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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 ...

We discuss two-armed clinical trials in which an ordinal response is observed on entry and at follow-up. In certain subject areas, such data are often reduced to the counts of patients who have deteriorated, remained the same, or improved. We develop asymptotic distribution theory to compare this approach to the use of nonparametric tests based on poststratification of the follow-up responses according to the entry levels. The theory for the latter approach is established by a general result for two-sample U statistics based on bivariate data. Application to data from clinical trials and evaluation of asymptotic relative efficiencies and powers for a variety of contexts illustrate the increased sensitivity of poststratified Wilcoxon tests.

A set of permutation tests that are both exact and distribution-free are proposed to simultaneously detect differences in marginal locations and/or scales in bivariate data. The tests take advantage of the fact that when the marginal means and variances are equal, the pairwise differences are symmetrically distributed about 0 and are uncorrelated with the pairwise sums. Two statistics for detecting the marginal location and scale differences are combined in a quadratic form. A permutation distribution for this quadratic form follows from considering all 2 n conditionally equally likely sign changes on the differences. Several methods of estimating the covariance matrix of the quadratic form are examined, including conditional and unconditional (plug-in) approaches. These new tests are compared with the standard tests in the literature and, through simulation for several families of bivariate distributions, are found to compare quite favorably. This article also brings to light the largely overloo...

This article outlines a general Bayesian approach to estimating a bivariate regression function in a nonparametric manner. It models the function using a bivariate regression spline basis with many terms. Binary indicator variables corresponding to these terms are introduced to explicitly model the uncertainty of whether or not the terms provide a significant contribution to the regression. The regression function is estimated using an estimate of its posterior mean, smoothing over the distribution of these binary indicator variables. To make the computations tractable, all estimates are obtained using Markov chain Monte Carlo sampling. Extensive simulated comparisons are provided that demonstrate the competitive performance of this approach against other data-driven bivariate surface estimators prominent in the literature. It is then shown how the procedure can be extended to provide a general approach to nonparametric bivariate surface estimation in two difficult regression settings. The first ...

In this article we construct a kernel estimate of the probability density function from bivariate data that have been randomly censored. We study the large-sample properties of the proposed estimator using a strong approximation result. We establish consistency and asymptotic normality and give a convenient representation of the kernel density estimator. Simulation studies show that the proposed procedure gives a good estimate of the true density function even when the sample size is moderate. We discuss various issues about implementation of the estimator, including bandwidth selection and boundary effects. The procedure can be generalized to higher dimensional variables in a straightforward manner.

A new family of bivariate densities with specified marginal densities is described. The bivariate dependence structure takes the form of a density weighting function constructed as follows. The space of continuous variates X and Y is mapped into a unit square through probability integral transformations. A polygonal partition of the unit square supports a bivariate covariance characteristic, which is structured as a functional of a univariate regression characteristic. This structure (a) exhibits all geometric features of positive (or negative) dependence implied by Frechet bounds, which it can attain, and (b) prescribes positive (or negative) mutual regression dependence between X and Y. To obtain a parametric family of bivariate densities, the regression characteristic is given a power form. The resultant two-parameter model has specified marginals and, within certain constraints, allows one to separately control the shape of the bivariate density and the degree of association between X and Y; ...

We show that the survival curve is identifiable in bivariate censored data problems under weaker independence assumptions than have commonly been made. The common assumption has been mutual independence of (T 1, T 2) and (Z 1, Z 2), where (T 1, T 2) is the true survival vector, (Z 1, Z 2) is a nuisance censoring vector, and bivariate right-censored data is observed. We show that the distribution of (T 1, T 2) is identifiable under weaker, conditional independence assumptions for distributions with full support. Bivariate survival analysis is a more powerful analysis tool than univariate analysis if multiple, possibly related, times are of interest. The mutual independence model has become popular as a nonparametric way of analyzing such data. Analysis of the bivariate problem and analogy with univariate models are used to show that the conditional independence model is more widely applicable as a general nonparametric model for bivariate survival data.

This article describes a new, biologically motivated model for bivariate data containing a trend line. The model is characterized by a mixture in which the component means lie on a line. Consistent methods for estimating the parameters of the mixture line are derived, as are procedures for obtaining a confidence interval for the slope. The methods are proved valid within a large class of models for bivariate data, including an errors-invariables model, and are illustrated with data on Canadian herring. This work provides one possible objective resolution of a (sometimes heated) debate in allometry and elsewhere on how to define and estimate a linear trend through bivariate data from a natural population.

A conditional test is proposed that uses the null hypothesis of bivariate symmetry to detect unequal marginal scales in a bivariate population. The asymptotic normality of the test statistic and the consistency of the test are established. Asymptotic relative efficiency comparisons are made with the standard parametric test in both the bivariate normal case and in a trivariate reduction model case and with two of Sen's (1967) tests in the bivariate normal case. The good performance of the test is also illustrated via a Monte Carlo study.

