Dependence Between Two Diagnostic Tests with Copula Function Approach: A Simulation Study

The study of the dependence between two medical diagnostic tests is an important issue in health research since it can modify the diagnosis and, therefore, the decision regarding a therapeutic treatment for an individual. In many practical situations, the diagnostic procedure includes the use of two tests, with outcomes on a continuous scale. For final classification, usually there is an additional “gold standard” or reference test. Considering binary test responses, we usually assume independence between tests or a joint binary structure for dependence. In this article, we introduce a simulation study assuming two dependent dichotomized tests using two copula function dependence structures in the presence or absence of verification bias. We compare the test parameter estimators obtained under copula structure dependence with those obtained assuming binary dependence or assuming independent tests.     

Optimal transitive procedures for comparing large numbers of parameters

Independent random samples are selected from each of a set of N independent populations, P₁,…,P_n. Interest centers around comparing N (unknown) scalar parameters θ₁,…,θ_N associated respectively with the N populations P₁,…,P_n. Procedures are constructed for estimating the magnitude of each of the differences δ_t,j = θ_i ? θ_j (1 ≤ i,j ≤ N) between pairs of populations. A loss function which adopts appropriate penalties for magnitude errors in estimation of differences is constructed. Magnitude estimators of differences are called transitive if they give rise to a transitive (i.e., consistent) relationship between pairwise differences of parameters. We show how to construct optimal effcient transitive magnitude–estimation procedures and demonstrate their usefulness through an example involving estimating the magnitude of the differences between disease incidence in paired towns for different pairs. Optimal transitive pairwise–comparison procedures are optimum (i.e., have the smallest posterior Bayes risks) in the class of all transitive pairwise–comparison procedures; as such they replace classical Bayes procedures which are usually not transitive when the number N of parameters compared is large. The posterior Bayes risk of optimal transitive pairwise comparison procedures are compared with that for alternative ‘adapted’ procedures, constructed from optimal simultaneous estimators and adapted for the purpose of pairwise comparisons. It is shown that the optimal transitive pairwise comparison procedures dominated the adapted procedures (in posterior Bayes risk) and typically represent only a small increase in posterior risk over the classical Bayes procedures which generally fail to be consistent. Optimal Bayes procedures are shown, for large numbers of parameters to be reasonably easy to construct using the algorithms outlined in this paper     

On concomitants of order statistics arising from the extended Farlie–Gumbel–Morgenstern bivariate logistic distribution and its application in estimation

In this paper, we consider concomitants of order statistics arising from the extended Farlie–Gumbel–Morgenstern bivariate logistic distribution and develop its distribution theory. Using ranked set sample obtained from the above distribution, unbiased estimators of the parameters associated with the study variate involved in it are generated. The best linear unbiased estimators (BLUEs) based on observations in the ranked set sample of those parameters as well have been derived. The efficiencies of the BLUEs relative to the respective unbiased estimators generated also have been evaluated.     

On the expected discounted penalty function for a risk model with dependence under a multi-layer dividend strategy

In this article, we consider a dependent risk model in the presence of a multi-laydividend strategy. We construct the dependence structure between the claim size and interclaim time by a Farlie–Gumbel–Morgenstern copula. A piecewise integro-differential equations for the expected discounted penalty function with boundary conditions are established. A renewal equation satisfied by the expected discounted penalty function is obtained via the translation operator. Then, we provide a recursive approach to derive the analytical solution of the expected discounted penalty function. Finally, a numerical example is presented to illustrate the solution procedure.     

Information Properties for Concomitants of Order Statistics in Farlie–Gumbel–Morgenstern (FGM) Family

Let (X _i , Y _i ), i = 1, 2,…, n be independent and identically distributed random variables from some continuous bivariate distribution. If X _(r) denotes the rth-order statistic, then the Y's associated with X _(r) denoted by Y _[r] is called the concomitant of the rth-order statistic. In this article, we derive an analytical expression of Shannon entropy for concomitants of order statistics in FGM family. Applying this expression for some well-known distributions of this family, we obtain the exact form of Shannon entropy, some of the information properties, and entropy bounds for concomitants of order statistics. Some comparisons are also made between the entropy of order statistics X _(r) and the entropy of its concomitants Y _[r]. In this family, we show that the mutual information between X _(r) and Y _[r], and Kullback–Leibler distance among the concomitants of order statistics are all distribution-free. Also, we compare the Pearson correlation coefficient between X _(r) and Y _[r] with the mutual information of (X _(r), Y _[r]) for the copula model of FGM family.     

