On generalised estimating equations for vector regression

Generalised estimating equations (GEE) for regression problems with vector‐valued responses are examined. When the response vectors are of mixed type (e.g. continuous–binary response pairs), the GEE approach is a semiparametric alternative to full‐likelihood copula methods, and is closely related to Prentice & Zhao's mean‐covariance estimation equations approach. When the response vectors are of the same type (e.g. measurements on left and right eyes), the GEE approach can be viewed as a ‘plug‐in’ to existing methods, such as the vglm function from the state‐of‐the‐art VGAM package in R. In either scenario, the GEE approach offers asymptotically correct inferences on model parameters regardless of whether the working variance–covariance model is correctly or incorrectly specified. The finite‐sample performance of the method is assessed using simulation studies based on a burn injury dataset and a sorbinil eye trial dataset. The method is applied to data analysis examples using the same two datasets, as well as to a trivariate binary dataset on three plant species in the Hunua ranges of Auckland.     

Conditional variance estimation in heteroscedastic regression models

First, we propose a new method for estimating the conditional variance in heteroscedasticity regression models. For heavy tailed innovations, this method is in general more efficient than either of the local linear and local likelihood estimators. Secondly, we apply a variance reduction technique to improve the inference for the conditional variance. The proposed methods are investigated through their asymptotic distributions and numerical performances.     

Estimating a sensitive proportion through randomized response procedures based on auxiliary information

Randomized response techniques are widely employed in surveys dealing with sensitive questions to ensure interviewee anonymity and reduce nonrespondents rates and biased responses. Since Warner’s (J Am Stat Assoc 60:63–69, 1965) pioneering work, many ingenious devices have been suggested to increase respondent’s privacy protection and to better estimate the proportion of people, π _A , bearing a sensitive attribute. In spite of the massive use of auxiliary information in the estimation of non-sensitive parameters, very few attempts have been made to improve randomization strategy performance when auxiliary variables are available. Moving from Zaizai’s (Model Assist Stat Appl 1:125–130, 2006) recent work, in this paper we provide a class of estimators for π _A , for a generic randomization scheme, when the mean of a supplementary non-sensitive variable is known. The minimum attainable variance bound of the class is obtained and the best estimator is also identified. We prove that the best estimator acts as a regression-type estimator which is at least as efficient as the corresponding estimator evaluated without allowing for the auxiliary variable. The general results are then applied to Warner and Simmons’ model.     

ESTIMATING A PARAMETER WHEN IT IS KNOWN THAT THE PARAMETER EXCEEDS A GIVEN VALUE

In some statistical problems a degree of explicit, prior information is available about the value taken by the parameter of interest, θ say, although the information is much less than would be needed to place a prior density on the parameter's distribution. Often the prior information takes the form of a simple bound, ‘θ > θ₁ ’ or ‘θ < θ₁ ’, where θ₁ is determined by physical considerations or mathematical theory, such as positivity of a variance. A conventional approach to accommodating the requirement that θ > θ₁ is to replace an estimator, , of θ by the maximum of and θ₁. However, this technique is generally inadequate. For one thing, it does not respect the strictness of the inequality θ > θ₁ , which can be critical in interpreting results. For another, it produces an estimator that does not respond in a natural way to perturbations of the data. In this paper we suggest an alternative approach, in which bootstrap aggregation, or bagging, is used to overcome these difficulties. Bagging gives estimators that, when subjected to the constraint θ > θ₁ , strictly exceed θ₁ except in extreme settings in which the empirical evidence strongly contradicts the constraint. Bagging also reduces estimator variability in the important case for which is close to θ₁, and more generally produces estimators that respect the constraint in a smooth, realistic fashion.     

Improved likelihood inferences for Weibull regression model

A general procedure is developed for bias-correcting the maximum likelihood estimators (MLEs) of the parameters of Weibull regression model with either complete or right-censored data. Following the bias correction, variance corrections and hence improved t-ratios for model parameters are presented. Potentially improved t-ratios for other reliability-related quantities are also discussed. Simulation results show that the proposed method is effective in correcting the bias of the MLEs, and the resulted t-ratios generally improve over the regular t-ratios.     

