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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.

Frailty models are used in the survival analysis to account for the unobserved heterogeneity in individual risks to disease and death. To analyze the bivariate data on related survival times (e.g., matched pairs experiment, twin or family data), the shared frailty models were suggested. These models are based on the assumption that frailty acts multiplicatively to hazard rate. In this paper, we assume that frailty acts additively to hazard rate. We introduce the positive stable shared frailty models with three different baseline distributions namely, the generalized log-logistic and the generalized Weibull distributions. We introduce the Bayesian estimation procedure using Markov Chain Monte Carlo (MCMC) technique to estimate the parameters involved in these models. We apply these models to a real-life bivariate survival data set of McGilchrist and Aisbett (Biometrics 47:461–466, 1991) related to the kidney infection data and a better model is suggested for the data.

In this work, we utilise a recently developed method of constructing a bivariate distributional model to generate a new bivariate distribution with Weibull and Burr type XII distributions as its marginals. Some general characteristics of this newly generated distribution together with some characterization results are further derived. The maximum likelihood method of estimation is applied for the estimation of the parameters of the constructed distribution. The closeness of the maximum likelihood estimators with the true values of the parameters have been illustrated through a simulation study. We have identified the generated model as a suitable model for a real life bivariate data set reported in the literature. We have also used another bivariate data set on geoelectrical variables X and Y in which X represents the aquifer thickness and Y represents the coefficient of anisotropy to illustrate the results described in this paper for bivariate distributional modelling.

In this work, we investigate the asymptotic behaviour of the intermediate and central order statistics of a bivariate data by using the Reduced Ordering Principle (R-ordering). When, the sup-norm i...

Methods to compare two samples of interval-valued data are discussed. In the interval-valued data, observations have the form of intervals and are often read as two-dimensional vector of lower and upper bounds. We consider four methods, two of which are the applications of existing methods for bivariate data, and the other two are based on the marginalization (univariate distributional representation) of the interval-valued data. We conduct a comprehensive numerical study and analysis of real data to understand the performance of four methods.

Frailty models are used in the survival analysis to account for the unobserved heterogeneity in individual risks to disease and death. To analyze the bivariate data on related survival times, the s...

Bivariate data consist of 2 random variables that are obtained from the same population. The relationship between 2 bivariate data can be measured by correlation coefficient. A correlation coefficient computed from the sample data is used to measure the strength and direction of a linear relationship between 2 variables. However, the classical correlation coefficient results are inadequate in the presence of outliers. Therefore, this study focuses on the performance of different correlation coefficient under contaminated bivariate data to determine the strength of their relationships. We compared the performance of 5 types of correlation, which are classical correlations such as Pearson correlation, Spearman correlation and Kendall’s Tau correlation with other robust correlations, such as median correlation and median absolute deviation correlation. Results show that when there is no contamination in data, all 5 correlation methods show a strong relationship between 2 random variables. However, under the condition of data contamination, median absolute deviation correlation denotes a strong relationship compared to other methods.

The definition of multivariate quantiles has gained considerable attention in previous years as a tool for understanding the structure of a multivariate data cloud. Due to the lack of a natural ordering for multivariate data, many approaches have either considered geometric generalisations of univariate quantiles or data depths that measure centrality of data points. Both approaches provide a centre-outward ordering of data points but do no longer possess a relation to the cumulative distribution function of the data generating process and corresponding tail probabilities. We propose a new notion of bivariate quantiles that is based on inverting the bivariate cumulative distribution function and therefore provides a directional measure of extremeness as defined by the contour lines of the cumulative distribution function which define the quantile curves of interest. To determine unique solutions, we transform the bivariate data to the unit square. This allows us to introduce directions along which the quantiles are unique. Choosing a suitable transformation also ensures that the resulting quantiles are equivariant under monotonically increasing transformations. We study the resulting notion of bivariate quantiles in detail, with respect to computation based on linear programming and theoretical properties including asymptotic behaviour and robustness. It turns out that our approach is especially useful for data situations that deviate from the elliptical shape typical for ‘normal-like’ bivariate distributions. Moreover, the bivariate quantiles inherit the robustness of univariate quantiles even in case of extreme outliers.

