From Conditional Independence to Conditionally Negative Association: Some Preliminary Results

Conditional moment estimates on the cumulative sum of conditionally independent random variables are derived, conditional prophet inequalities for conditionally independent random variables are established, a comparison theorem on conditional moment inequalities between conditionally independent and conditionally negatively associated random variables is established. As applications of these results, a conditional Rosenthal type inequality and two conditional Kolmogorov exponential inequalities for conditionally negatively associated random variables are obtained.     

PARTIAL CORRELATION AND CONDITIONAL CORRELATION AS MEASURES OF CONDITIONAL INDEPENDENCE

This paper investigates the roles of partial correlation and conditional correlation as measures of the conditional independence of two random variables. It first establishes a sufficient condition for the coincidence of the partial correlation with the conditional correlation. The condition is satisfied not only for multivariate normal but also for elliptical, multivariate hypergeometric, multivariate negative hypergeometric, multinomial and Dirichlet distributions. Such families of distributions are characterized by a semigroup property as a parametric family of distributions. A necessary and sufficient condition for the coincidence of the partial covariance with the conditional covariance is also derived. However, a known family of multivariate distributions which satisfies this condition cannot be found, except for the multivariate normal. The paper also shows that conditional independence has no close ties with zero partial correlation except in the case of the multivariate normal distribution; it has rather close ties to the zero conditional correlation. It shows that the equivalence between zero conditional covariance and conditional independence for normal variables is retained by any monotone transformation of each variable. The results suggest that care must be taken when using such correlations as measures of conditional independence unless the joint distribution is known to be normal. Otherwise a new concept of conditional independence may need to be introduced in place of conditional independence through zero conditional correlation or other statistics.     

Robust generalized confidence intervals

Most interval estimates are derived from computable conditional distributions conditional on the data. In this article, we call the random variables having such conditional distributions confidence distribution variables and define their finite-sample breakdown values. Based on this, the definition of breakdown value of confidence intervals is introduced, which covers the breakdowns in both the coverage probability and interval length. High-breakdown confidence intervals are constructed by the structural method in location-scale families. Simulation results are presented to compare the traditional confidence intervals and their robust analogues.     

Approximate conditional inference and the linear functional model

Approximate conditional inference is developed for the slope parameter of the linear functional model with two variables. It is shown that the model can be transformed so that the slope parameter becomes an angle and nuisance parameters are radial distances. If the nuisance parameters are known an exact confidence interval based on a location-type conditional distribution is available for the angle. More gen¬erally, confidence distributions are used to average the conditional distribution over the nuisance parameters yielding an approximate conditional confidence interval that reflects the precision indicated by the data. An example is analyzed.     

Bootstrapping the conditional copula

This paper is concerned with inference about the dependence or association between two random variables conditionally upon the given value of a covariate. A way to describe such a conditional dependence is via a conditional copula function. Nonparametric estimators for a conditional copula then lead to nonparametric estimates of conditional association measures such as a conditional Kendall's tau. The limiting distributions of nonparametric conditional copula estimators are rather involved. In this paper we propose a bootstrap procedure for approximating these distributions and their characteristics, and establish its consistency. We apply the proposed bootstrap procedure for constructing confidence intervals for conditional association measures, such as a conditional Blomqvist beta and a conditional Kendall's tau. The performances of the proposed methods are investigated via a simulation study involving a variety of models, ranging from models in which the dependence (weak or strong) on the covariate is only through the copula and not through the marginals, to models in which this dependence appears in both the copula and the marginal distributions. As a conclusion we provide practical recommendations for constructing bootstrap-based confidence intervals for the discussed conditional association measures.     

Testing the Granger Noncausality Hypothesis in Stationary Nonlinear Models of Unknown Functional Form

In this article, we propose a general method for testing the Granger noncausality hypothesis in stationary nonlinear models of unknown functional form. These tests are based on a Taylor expansion of the nonlinear model around a given point in the sample space. We study the performance of our tests by a Monte Carlo experiment and compare these to the most widely used linear test. Our tests appear to be well-sized and have reasonably good power properties.     

A hybrid approach to achieving both marginal and conditional balances for stratification variables in sequential clinical trials

Various methods exist in the literature for achieving marginal balance for baseline stratification variables in sequential clinical trials. One major limitation with balancing on the margins of the stratification variables is that there is an efficiency loss when the primary analysis is stratified. To preserve the efficiency of a stratified analysis one recently proposed approach balances on the crossing of the stratification variables included in the analysis, which achieves conditional balance for the variables. A hybrid approach to achieving both marginal and conditional balances in sequential clinical trials is proposed, which is applicable to both continuous and categorical stratification variables. Numerical results based on extensive simulation studies and a real dataset show that the proposed approach outperforms the existing ones and is particularly useful when both additive and stratified models are planned for a trial. Copyright © 2013 John Wiley & Sons, Ltd.     