The infinite-time ruin probability for a bidimensional renewal risk model with constant force of interest and dependent claims

This article studies a continuous-time bidimensional risk model, in which an insurer simultaneously confronts two kinds of claim sharing a common renewal claim-number process. Under the assumption that the claim size vectors form a sequence of independent and identically distributed random vectors following a common bivariate Farlie–Gumbel–Morgenstern distribution with extended regularly varying margins, we derive an explicit asymptotic formula for the corresponding infinite-time ruin probability.     

Bivariate short distributions

The method of MML estimation for a univariate normal (Tiku 1967, 1973) is extended to a bivariate normal population. Thus, a theoretical foundation is given to the robust correlation coefficient proposed by Tiku and Balakrishnan (1986).     

Bivariate compound poisson distributions

This paper discusses four alternative methods of forming bivariate distributions with compound Poisson marginals. Basic properties of each bivariate version are given. A new bivariate negative binomial distribution, and four bivariate versions of the Sichel distribution, are defined and their properties given.     

Role of concomitants of order statistics in determining parent bivariate distributions

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.     

Use of asymmetric loss functions in sequential estimation problems for multiple linear regression

When estimating in a practical situation, asymmetric loss functions are preferred over squared error loss functions, as the former is more appropriate than the latter in many estimation problems. We consider here the problem of fixed precision point estimation of a linear parametric function in beta for the multiple linear regression model using asymmetric loss functions. Due to the presence of nuissance parameters, the sample size for the estimation problem is not known beforehand and hence we take the recourse of adaptive multistage sampling methodologies. We discuss here some multistage sampling techniques and compare the performances of these methodologies using simulation runs. The implementation of the codes for our proposed models is accomplished utilizing MATLAB 7.0.1 program run on a Pentium IV machine. Finally, we highlight the significance of such asymmetric loss functions with few practical examples.     

Multivariate extreme-value distributions with applications to environmental data

Some parametric families of multivariate extreme-value distributions have been proposed in recent years; several additional parametric families are derived here. The parametric models are fitted, using numerical maximum likelihood, to some environmental multivariate extreme data sets consisting of extreme concentrations of a pollutant at several monitoring stations in a region. Some multivariate nonnormal data analysis techniques are proposed to aid in the likelihood analysis. The new models, together with previous models, appear to be adequate for inferences in that they cover a wide range of possible dependence patterns.     

Bivariate distributions with transmuted conditionals: Models and applications

Abstract

The class of transmuted distributions has received a lot of attention in the recent statistical literature. In this paper, we propose a rich family of bivariate distribution whose conditionals are transmuted distributions. The new family of distributions depends on the two baseline distributions and three dependence parameters. Apart from the general properties, we also study the distribution of the concomitance of order statistics. We study specific bivariate models. Estimation methodologies are proposed. A simulation study is conducted. The usefulness of this family is established by fitting well analyzed real life time data.     

Joint regression modeling for missing categorical covariates in generalized linear models

Missing covariates data is a common issue in generalized linear models (GLMs). A model-based procedure arising from properly specifying joint models for both the partially observed covariates and the corresponding missing indicator variables represents a sound and flexible methodology, which lends itself to maximum likelihood estimation as the likelihood function is available in computable form. In this paper, a novel model-based methodology is proposed for the regression analysis of GLMs when the partially observed covariates are categorical. Pair-copula constructions are used as graphical tools in order to facilitate the specification of the high-dimensional probability distributions of the underlying missingness components. The model parameters are estimated by maximizing the weighted log-likelihood function by using an EM algorithm. In order to compare the performance of the proposed methodology with other well-established approaches, which include complete-cases and multiple imputation, several simulation experiments of Binomial, Poisson and Normal regressions are carried out under both missing at random and non-missing at random mechanisms scenarios. The methods are illustrated by modeling data from a stage III melanoma clinical trial. The results show that the methodology is rather robust and flexible, representing a competitive alternative to traditional techniques.     