In this paper, we investigate the linkages between global crude oil market volatilities and financial markets and the degree of synchrony of crude oil markets and stock markets. A nonlinear cross-analysis of bivariate data method called CRP is applied to study the probability distribution of occurrence of similar states and the time span of occurrence of synchronization dynamics for crude oil return series and stock return series. Further, a new composite multiscale complexity invariance distance is introduced to measure the similarity of complexity between crude oil markets and stock markets. The results of this study show that there is a synchronization in the crude oil markets and stock markets, and those two systems have similar complexity from composite multiscale perspective.

Statistical analyses of directional or angular data have applications in a variety of fields, such as geology, meteorology and bioinformatics. There is substantial literature on descriptive and inferential techniques for univariate angular data, with the bivariate (or more generally, multivariate) cases receiving more attention in recent years. More specifically, the bivariate wrapped normal, von Mises sine and von Mises cosine distributions, and mixtures thereof, have been proposed for practical use. However, there is a lack of software implementing these distributions and the associated inferential techniques. In this article, we introduce BAMBI, an R package for analyzing bivariate (and univariate) angular data. We implement random data generation, density evaluation, and computation of theoretical summary measures (variances and correlation coefficients) for the three aforementioned bivariate angular distributions, as well as two univariate angular distributions: the univariate wrapped normal and the univariate von Mises distribution. The major contribution of BAMBI to statistical computing is in providing Bayesian methods for modeling angular data using finite mixtures of these distributions. We also provide functions for visual and numerical diagnostics and Bayesian inference for the fitted models. In this article, we first provide a brief review of the distributions and techniques used in BAMBI, then describe the capabilities of the package, and finally conclude with demonstrations of mixture model fitting using BAMBI on the two real data sets included in the package, one univariate and one bivariate.

Many clinical decisions are taken based on the results of continuous diagnostic tests. Usually, only the results of one single test is taken into consideration, the interpretation of which requires a reference range for the healthy population. However, the use of two different tests, can be necessary in the diagnosis of certain diseases. This obliges a bivariate reference region be available for their interpretation. It should also be remembered that reference regions may depend on patient variables (eg, age and sex) independent of the suspected disease. However, few proposals have been made regarding the statistical modeling of such reference regions, and those put forward have always assumed a Gaussian distribution, which can be rather restrictive. The present work describes a new statistical method that allows such reference regions to be estimated with no insistence on the results being normally distributed. The proposed method is based on a bivariate location-scale model that provides probabilistic regions covering a specific percentage of the bivariate data, dependent on certain covariates. The reference region is estimated nonparametrically and the nonlinear effects of continuous covariates via polynomial kernel smoothers in additive models. The bivariate model is estimated using a backfitting algorithm, and the optimal smoothing parameters of the kernel smoothers selected by cross-validation. The model performed satisfactorily in simulation studies under the assumption of non-Gaussian conditions. Finally, the proposed methodology was found to be useful in estimating a reference region for two continuous diagnostic tests for diabetes (fasting plasma glucose and glycated hemoglobin), taking into account the age of the patient Óscar Lado-Baleato is funded by a predoctoral grant (ED481A-2018) from the Galician Government (Plan I2C)-Xunta de Galicia. This research was also supported by grants from the Carlos III Health Institute, Spain (PI16/01404 and RD16/0017/0018), and by the project MTM2017-83513-R cofinanced by the Ministry of Economy and Competitiveness (SPAIN) and the European Regional Development Fund (FEDER). This work was also supported by grants from the Galician Government: RED INBIOEST (ED341D-R2016/032), Grupo de Referencia Competitiva (ED431C 2016-025), and Grupo de Potencial Crecimiento (IN607B 2018-1). Javier Roca-Pardiñas acknowledges financial support by the Grant MTM2017-89422-P (MINECO/AEI/FEDER, UE) SI

A novel, general two-sample hypothesis testing procedure is established for testing the equality of tail copulas associated with bivariate data. More precisely, using an ingenious transformation of a natural two-sample tail copula process, a test process is constructed, which is shown to converge in distribution to a standard Wiener process. Hence, from this test process a myriad of asymptotically distribution-free two-sample tests can be obtained. The good finite-sample behavior of our procedure is demonstrated through Monte Carlo simulations. Using the new testing procedure, no evidence of a difference in the respective tail copulas is found for pairs of negative daily log-returns of equity indices during and after the global financial crisis.