PERFORMANCE OF THE LOCATION LINEAR DISCRIMINANT FUNCTION UNDER ACROSS-LOCATION HETEROSCEDASTICITY

ABSTRACT

Classification of data consisting of both categorical and continuous variables between two groups is often handled by the sample location linear discriminant function confined to each of the locations specified by the observed values of the categorical variables. Homoscedasticity of across-location conditional dispersion matrices of the continuous variables is often assumed. Quite often, interactions between continuous and categorical variables cause across-location heteroscedasticity. In this article, we examine the effect of heterogeneous across-location conditional dispersion matrices on the overall expected and actual error rates associated with the sample location linear discriminant function. Performance of the sample location linear discriminant function is evaluated against the results for the restrictive classifier adjusted for across-location heteroscedasticity. Conclusions based on a Monte Carlo study are reported.     

On some characteristics of bivariate chi-square distribution

Product moments of bivariate chi-square distribution have been derived in closed forms. Finite expressions have been derived for product moments of integer orders. Marginal and conditional distributions, conditional moments, coefficient of skewness and kurtosis of conditional distribution have also been discussed. Shannon entropy of the distribution is also derived. We also discuss the Bayesian estimation of a parameter of the distribution. Results match with the independent case when the variables are uncorrelated.     

Bayesian Model Choice in Exponential Survival Models

ABSTRACT

Log-linear models for the distribution on a contingency table are represented as the intersection of only two kinds of log-linear models. One assuming that a certain group of the variables, if conditioned on all other variables, has a jointly independent distribution and another one assuming that a certain group of the variables, if conditioned on all other variables, has no highest order interaction. The subsets entering into these models are uniquely determined by the original log-linear model. This canonical representation suggests considering joint conditional independence and conditional no highest order association as the elementary building blocks of log-linear models.     

The effect of conditional heteroskedasticity on common statistical procedures for means and variances

Summary: Commonly used standard statistical procedures for means and variances (such as the t–test for means or the F–test for variances and related confidence procedures) require observations from independent and identically normally distributed variables. These procedures are often routinely applied to financial data, such as asset or currency returns, which do not share these properties. Instead, they are nonnormal and show conditional heteroskedasticity, hence they are dependent. We investigate the effect of conditional heteroskedasticity (as modelled by GARCH(1,1)) on the level of these tests and the coverage probability of the related confidence procedures. It can be seen that conditional heteroskedasticity has no effect on procedures for means (at least in large samples). There is, however, a strong effect of conditional heteroskedasticity on procedures for variances. These procedures should therefore not be used if conditional heteroskedasticity is prevalent in the data.*We are grateful to the referees for their useful and constructive comments.     

Factor Stochastic Volatility in Mean Models: A GMM Approach

This paper provides a semiparametric framework for modeling multivariate conditional heteroskedasticity. We put forward latent stochastic volatility (SV) factors as capturing the commonality in the joint conditional variance matrix of asset returns. This approach is in line with common features as studied by Engle and Kozicki (1993), and it allows us to focus on identication of factors and factor loadings through first- and second-order conditional moments only. We assume that the time-varying part of risk premiums is based on constant prices of factor risks, and we consider a factor SV in mean model. Additional specification of both expectations and volatility of future volatility of factors provides conditional moment restrictions, through which the parameters of the model are all identied. These conditional moment restrictions pave the way for instrumental variables estimation and GMM inference.     

Alternative estimators in logistic regression when the data are collinear

Logistic regression using conditional maximum likelihood estimation has recently gained widespread use. Many of the applications of logistic regression have been in situations in which the independent variables are collinear. It is shown that collinearity among the independent variables seriously effects the conditional maximum likelihood estimator in that the variance of this estimator is inflated in much the same way that collinearity inflates the variance of the least squares estimator in multiple regression. Drawing on the similarities between multiple and logistic regression several alternative estimators, which reduce the effect of the collinearity and are easy to obtain in practice, are suggested and compared in a simulation study.     