Bivariate kurtotic distributions of garment fibre data

Summary. A bivariate and unimodal distribution is introduced to model an unconventionally distributed data set collected by the Forensic Science Service. This family of distributions allows for a different kurtosis in each orthogonal direction and has a constructive rather than probability density function definition, making conventional inference impossible. However, the construction and inference work well with a Bayesian Markov chain Monte Carlo analysis.     

A strategy for constructing multivariate distributions

Given a random vector (X₁,…, X_n) for which the univariate and bivariate marginal distributions belong to some specified families of distributions, we present a procedure for constructing families of multivariate distributions with the specified univariate and bivariate margins. Some general properties of the resulting families of multivariate distributions are reviewed. This procedure is illustrated by generalizing the bivariate Plackett (1965) and Clayton (1978) distributions to three dimensions. In addition to providing rich families of models for data analysis, this method of construction provides a convenient way of simulating observations from multivariate distributions with specific types of univariate and bivariate marginal distributions. A general algorithm for simulating random observations from these families of multivariate distributions is presented     

Bivariate prior distributions via branching exchangeable sequences

This paper provides a novel approach to constructing bivariate prior distributions. The idea is based on the notion of partial exchangeability. In particular, in a simple extension of the exchangeable sequence, we create two dependent exchangeable sequences via a branching mechanism. This implies the existence of a bivariate prior distribution.     

A bivariate extension of the beta generated distribution derived from copulas

In this paper, we introduce a new class of bivariate distributions whose marginals are beta-generated distributions. Copulas are employed to construct this bivariate extension of the beta-generated distributions. It is shown that when Archimedean copulas and convex beta generators are used in generating bivariate distributions, the copulas of the resulting distributions also belong to the Archimedean family. The dependence of the proposed bivariate distributions is examined. Simulation results for beta generators and an application to financial risk management are presented.     

Bayesian quantile regression for hierarchical linear models

The paper proposes a Bayesian quantile regression method for hierarchical linear models. Existing approaches of hierarchical linear quantile regression models are scarce and most of them were not from the perspective of Bayesian thoughts, which is important for hierarchical models. In this paper, based on Bayesian theories and Markov Chain Monte Carlo methods, we introduce Asymmetric Laplace distributed errors to simulate joint posterior distributions of population parameters and across-unit parameters and then derive their posterior quantile inferences. We run a simulation as the proposed method to examine the effects on parameters induced by units and quantile levels; the method is also applied to study the relationship between Chinese rural residents' family annual income and their cultivated areas. Both the simulation and real data analysis indicate that the method is effective and accurate.     

A non-iterative Bayesian sampling algorithm for censored Student-t linear regression models

ABSTRACT

In this paper, we consider an effective Bayesian inference for censored Student-t linear regression model, which is a robust alternative to the usual censored Normal linear regression model. Based on the mixture representation of the Student-t distribution, we propose a non-iterative Bayesian sampling procedure to obtain independently and identically distributed samples approximately from the observed posterior distributions, which is different from the iterative Markov Chain Monte Carlo algorithm. We conduct model selection and influential analysis using the posterior samples to choose the best fitted model and to detect latent outliers. We illustrate the performance of the procedure through simulation studies, and finally, we apply the procedure to two real data sets, one is the insulation life data with right censoring and the other is the wage rates data with left censoring, and we get some interesting results.     

Bivariate generalized gamma distributions of Kibble's type

In this paper, a new type of bivariate generalized gamma (BGG) distribution derived from the bivariate gamma distribution of Kibble [Two-variate gamma-type distribution. Sankh?a 1941;5:137–150] by means of a power transformation is presented. The explicit expressions of statistical properties of the BGG distribution are presented. The estimation of marginal and dependence parameters using the method of moments and the method of inference functions for margins are discussed, and their performance through a Monte Carlo simulation study is assessed. Finally, an example is given to illustrate the applicability of the distributions introduced here.