In this article we describe a new Hermite series based sequential estimator for the Spearman rank correlation coefficient and provide algorithms applicable in both the stationary and non-stationary settings. To treat the non-stationary setting, we introduce a novel, exponentially weighted estimator for the Spearman rank correlation, which allows the local nonparametric correlation of a bivariate data stream to be tracked. To the best of our knowledge this is the first algorithm to be proposed for estimating a time varying Spearman rank correlation that does not rely on a moving window approach. We explore the practical effectiveness of the Hermite series based estimators through real data and simulation studies demonstrating good practical performance. The simulation studies in particular reveal competitive performance compared to an existing algorithm. The potential applications of this work are manifold. The Hermite series based Spearman rank correlation estimator can be applied to fast and robust online calculation of correlation which may vary over time. Possible machine learning applications include, amongst others, fast feature selection and hierarchical clustering on massive data sets., Revised article incorporating comments by the reviewers, associate editor and editor of the Journal of Multivariate Analysis (accepted manuscript). Changes include a modified title and a significantly shortened article

This paper proposes a new class of covariance functions for bivariate random fields on spheres, having the same properties as the bivariate Matern model proposed in Euclidean spaces. The new class depends on the geodesic distance on a sphere; it allows for indexing differentiability (in the mean square sense) and fractal dimensions of the components of any bivariate Gaussian random field having such covariance structure. We find parameter conditions ensuring positive definiteness. We discuss other possible models and illustrate our findings through a simulation study, where we explore the performance of maximum likelihood estimation method for the parameters of the new covariance function. A data illustration then follows, through a bivariate data set of temperatures and precipitations, observed over a large portion of the Earth, provided by the National Oceanic and Atmospheric Administration Earth System Research Laboratory.

In this work, we investigate the asymptotic behavior of the extremes of a multivariate data by using the Reduced Ordering Principle (R-ordering). When, the sup-norm is used, we reveal the interrelation between the R-ordering principle and Marginal Ordering Principle (M-ordering). The asymptotic behavior of the maximum sup-norms corresponding to the bivariate data is completely determined. Finally, an application to real data illustrates and corroborates the theoretical results.

Copula functions have been extensively used in applied statistics, becoming a good alternative for modeling the dependence of multivariate data. Each copula function has a different dependence structure. An important issue in these applications is the choice of an appropriate copula function model for each one; thus common classical or Bayesian discrimination methods might not be appropriate for determining the best copula. Considering only the special case of bivariate data, we propose a procedure obtained from a recently introduced dependence measure for selecting an appropriate copula for the statistical data analyses.

Recently, a procedure of impounding a given marginal distribution, with the auxiliary density function generated from the assumed form of the distribution for the concomitant of an extreme order statistic of a sample, has been developed to create a bivariate distribution. In this paper we consider a marginal Rayleigh distribution and impound it with the pdf of an extended form of Lindley distribution to generate a new bivariate distribution. Some important properties of this new bivariate model together with some characterization theorems have been also derived. Most commonly used reliability functions associated with the generated bivariate distribution as well have been obtained. The well-known maximum likelihood (ML) method of estimation is applied for the estimation of the parameters of the generated bivariate distribution. A simulation study is also carried out to illustrate the closeness of ML estimates with the true value of the parameters. A real-life bivariate data set is used to demonstrate the application of the results of this paper in generating the appropriate bivariate distributional model in an innovative manner.

Frailty models are used in the survival analysis to account for the unobserved heterogeneity in individual risks to disease and death. To analyze the bivariate data on related survival time...

A simple sorting approach for inducing any desired Pearson or Spearman correlation to independent bivariate data, whose marginals can be of any distributional type and nature is described and illus...

Cylindrical data are bivariate data from the combination of circular and linear variables. However, up to now no work has been done on the detection of outlier in cylindrical data. We intro...

In this study we introduce a new method of adding two shape parameters to any baseline bivariate distribution function (df) to get a more flexible family of bivariate df's. Through the additional parameters we can fully control the type of the resulting family. This method is applied to yield a new two-parameter extension of the bivariate standard normal distribution, denoted by BSSN. The statistical properties of the BSSN family are studied. Moreover, via a mixture of the BSSN family and the standard bivariate logistic df, we get a more capable family, denoted by FBSSN. Theoretically, each of the marginals of the FBSSN contains all the possible types of df's with respect to the signs of skewness and excess kurtosis. In addition, each possesses very wide range of the indices of skewness and kurtosis. Finally, we compare the families BSSN and FBSSN with some important competitors (i.e., some generalized families of bivariate df's) via real data examples. AMS 2010 Subject Classification: 62-07; 62E10; 62F99.