An estimated‐score approach for dealing with missing covariate data in matched case–control studies

Matched case–control designs are commonly used in epidemiological studies for estimating the effect of exposure variables on the risk of a disease by controlling the effect of confounding variables. Due to retrospective nature of the study, information on a covariate could be missing for some subjects. A straightforward application of the conditional logistic likelihood for analyzing matched case–control data with the partially missing covariate may yield inefficient estimators of the parameters. A robust method has been proposed to handle this problem using an estimated conditional score approach when the missingness mechanism does not depend on the disease status. Within the conditional logistic likelihood framework, an empirical procedure is used to estimate the odds of the disease for the subjects with missing covariate values. The asymptotic distribution and the asymptotic variance of the estimator when the matching variables and the completely observed covariates are categorical. The finite sample performance of the proposed estimator is assessed through a simulation study. Finally, the proposed method has been applied to analyze two matched case–control studies. The Canadian Journal of Statistics 38: 680–697; 2010 © 2010 Statistical Society of Canada     

Bayesian inference over ICA models: application to multibiometric score fusion with quality estimates

Bayesian networks are not well-formulated for continuous variables. The majority of recent works dealing with Bayesian inference are restricted only to special types of continuous variables such as the conditional linear Gaussian model for Gaussian variables. In this context, an exact Bayesian inference algorithm for clusters of continuous variables which may be approximated by independent component analysis models is proposed. The complexity in memory space is linear and the overfitting problem is attenuated, while the inference time is still exponential. Experiments for multibiometric score fusion with quality estimates are conducted, and it is observed that the performances are satisfactory compared to some known fusion techniques.     

Vine copula approximation: a generic method for coping with conditional dependence

Pair-copula constructions (or vine copulas) are structured, in the layout of vines, with bivariate copulas and conditional bivariate copulas. The main contribution of the current work is an approach to the long-standing problem: how to cope with the dependence structure between the two conditioned variables indicated by an edge, acknowledging that the dependence structure changes with the values of the conditioning variables. The changeable dependence problem, though recognized as crucial in the field of multivariate modelling, remains widely unexplored due to its inherent complication and hence is the motivation of the current work. Rather than resorting to traditional parametric or nonparametric methods, we proceed from an innovative viewpoint: approximating a conditional copula, to any required degree of approximation, by utilizing a family of basis functions. We fully incorporate the impact of the conditioning variables on the functional form of a conditional copula by employing local learning methods. The attractions and dilemmas of the pair-copula approximating technique are revealed via simulated data, and its practical importance is evidenced via a real data set.     

The effect of omitted covariates in marginal and partially conditional recurrent event analyses

There have been many advances in statistical methodology for the analysis of recurrent event data in recent years. Multiplicative semiparametric rate-based models are widely used in clinical trials, as are more general partially conditional rate-based models involving event-based stratification. The partially conditional model provides protection against extra-Poisson variation as well as event-dependent censoring, but conditioning on outcomes post-randomization can induce confounding and compromise causal inference. The purpose of this article is to examine the consequences of model misspecification in semiparametric marginal and partially conditional rate-based analysis through omission of prognostic variables. We do so using estimating function theory and empirical studies.

     

A note on nonparametric estimation of circular conditional densities

ABSTRACT

The conditional density offers the most informative summary of the relationship between explanatory and response variables. We need to estimate it in place of the simple conditional mean when its shape is not well-behaved. A motivation for estimating conditional densities, specific to the circular setting, lies in the fact that a natural alternative of it, like quantile regression, could be considered problematic because circular quantiles are not rotationally equivariant. We treat conditional density estimation as a local polynomial fitting problem as proposed by Fan et al. [Estimation of conditional densities and sensitivity measures in nonlinear dynamical systems. Biometrika. 1996;83:189–206] in the Euclidean setting, and discuss a class of estimators in the cases when the conditioning variable is either circular or linear. Asymptotic properties for some members of the proposed class are derived. The effectiveness of the methods for finite sample sizes is illustrated by simulation experiments and an example using real data.     

The Construction and Use of Approximations for Missing Quarterly Observations: A Model-Based Approach

We use a model-based approach to derive quarterly figures on several variables for the aggregate labor market in the Netherlands that are only observed annually. These approximations are conditional expectations derived from univariate and bivariate quarterly time series models for the series under consideration. They are subsequently used as proxies to estimate and analyze the structural labor market equations. Attention is given to the properties of estimation procedures based on proxy variables.     

Statistical Inference in Context Specific Interaction Models for Contingency Tables

Abstract.  Context specific interaction models is a class of interaction models for contingency tables in which interaction terms are allowed to vanish in specific contexts given by the levels of sets of variables. Such restrictions can entail conditional independencies which only hold for some values of the conditioning variables and allows also for irrelevance of some variables in specific contexts. A Markov property is established and so is an iterative proportional scaling algorithm for maximum likelihood estimation. Decomposition of the estimation problem is treated and model selection is discussed.