Family Of Bivariate Distributions Research Articles

On proportional odds models

Recently, Marshall and Olkin (Biometrika 84(3):641–652 1997) introduced a family of distributions by adding a new parameter to a survival function. In this paper, we give physical interpretation of the family using odds function. It is shown that the family of distributions satisfies the property of proportional odds function. We, then, develop a generalized family and study its properties. Further, we give various definitions of proportional odds model in the bivariate set up. Based on these, we introduce new families of bivariate distributions and study their properties.

On Bivariate Reversed Hazard Rates

In this paper we discuss various definitions of bivariate reversed hazard rate and their properties. An exponential representation of bivariate distribution using reversed hazard rates is given and we also develop a new family of bivariate distributions using bivariate reversed hazard rate. Finally we give a local dependence measure using bivariate reversed hazard rates and study its properties. Various applications of the models are pointed out.

Monotonicity of the (reversed) hazard rate of the (maximum) minimum in bivariate distributions

It is well known that in the case of independent random variables, the (reversed) hazard rate of the (maximum) minimum of two random variables is the sum of the individual (reversed) hazard rates and hence the onotonicity of the (reversed) hazard rate of the marginals is preserved by the monotonicity of the (reversed) hazard rate of the (maximum) minimum. However, for the bivariate distributions this property is not always preserved. In this paper, we study the monotonicity of the (reversed) hazard rate of the (maximum) minimum for two well known families of bivariate distributions viz the Farlie-Gumbel-Morgenstern (FGM) and Sarmanov family. In case of the FGM family, we obtain the (reversed) hazard rate of the (maximum) minimum and provide several examples in some of which the (reversed) hazard rate is monotonic and in others it is non-monotonic. In the case of Sarmanov family the (reversed) hazard rate of the (maximum) minimum may not be expressed in a compact form in general. We consider some examples to illustrate the procedure

Estimation of a bivariate symmetric distribution function

We consider the efficient estimation of a bivariate distribution function (DF) under the class of radially symmetric distributions and propose an estimator based on the mean of the empirical distribution and survival functions. We obtain the mean and variance of the estimator and show that it has an asymptotic normal distribution. We also show that the nonparametric maximum likelihood estimator of the bivariate DF coincides with the new estimator under radial symmetry. We study the asymptotic relative efficiency of this estimator and show that it results in a minimum of 50% reduction in sample size over the empirical DF at any point ( x, y) in R 2 . A bootstrap procedure to test whether the data support a radially symmetric model is examined. A simulation study compares the size and power of this test under bivariate normality, against alternatives in the Plackett's family of bivariate distributions, to two other procedures based on Kolmogorov–Smirnov distance.

Efficient nonparametric estimation of a distribution function

We consider efficient estimation of a distribution function F under several models based on the nonparametric likelihood principal. Under the symmetry model, we derive the nonparametric MLE of F and show that it coincides with the symmetrized estimator. Under the auxiliary-sample model, we discuss an estimator based on the total law of probability and show that it coincides with the nonparametric MLE of F. Assuming quadrant dependence, we show that the estimator has a minimum asymptotic relative efficiency of one with respect to the empirical distribution function. We consider the intersection of the two models and present an efficient hybrid estimator. The estimator is based on the conditional distribution function after post stratification of the sample. We show that the hybrid estimator has an asymptotic normal distribution and converges to the nonparametric MLE of F under the assumption of conditional symmetry. A Monte Carlo simulation assesses the small sample efficiency of the proposed estimators under the Plackett family of bivariate distributions. A bootstrap algorithm to obtain the power of the Wilcoxon signed rank test is presented.

The Mixture Approach for Simulating Bivariate Distributions With Specified Correlations

The Mixture approach is an exact methodology for simulating families of bivariate distributions with specified correlation coefficients, some of which are new. It can accommodate the entire range of correlation coefficients, produces bivariate surfaces that are intuitively appealing, and is often remarkably easy to implement. The approach is introduced in a Bayesian context and demonstrated for the conjugate families of beta and gamma distributions, with special attention given to the bivariate uniform. For these distributions, formulas for correlations have simple closed forms and computations are easy.

A family of bivariate distributions with non-elliptical contours

In this paper, a family of copulas with two parameters is proposed and its dependence analysis is performed. The corresponding family of bivariate distributions with specified marginals is constructed. For normal marginals, the new distributions are non-elliptical and can be applied in data analysis. They provide various alternative hypotheses for testing normality. Finally, an example is given.

A Class of Permutation Tests of Bivariate Interchangeability

Abstract 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 overlooked likelihood ratio test for equal means and variances in the bivariate normal and shows its relationship to more recent approaches, including those presented here.

A Class of Permutation Tests of Bivariate Interchangeability

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 overlooked likelihood ratio test for equal means and variances in the bivariate normal and shows its relationship to more recent approaches, including those presented here.