Measures of direction dependence enable researchers to determine the directionality of linear effects in bivariate data. Existing fourth moment-based approaches assume that regression error...

Financial time series are commonly modeled with methods assuming data normality. However, the real distribution can be nontrivial, also not having an explicitly formulated probability density function. In this work we introduce novel parameter estimation and high-powered distribution testing methods which do not rely on closed form densities, but use the characteristic functions for comparison. The approach applied to a pair of stock index returns demonstrates that such a bivariate vector can be a sample coming from a bivariate sub-Gaussian distribution. The methods presented here can be applied to any nontrivially distributed financial data, among others.

Weighted distributions (univariate and bivariate) have received widespread attention over the last two decades because of their flexibility for analyzing skewed data. In this paper, we propose an alternative method to construct a new family of bivariate and multivariate weighted distributions. For illustrative purposes, some examples of the proposed method are presented. Several structural properties of the bivariate weighted distributions including marginal distributions together with distributions of the minimum and maximum, evaluation of the reliability parameter and verification of total positivity of order two are also presented. In addition, we provide some multivariate extensions of the proposed models. A real life data set is used to show the applicability of these bivariate weighted distributions.

Recently, Mahmoudi and Mahmoodian [7] introduced a new class of distributions which contains univariate normal–geometric distribution as a special case. This class of distributions are very flexible and can be used quite effectively to analyze skewed data. In this paper we propose a new bivariate distribution with the normal–geometric distribution marginals. Different properties of this new bivariate distribution have been studied. This distribution has five unknown parameters. The EM algorithm is used to determine the maximum likelihood estimates of the parameters. We analyze one series of real data set for illustrative purposes.

Bivariate counts are collected in many sectors of research but the analysis of such data is often challenging because each series of counts may exhibit different levels and types of dispersion. This paper addresses this problem by proposing a flexible bivariate COM-Poisson model that may handle any combination of over-, equi- and under-dispersion at any levels. In this paper, the bivariate COM-Poisson is developed via Archimedean copulas. The Generalized Quasi-Likelihood (GQL) approach is used to estimate the unknown mean parameters in the copula-based bivariate COM-Poisson model while the dependence parameter is estimated using the copula likelihood. We further introduce a Monte Carlo experiment to generate bivariate COM-Poisson data under different dispersion levels. The performance of the GQL approach is assessed on the simulated data. The model is applied to analyze real-life epileptic seizures data.

Summary This research is motivated by discovering and underpinning genetic causes for the progression of a bilateral eye disease, age-related macular degeneration (AMD), of which the primary outcomes, progression times to late-AMD, are bivariate and interval-censored due to intermittent assessment times. We propose a novel class of copula-based semiparametric transformation models for bivariate data under general interval censoring, which includes the case 1 interval censoring (current status data) and case 2 interval censoring. Specifically, the joint likelihood is modeled through a two-parameter Archimedean copula, which can flexibly characterize the dependence between the two margins in both tails. The marginal distributions are modeled through semiparametric transformation models using sieves, with the proportional hazards or odds model being a special case. We develop a computationally efficient sieve maximum likelihood estimation procedure for the unknown parameters, together with a generalized score test for the regression parameter(s). For the proposed sieve estimators of finite-dimensional parameters, we establish their asymptotic normality and efficiency. Extensive simulations are conducted to evaluate the performance of the proposed method in finite samples. Finally, we apply our method to a genome-wide analysis of AMD progression using the Age-Related Eye Disease Study data, to successfully identify novel risk variants associated with the disease progression. We also produce predicted joint and conditional progression-free probabilities, for patients with different genetic characteristics.

Interval-censored failure time data arise in a number of fields and many authors have discussed various issues related to their analysis. However, most of the existing methods are for univariate data and there exists only limited research on bivariate data, especially on regression analysis of bivariate interval-censored data. We present a class of semiparametric transformation models for the problem and for inference, a sieve maximum likelihood approach is developed. The model provides a great flexibility, in particular including the commonly used proportional hazards model as a special case, and in the approach, Bernstein polynomials are employed. The strong consistency and asymptotic normality of the resulting estimators of regression parameters are established and furthermore, the estimators are shown to be asymptotically efficient. Extensive simulation studies are conducted and indicate that the proposed method works well for practical situations. Supplementary materials for this article are ...

