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A class of absolutely continuous bivariate exponential distributions is constructed using the product form of a first order autoregressive model. Inference methods are proposed for parameter estimation and diagnosis. Data analysis is carried out to illustrate the applications.

Non-exchangeable dependence structures exist in the real world. In particular, when dependence patterns in joint distributional tails are important, such as in the fields of engineering, environmetrics and econometrics, one may need to detect the existence, and assess the strength of non-exchangeable dependence patterns in the tails. In this paper, we propose a sensible metric to quantify the degree of tail non-exchangeability of bivariate copulas. Based on the metric, we propose a practical method of assessing tail non-exchangeable dependence with uniform scores of bivariate data. An empirical example is used to demonstrate the usefulness of the proposed method.

Let { X i , Y i } be jointly distributed second-order random variables with correlation coefficient r . The estimation of r from the observations { X i , Y i } i = 1 n is a classical problem which has been examined under the assumption of an i.i.d. setting. In this paper we examine the statistical properties of the correlation coefficient estimate when the process { X i , Y i } is dependent, constituting either a strongly mixing process or asymptotically uncorrelated. We establish convergence in probability (with rates) as well as asymptotic normality for the estimation error and present an explicit expression for the asymptotic variance.

In this paper, we derive distributions of order statistics and a linear combination of order statistics arising from a bivariate selection normal distribution. We show that they are mixtures of univariate selection normal distributions. We then illustrate the usefulness of these results with a real-life data relating to visual acuity analysis. Finally, some concluding remarks are made.

We prove the following fact in this paper: assuming that the bivariate data follows some bivariate frailty model [Oakes, D., 1989. Bivariate survival models induced by frailties. J. Amer. Statist. Assoc. 84, 487–493], the original bivariate data follows the Clayton model if and only if one of its truncated samples follows the Clayton model. On the basis of this fact, we can establish simple conditions for checking the Clayton model assumption when the bivariate frailty model assumption is assumed for the original data.

We propose a bivariate Weibull regression model with heterogeneity (frailty or random effect) which is generated by log-normal distribution. We assume that the bivariate survival data follow bivariate Weibull of [Hanagal, D.D., 2004. Parametric bivariate regression analysis based on censored samples: A Weibull model. Economic Quality Control 19, 83—90]. There are some interesting situations like survival times in genetic epidemiology, dental implants of patients and twin births (both monozygotic and dizygotic) where genetic behaviour (which is unknown and random) of patients follows known frailty distribution. These are the situations which motivate to study this particular model. We propose two-stage maximum likelihood estimation for hierarchical likelihood in the proposed model. We present a small simulation study to compare these estimates with the true value of the parameters and it is observed that these estimates are very close to the true values of the parameters. We also compare theoretical standard errors with Monte Carlo standard errors and theoretical coverage probabilities with Monte Carlo coverage probabilities.

A new Evidence Test is applied to the problem of testing whether two Poisson random variables are dependent. The dependence structure is that of Holgate’s bivariate distribution. These bivariate distribution depends on three parameters, 0i� 1;� 2 i ∞, and 0 6� 3 6 min(� 1;� 2). The Evidence Test was originally developed as a Bayesian test, but in the present paper it is compared to the best known test of the hypothesis of independence in a frequentist framework. It is shown that the Evidence Test is considerably more powerful when the correlation is not too close to zero, even for small samples. c � 2002 Published by Elsevier Science B.V. MSC: 62F03; 62F04

In this note we develop a new analytic bootstrap-type method for estimating the covariance between the bivariate order statistics Xr:n and Ys:n. Correlation and regression estimators follow straightforward from this result. An easy-to-use method for calculating the nonparametric percentile confidence regions for bivariate quantiles is developed and illustrated using biological data. A specific case of interest is the application to bivariate medians, i.e., correlated marginal medians.

The problem of estimating a marginal density from incomplete bivariate data is considered. A kernel estimator is proposed. Strong consistency of the estimator is proved, and asymptotic formulae for its mean and variance derived. A method of bandwidth selection is suggested. Application of the estimator is then illustrated on example data sets. Possible extensions and improvements are discussed.

The maximum likelihood estimator (MLE) of the correlation coefficient and its asymptotic properties are well-known for bivariate normal data when no observations are missing. The situation in which one of the two variates is not observed in some of the data is examined herein. The MLE of the correlation coefficient in the missing data case is shown to be asymptotically normal with variance smaller than that when using only the complete bivariate data. The asymptotic results are valid when the number of incomplete data pairs tends to infinity at a faster rate than the number of complete data pairs. The proofs require some mathematical detail since the sample sizes tend to infinity simultaneously rather than individually. A small simulation study corroborates the theoretical results.

The bivariate Cramer—von Mises statistic has been proposed as a basis for goodness-of-fit testing for continuous bivariate distributions. This statistic can be expressed as the integral, over the unit square I2, of the square of the bivariate uniform empirical process. For some time it apparently went unnoticed that this statistic's value is not ‘origin-invariant’, i.e., its value depends on which corner of I2 is identified as the origin. Zimmerman (1993) proposed a modification of the bivariate Cramer—von Mises statistic that does not suffer from this defect. In this paper this statistic's limiting null distribution is derived and tabulated using the principal component decomposition method.

Suppose that we observe bivariate data (Xi, Yi) only when Yi ⩽ Xi (left truncation). Denote with F the marginal d.f. of the X's. In this paper we derive a Bahadur-type representation for the quantile function of the pertaining product-limit estimator of F. As an application we obtain confidence intervals and bands for quantiles of F.

We establish the consistency of a non-parametric estimator of bivariate extreme-value distributions.

We investigate the use of power transformations when data on two quantitative variables are presented in a two-way table. Given a suitable transformation we can model theunderlying continuous variables. Regressions and the correlation are obtained from the transformed grouped data. Also, by transforming back to the original scale, we obtain a smoothed version of the data.