Variations on a Bhattacharyya Theme

Bhattacharyya introduced a family of bivariate distributions with normal conditions in 1943. He discussed what additional conditions are required to guarantee that the joint distribution is a classical bivariate normal distribution. The family of distributions with normal conditionals is , in itself, an interesting exponential family. Its structure is discussed in the present paper. Extensions to higher dimensions and to cases in which the conditional distributions are prescribed to belong to non normal families are discussed. These conditionally specified families of distributions are shown to be particularly suitable for use as prior distributions in multiparameter Bayesian Inference settings.

Garvey: A Family of Joint Probability Models for Cost and Schedule Uncertainties

Abstract This paper discusses a family of analytical models from which the joint probability of total program cost and schedule can be calculated, analyzed, and presented to decision-makers. Specifically, the classical bivariate normal and two lesser known joint distributions, the normal-lognormal and the bivariate lognormal distributions are discussed. Experiences from Monte Carlo simulations suggest that this family of bivariate distributions are candidate models for computing joint and conditional cost and schedule probabilities. In particular, the discussion on the normal-lognormal distribution as a joint cost-schedule probability model extends research on the applicability of the bivariate lognormal presented last year.

On some models and tests for heteroscedasticity

In the present paper a family of bivariate distributions characterized by standardized symmetric conditional distributions and by a simple scedastic curve (i.e. conditional Variance) is considered. A particular case arises from the first order autoregressive Conditionally Heteroscedastic (ARCH) model of time series analysis; therefore, for this family, inference based on data from simple random sampling and from ARCH models is compared and unified to some extent. Despite the fact that the marginal density is not known is closed form even for typical cases (as conditional Normality), its moments (computed for the general case) show that the marginal distribution has higher kurtosis than the conditional distributions; hence heavy tails of rest data distribution may be disturbing factors in testing heteroscedasticity, especially if one uses a gaussian-score based test. The score based test for independence vs. scedastic regression is shown to be of square correlation type under conditional normal distribution. Moreover, the score based test under Student's t conditional normal distribution is also introduced and proposed as an alternative to the above Normal case test. Although the Student's t test converges to the Normal case for increasing degrees of freedom, it is shown that, using the Normal case test when in fact the conditional is leptokurtic Student's t (e.g. with degrees of freedom between 9 and 12) entails a relevant loss in asymptotic efficiency.

Bivariate Distributions With Nonmonotone Dependence Structure

A new ordering for nonmonotone dependence is proposed, and a method is presented for describing parametric families of bivariate distributions with fixed marginals and specified dependence structures. The method is easy to use and allows for arbitrary marginals and arbitrary dependence structures (e.g., structures of nonmonotone dependence). Any parametric family of bivariate distributions is ordered by the nonmonotone dependence ordering constructed from the desirable dependence structure. The construction is based on a generalization of the copula of distribution functions to a copula of probability measures on product spaces. Examples are shown for various bivariate distributions.

Families of Multivariate Distributions

Abstract For many years there has been an interest in families of bivariate distributions with marginals as parameters. Genest and MacKay (1986a,b) showed that several such families that appear in the literature can be derived by a unified method. A similar conclusion is obtained in this article through the use of mixture models. These models might be regarded as multivariate proportional hazards models with random constants of proportionality. The mixture models are useful for two purposes. First, they make some properties of the derived distributions more transparent; the positive-dependency property of association is sometimes exposed, and a method for simulation of data from the distributions is suggested. But the mixture models also allow derivation of several new families of bivariate distributions with marginals as parameters, and they indicate obvious multivariate extensions. Some of the new families of bivariate distributions given in this article extend known distributions by adding a parameter to make them more flexible. Other families are derived that appear to be entirely new.

Frank's Family of Bivariate Distributions

This paper examines the properties of a new class of bivariate distributions whose members are stochastically ordered and likelihood ratio dependent. The proposed class can be used to construct bivariate families of distributions whose marginals are arbitrary and which include the Fréchet bounds as well as the distribution corresponding to independent variables. Three nonparametric estimators of the association parameter are suggested and Monte Carlo experiments are used to compare their small-sample behaviour to that of the maximum likelihood estimate.

Frank's family of bivariate distributions

SUMMARY This paper examines the properties of a new class of bivariate distributions whose members are stochastically ordered and likelihood ratio dependent. The proposed class can be used to construct bivariate families of distributions whose marginals are arbitrary and which include the Frechet bounds as well as the distribution corresponding to independent variables. Three nonparametric estimators of the association parameter are suggested and Monte Carlo experiments are used to compare their small-sample behaviour to that of the maximum likelihood estimate.

Multivariate Distributions Which Have Beta Conditional Distributions

Abstract Distributions having beta conditional distributions arise in connection with the generation of distributions for random proportions which do not necessarily possess neutrality properties (see [3]). The family of bivariate distributions for which both conditional distributions are beta is evaluated, and some multivariate extensions are briefly discussed. The bivariate Dirichlet distribution is characterized by the assumptions that both conditional distributions and one marginal distribution are beta.

Families of Bivariate Distributions.

Families of Bivariate Distributions.