Frailty models are used in the survival analysis to account for the unobserved heterogeneity in individual risks to disease and death. To analyze the bivariate data on related survival times (e.g.,...

A new bivariate beta distribution capable of providing better fits than all its competitors is introduced. Various representations are derived for its product moments, marginal densities, marginal moments, conditional densities and conditional moments. The method of maximum likelihood is used to derive the associated estimation procedure. Applications to six bivariate data sets are illustrated.

The stress-strength models have been intensively investigated in the literature in regards of estimating the reliability = P (X > Y) using parametric and nonparametric approaches under different sampling schemes when X and Y are independent random variables. In this paper, we consider the problem of estimating when (X; Y) are dependent random variables with a bivariate underlying distribution. The empirical and kernel estimates of = P (X > Y), based on bivariate ranked set sampling (BVRSS) are considered, when (X; Y) are paired dependent continuous random variables. The estimators obtained are compared to their counterpart, bivariate simple random sampling (BVSRS), via the bias and mean square error (MSE). We demonstrate that the suggested estimators based on BVRSS are more efficient than those based on BVSRS. A simulation study is conducted to gain insight into the performance of the proposed estimators. A real data example is provided to illustrate the process.

The bivariate Poisson distribution is a popular distribution for modeling bivariate count data. Its basic assumptions and marginal equi-dispersion, however, may prove limiting in some contexts. To allow for data dispersion, we develop here a bivariate Conway-Maxwell-Poisson (COM-Poisson) distribution that includes the bivariate Poisson, bivariate Bernoulli, and bivariate geometric distributions all as special cases. As a result, the bivariate COM-Poisson distribution serves as a flexible alternative and unifying framework for modeling bivariate count data, especially in the presence of data dispersion.

In this paper, we establish the role of concomitants of order statistics in the unique identification of the parent bivariate distribution. From the results developed, we have illustrated by examples the process of determination of the parent bivariate distribution using a marginal pdf and the pdf of either of the concomitant of largest or smallest order statistic on the other variable. An application of the results derived in modeling of a bivariate distribution for data sets drawn from a population as well is discussed.

This paper proposes a bivariate version of the univariate discrete generalized geometric distribution considered by Gomez–Deniz (2010). The proposed bivariate distribution can have a positive or negative correlation coefficient which can be used for modeling bivariate-dependent count data. After discussing some of its properties, maximum likelihood estimation is discussed. Two illustrative examples are given for fitting and demonstrating the usefulness of the new bivariate distribution proposed here.

Bivariate zero-inflated Poisson regression models have recently been used in various medical and biological settings to model excess zeros. However, there has not been any definite approach to deal with the same in the event of missing responses. The model itself is complex and as the responses are paired, missing values can occur in either or both coordinates. We propose a flexible Monte Carlo expectation maximization based approach to handle bivariate zero inflated count data with missing responses. We report the results of a simulation study designed to evaluate the performance of the proposed approach. To illustrate the application of our model and methodology, we consider a bivariate data concerning the demand for health care in Australia.

In this work, we propose a bivariate Birnbaum-Saunders regression model through the use of bivariate Sinh-normal distribution. The proposed regression model has its marginal as the Birnbaum-Saunders regression model of Rieck and Nedelman (1991), which has been discussed extensively by various authors with natural applications in survival and reliability studies. This bivariate regression model can be used to analyze correlated log-lifetimes of two units, in which the dependence structure between observations arises from the bivariate normal distribution.The main aim of this paper is to propose a bivariate Birnbaum-Saunders regression model and discuss some of its properties. Specifically, we have developed the moment estimation, the maximum likelihood estimation and the observed Fisher information matrix. Hypothesis testing is also performed by the use of the asymptotic normality of the maximum-likelihood estimators. Finally, the results of simulation studies as well as an application to a real data set are presented to illustrate the model and all the inferential methods developed here.

