPLR models is that the BGPR can be used to model bivariate count data with either over-dispersion or under-dispersion. In this paper, we carry out a simulation study to compare the three regression models when the true data-generating process exhibits over-dispersion. In the simulation experiment, we observe that the bivariate generalized Poisson regression model performs better than the bivariate negative binomial regression model and the BPLR model.

Interval-valued variables have become very common in data analysis. Up until now, symbolic regression mostly approaches this type of data from an optimization point of view, considering neither the probabilistic aspects of the models nor the nonlinear relationships between the interval response and the interval predictors. In this article, we formulate interval-valued variables as bivariate random vectors and introduce the bivariate symbolic regression model based on the generalized linear models theory which provides much-needed exibility in practice. Important inferential aspects are investigated. Applications to synthetic and real data illustrate the usefulness of the proposed approach.

This article is concerned with finding trends, maxima, and minima in a curve underlying a scatter plot of observations. When the data exhibit large variation, the detection of such features can be difficult because it is not easy to say which of the seeming features are statistically significant. Additional difficulties emerge when the values of the predictor variable may contain errors and when errors in the response variable cause dependence in their observed values. We propose a Bayesian approach for finding statistically significant features in a scatter plot under such challenging conditions. The method extends in several ways the BSiZer approach earlier introduced by the authors.

Heterogeneity in lifetime data may be modelled by multiplying an individual's hazard by an unobserved frailty. We test for the presence of frailty of this kind in univariate and bivariate data with Weibull distributed lifetimes, using statistics based on the ordered Cox-Snell residuals from the null model of no frailty. The form of the statistics is suggested by outlier testing in the gamma distribution. We find through simulation that the sum of the k largest or k smallest order statistics, for suitably chosen k, provides a powerful test when the frailty distribution is assumed to be gamma or positive stable, respectively. We provide recommended values of k for sample sizes up to 100 and simple formulae for estimated critical values for tests at the 5% level.

The polychoric correlation and an approximate maximum likelihood estimator of correlation are compared for count data that are assumed to be derived from an underlying bivariate normal distribution. A related chi squared test for bivariate normality is also examined.

A simple test is given for comparing three populations when one has bivariate data (X, Y) and the primary comparison of interest involves Y.The alternative of interest is an ordered one, e.g. for all x. The distribution of the test statistic is estimated by Monte Carlo simulation when the data is bivariate normal. Further simulation studies indicate that the distribution of the test statistic is robust with respect to departures from normality. Examples are given to illustrate the value of this test procedure compared to one which considers only the Y values and not the X values.

Various simple and exact methods for computer generation of bivariate Beta random variables are considered. Comparative timings over a range of values of the distribution parameters are given and some recommendations are made.

The problem of unbiased estimation of the common mean in sampling from a bivariate normal distribution is considered. Performance of several estimators is compared.