There has been a lot of work on measures of dependence or association for bivariate probability distributions or bivariate data. These measures usually assume that the variables are both continuous or both categorical. In comparison, there is very little work on multivariate or conditional measures of dependence. The purpose of this article is to discuss measures of multivariate dependence and measures of conditional dependence based on relative entropies. These measures are conceptually very general, as they can be used for a set of variables that can be a mixture of continuous, ordinal-categorical, and nominal-categorical variables. For continuous or ordinal-categorical variables, a certain transformation of relative entropy to the interval [0, 1] leads to generalizations of the correlation, multiple-correlation, and partial-correlation coefficients. If all variables are nominal categorical, the relative entropies are standardized to take a maximum of 1 and then transformed so that in the bivar...

Two methods, one based upon grouping, are proposed for fltting a regression line to bivariate data when the means of the observations have known order along the line. The two procedures are compared with respect to their limiting distributions and asymptotic efficiency. Criteria are given for choosing one of these procedures to fit the line.

By developing an analogy with the measure of association in a fourfold contingency table, a method is given of constructing a one-parameter class of bivariate distributions from given margins. This class contains the known boundary distributions and the member corresponding to independent random variables. The class resulting from standard normal margins is compared with the standard normal bivariate distribution, and it is found that the two joint distributions and the two conditional distributions each agree tolerably well. Estimation of the parameter is illustrated by an example showing a computational advantage of this class of distributions.

The logistic distribution closely resembles the normal one. Both are symmetrical. Here two logistic bivariate distributions are studied. In both cases the curves of equal probability density are not ellipses, the regression curves are not linear and the conditional expectations are limited. The first distribution analyzed with the help of the bivariate moment generating function is asymmetrical and therefore departs considerably from the normal one. The coefficient of correlation is constant and equal to one half. The second bivariate logistic distribution is symmetrical. The regression curves are linear in probability scale and the coefficient of correlation varies in the interval ± .30396.

A continuous bivariate distribution is grouped by intervals on the margins so as to obtain a discrete bivariate distribution. Marginal variates are considered which are the scores (or mean values) on the intervals of the first canonical variables associated with the ungrouped distribution. A method is suggested for choosing the end points of the intervals so as to obtain the maximum possible correlation between these marginal variates. The solution is exact when only one margin is grouped and approximate when both are. Tables of interval end points (in terms of the c.d.f.) and of the resulting correlation are given for the case of a bivariate normal distribution.

This paper uses a family of bivariate extreme-value distributions to estimate the probability of a large exceedance of a random variable given that a certain other random variable not independent of the first has exceeded a certain value. A simple method of reasonably good efficiency is given for estimating a bivariate extreme-value distribution from independent bivariate samples. The method is used to analyze the performance of a spacecraft command receiver which has an indication of data quality so that commands likely to be in error can be rejected.

Some large-sample properties of the three nonparametric tests of David and Fix for the two-sample bivariate location problem are given. In particular the Pitman A.R.E.s of the tests against some ‘normal-theory’ competitors are shown to be satisfactory in most cases for a number of ‘near-normal’ bivariate distributions. Their performances in some mixtures of bivariate normal distributions prove to be particularly convincing. A useful relationship between the three test statistics is indicated.

This paper presents an extension of the mathematical model used to justify accident proneness. It assumes that the distribution of accidents incurred by an individual in non-overlapping intervals is a correlated bivariate Poisson (C.B.P.). On compounding this correlated bivariate Poisson through a Gamma distribution an extended bivariate negative binomial or, more precisely, a compound correlated bivariate Poisson (C.C.B.P.) distribution is obtained. Recurrence relations and expressions for the required probabilities are illustrated for two sets of data. The C.C.B.P. proved to fit as well as the bivariate negative binomial when the estimate of B12 was close to zero, and much better than the latter distribution when the estimate of B12 was not close to zero.

THE present note is concerned with a special case of the general problem of obtaining efficient estimators for the parameters of a normal multivariate population when the available sample data are incomplete in the sense that measures on all variables are not available for all individuals in the sample. Such fragmentary data may arise because part of the data are irretrievably lost (e.g., in an archaeological find), or because certain data were purposely not collected. The decision not to measure all individuals in the sample on all variables may be reached because of the cost of measurement, because of limited time, because the measurement of a certain variable alters or destroys the individual measured (e.g., in mental testing, testing explosives), and so forth. The general problem for normal bivariate populations has been treated by Wilks [3]. This treatment has been further generalized to normal multivariate populations by Matthai [2], who deals explicitly with the trivariate case. Unfortunately, the general maximum likelihood equations have proved rather intractable, and no simple formulas for the maximum likelihood estimators are available in the general case, even for a sample from a bivariate population. In the present paper, the problem of estimating the parameters of a normal trivariate population from incomplete data is dealt with in a special case for which explicit solutions to the maximum likelihood equations are readily obtained. This special case is described in the following section. Formulas for the maximum likelihood estimators are given; their application is illustrated by a numerical example. The sampling variances and covariances of the maximum likelihood estimators are derived. An examination is made of the efficiency of the usual methods that utilize only that portion of the data that is complete. The foregoing results are specialized to apply to a commonly encountered bivariate (rather than trivariate) situation.

A bivariate failure model is proposed in which the residual lifetime of one component is dependent on the working status of the other. General properties of the model are discussed, and the maximum likelihood estimates of the parameters are found in a bivariate exponential-life special case.

The asymptotic efficiencies relative to Hotelling's T 2 test of two nonparametric tests for the bivariate two-sample location problem are compared for some mixtures of bivariate normal distributions.