Frailty models are used in the survival analysis to account for the unobserved heterogeneity in individual risks to disease and death. To analyze the bivariate data on related survival times, the shared frailty models were suggested. In this paper, we introduce the shared inverse Gaussian frailty model based on the reversed hazard rate with the three baseline distributions, namely, log-logistic distribution, the inverse Weibull distribution and the generalized Weibull distribution. We introduce the Bayesian estimation procedure using the Markov Chain Monte Carlo technique to estimate the parameters involved in the model. We present a simulation study and show that the estimates of the parameters are very close to true values of the parameters. We apply the proposed models to the Australian twin data set and suggest a better model from the proposed eight models for Australian twin data set.

The problem of evaluating a military or GPS/GSM system’s precision quality is considered in this article, where one sequentially observes bivariate normal data ( X i , Y i ) ’s and wants to test hypotheses on the circular error probability (CEP) or the probability of nonconforming, i.e., the probabilities of the system hitting or missing a pre-specified disk target. In such a problem, we first consider a sequential probability ratio test (SPRT) developed under the erroneous assumption of the correlation coefficient ρ = 0 , and investigate its properties when the true ρ ≠ 0 . It was shown that at least one of the Type I and Type II error probabilities would be larger than the required ones if the true ρ ≠ 0 , and for the detailed effects, exp { − 2 } ≈ 0.1353 turns out to be a critical value for the hypothesized probability of nonconforming. Moreover, we propose several sequential tests when the correlation coefficient ρ is unknown, and among these tests, the method of generalized sequential likelihood ratio test (GSLRT) in Bangdiwala (1982) seems to work well.

We develop an empirical likelihood (EL) approach to test independence of two univariate random variables X and Y versus the alternative that X and Y are strictly positive quadrant dependent (PQD). Establishing this type of ordering between X and Y is of interest in many applications, including finance, insurance, engineering, and other areas. Adopting the framework in Einmahl and McKeague, we create a distribution-free test statistic that integrates a localized EL ratio test statistic with respect to the empirical joint distribution of X and Y. When compared to well-known existing tests and distance-based tests we develop by using copula functions, simulation results show the EL testing procedure performs well in a variety of scenarios when X and Y are strictly PQD. We use three datasets for illustration and provide an online R resource practitioners can use to implement the methods in this article. Supplementary materials for this article are available online.

This paper proposes a new minimum description length procedure to detect multiple changepoints in time series data when some times are a priori thought more likely to be changepoints. This scenario arises with temperature time series homogenization pursuits, our focus here. Our Bayesian procedure constructs a natural prior distribution for the situation, and is shown to estimate the changepoint locations consistently, with an optimal convergence rate. Our methods substantially improve changepoint detection power when prior information is available. The methods are also tailored to bivariate data, allowing changes to occur in one or both component series.

We propose measures of copula reflection and permutation asymmetry for data with positive quadrant dependence. We first define the measures of reflection asymmetry using a weighting function and then extend this approach to construct measures of permutation asymmetry for bivariate data. We define the corresponding statistical tests based on these measures and find that the proposed tests have higher statistical power comparing to some other tests for permutation and reflection symmetry studied in the literature. In addition, the measures can be used to summarize dependence structure of a multivariate data set in a few numbers and to select a more appropriate copula in the model.

This paper introduces several forms of bivariate generalized Poisson regression model (BGPR) which can be fitted to bivariate and correlated count data with covariates. The main advantage of these forms of BGPR is that they are nested and thus they allow likelihood ratio tests to be performed to choose the best model. The BGPR can be fitted not only to bivariate count data with positive, zero, or negative correlations, but also to under- or overdispersed bivariate count data with flexible form of mean–variance relationship. Applications of several forms of the BGPR are illustrated on two sets of count data: the Australian health survey data and the US National Medical Expenditure Survey data.

Kendall's tau statistic has been applied to test an association of bivariate random variables. However, incomplete bivariate data with a truncation and a censoring results in incomparable or unorderable pairs. With such a partial information, Tsai (1990) suggested a conditional tau statistic and a test procedure for a quasi independence that was extended to more diverse cases such as double truncation and a semi-competing risk data. In this paper, we also employed a conditional tau statistic to estimate an association of bivariate interval censored data. The suggested method shows a better result in simulation studies than Betensky and Finkelstein's multiple imputation method except a case in cases with strong associations. The association of incubation time and infection time from an AIDS cohort study is estimated as a real data example.