Difference-based matrix perturbation method for semi-parametric regression with multicollinearity

This paper addresses the collinearity problems in semi-parametric linear models. Under the difference-based settings, we introduce a new diagnostic, the difference-based variance inflation factor (DVIF), for detecting the presence of multicollinearity in semi-parametric models. The DVIF is then used to device a difference-based matrix perturbation method for solving the problem. The electricities distribution data set is analyzed, and numerical evidences validate the effectiveness of the proposed method.     

Empirical Uncertain Bayes Methods in Area‐level Models

Random effects model can account for the lack of fitting a regression model and increase precision of estimating area‐level means. However, in case that the synthetic mean provides accurate estimates, the prior distribution may inflate an estimation error. Thus, it is desirable to consider the uncertain prior distribution, which is expressed as the mixture of a one‐point distribution and a proper prior distribution. In this paper, we develop an empirical Bayes approach for estimating area‐level means, using the uncertain prior distribution in the context of a natural exponential family, which we call the empirical uncertain Bayes (EUB) method. The regression model considered in this paper includes the Poisson‐gamma and the binomial‐beta, and the normal‐normal (Fay–Herriot) model, which are typically used in small area estimation. We obtain the estimators of hyperparameters based on the marginal likelihood by using a well‐known expectation‐maximization algorithm and propose the EUB estimators of area means. For risk evaluation of the EUB estimator, we derive a second‐order unbiased estimator of a conditional mean squared error by using some techniques of numerical calculation. Through simulation studies and real data applications, we evaluate a performance of the EUB estimator and compare it with the usual empirical Bayes estimator.     

A study of quadratic inference functions with alternative weighting matrices

Quadratic inference function (QIF) is an alternative methodology to the popular generalized estimating equations (GEE) approach, it does not involve direct estimation of the correlation parameter, and thus remains optimal even if the working correlation structure is misspecified. The idea is to represent the inverse of the working correlation matrix by a linear combination of some basis matrices. In this article, we present a modification of QIF with a robust variance estimator of the extended score function. Theoretical and numerical results show that the modified QIF attains better efficiency and achieves better small sample performance than the original QIF method.     

Estimation of P{X < Y} for geometric—exponential model based on complete and censored samples

This article deals with the estimation of R = P{X < Y}, where X and Y are independent random variables from geometric and exponential distribution, respectively. For complete samples, the MLE of R, its asymptotic distribution, and confidence interval based on it are obtained. The procedure for deriving bootstrap-p confidence interval is presented. The UMVUE of R and UMVUE of its variance are derived. The Bayes estimator of R is investigated and its Lindley's approximation is obtained. A simulation study is performed in order to compare these estimators. Finally, all point estimators for right censored sample from the exponential distribution, are obtained.     

Evaluating modified generalized information criterion in presence of multicollinearity

When there are many explanatory variables in the regression model, there is a chance that some of these are intercorrelated. This is where the problem of multicollinearity creeps in due to which precision and accuracy of the coefficients is marred, and the quest to find the best model becomes tedious. To tackle such a situation, Model selection criteria are applied for selecting the best model that fits the data. Current study focuses on the evaluation of the four unmodified and four modified versions of generalized information criteria—Akaike Information Criterion, Schwarz's Bayes Information Criteria, Hannan-Quinn Information Criterion, and Akaike Information Criterion corrected for small samples. A simulation study using SAS software was carried out in order to compare the unmodified and modified versions of the generalized information criteria and to discover the best version amongst the four modified model selection criteria, for identifying the best model, when the collinearity assumption is violated. For the proposed simulation, two samples of size 50 and 100, for three explanatory variables X₁, X₂, and X₃, are drawn from Normal distribution. Two situations of collinearity violations between X₁ and X₂ are looked into, first when ρ = 0.6 and second when ρ = 0.8. The outcomes of the simulations are displayed in the tables along with visual representations. The results revealed that modified versions of the generalized information criteria are more sensitive in identifying models marred with high multicollinearity as compared to the unmodified generalized information criteria.