As a multivariate generalization of the univariate median, projection depth median (PM) is unique, and enjoys a very high breakdown point, much higher than its affine equivariant competitors such as halfspace depth median. Nevertheless, its computation is challenging. Until now PM can only be exactly computed efficiently for bivariate data. In this article, we develop an algorithm to approximate PM in higher dimensions. Some data examples indicate that the proposed algorithm performs well in terms of both accuracy and efficiency. As an application, we investigate the finite sample relative efficiency of PM by utilizing the Matlab implementation of this algorithm.

Recently, Louzada et al. proposed a new three-parameter distribution with decreasing, increasing, and unimodal hazard functions. They have provided several properties of the distribution, and discussed different inferential issues. In this article we discuss a generalized version of the model following the approach of Marshall and Olkin. The proposed model is more flexible than the Louzada–Marchi–Roman model, although they have the same number of parameters. The model has three unknown parameters, and the hazard function can take different shapes. We propose using the expectation–maximization (EM) algorithm to compute the maximum likelihood estimators of the unknown parameters. We further consider the bivariate generalization of the proposed model and discuss its different properties. The EM algorithm can be used to estimate the unknown parameters in case of the bivariate model also. One bivariate data set has been analyzed for illustrative purposes, and the performance is quite satisfactory.

Frailty models are used in the survival analysis to account for the unobserved heterogeneity in individual risks to disease and death. To analyze the bivariate data on related survival times (e.g.,...

The cluster-weighted model (CWM) is a mixture model with random covariates that allows for flexible clustering/classification and distribution estimation of a random vector composed of a response variable and a set of covariates. Within this class of models, the generalized linear exponential CWM is here introduced especially for modeling bivariate data of mixed-type. Its natural counterpart in the family of latent class models is also defined. Maximum likelihood parameter estimates are derived using the expectation-maximization algorithm and some computational issues are detailed. Through Monte Carlo experiments, the classification performance of the proposed model is compared with other mixture-based approaches, consistency of the estimators of the regression coefficients is evaluated, and several likelihood-based information criteria are compared for selecting the number of mixture components. An application to real data is also finally considered.

Marshall and Olkin (1997, Biometrika, 84, 641 - 652) introduced a very powerful method to introduce an additional parameter to a class of continuous distribution functions and hence it brings more flexibility to the model. They have demonstrated their method for the exponential and Weibull classes. In the same paper they have briefly indicated regarding its bivariate extension. The main aim of this paper is to introduce the same method, for the first time, to the class of discrete generalized exponential distributions both for the univariate and bivariate cases. We investigate several properties of the proposed univariate and bivariate classes. The univariate class has three parameters, whereas the bivariate class has five parameters. It is observed that depending on the parameter values the univariate class can be both zero inflated as well as heavy tailed. We propose to use EM algorithm to estimate the unknown parameters. Small simulation experiments have been performed to see the effectiveness of the proposed EM algorithm, and a bivariate data set has been analyzed and it is observed that the proposed models and the EM algorithm work quite well in practice., Comment: arXiv admin note: text overlap with arXiv:1701.03569

Multimodal and asymmetric bivariate circular data arise in several different disciplines and fitting appropriate distribution plays an important role in the analysis of such data. In this paper, we propose a new bivariate circular distribution which can be used to model both asymmetric and multimodal bivariate circular data simultaneously. In fact the proposed density covers unimodality as well as multimodality, symmetry as well as asymmetry of circular bivariate data. A number of properties of the proposed density are presented. A Bayesian approach with MCMC scheme is employed for statistical inference. Three real datasets and a simulation study are provided to illustrate the performance of the proposed model in comparison with alternative models such as finite mixture Cosine model.

The unknown or unobservable risk factors in the survival analysis cause heterogeneity between individuals. Frailty models are used in the survival analysis to account for the unobserved heterogeneity in individual risks to disease and death. To analyze the bivariate data on related survival times, the shared frailty models were suggested. The most common shared frailty model is a model in which frailty act multiplicatively on the hazard function. In this paper, we introduce the shared gamma frailty model and the inverse Gaussian frailty model with the reversed hazard rate. We introduce the Bayesian estimation procedure using Markov chain Monte Carlo (MCMC) technique to estimate the parameters involved in the model. We present a simulation study to compare the true values of the parameters with the estimated values. We also apply the proposed models to the Australian twin data set and a better model is suggested.