Correlated Binary Variables and Multi-level Probability Assessments

Abstract.  We introduce a fully parametric approach for updating beliefs regarding correlated binary variables, after marginal probability assessments based on information of varying quality are provided by an expert. This approach allows for the calculation of a predictive joint density for future assessments. The proposed methodology offers new insight into the parameters that control the dependence of the binary variables, and the relation of these parameters to the joint density of the probability assessments. A comprehensible elicitation procedure for the model parameters is put forward. The approach taken is motivated and illustrated through a practical application.     

GMM nonparametric correction methods for logistic regression with error‐contaminated covariates and partially observed instrumental variables

We consider logistic regression with covariate measurement error. Most existing approaches require certain replicates of the error‐contaminated covariates, which may not be available in the data. We propose generalized method of moments (GMM) nonparametric correction approaches that use instrumental variables observed in a calibration subsample. The instrumental variable is related to the underlying true covariates through a general nonparametric model, and the probability of being in the calibration subsample may depend on the observed variables. We first take a simple approach adopting the inverse selection probability weighting technique using the calibration subsample. We then improve the approach based on the GMM using the whole sample. The asymptotic properties are derived, and the finite sample performance is evaluated through simulation studies and an application to a real data set.     

Count Data Models with Correlated Unobserved Heterogeneity

Abstract. As previously argued, the correlation between included and omitted regressors generally causes inconsistency of standard estimators for count data models. Non‐linear instrumental variables estimation of an exponential model under conditional moment restrictions is one of the proposed remedies. This approach is extended here by fully exploiting the model assumptions and thereby improving efficiency of the resulting estimator. Empirical likelihood in particular has favourable properties in this setting compared with the two‐step generalized method of moments, as demonstrated in a Monte Carlo experiment. The proposed method is applied to the estimation of a cigarette demand function.     

Estimation and variable selection in partial linear single index models with error-prone linear covariates

We study the estimation and variable selection for a partial linear single index model (PLSIM) when some linear covariates are not observed, but their ancillary variables are available. We use the semiparametric profile least-square based estimation procedure to estimate the parameters in the PLSIM after the calibrated error-prone covariates are obtained. Asymptotic normality for the estimators are established. We also employ the smoothly clipped absolute deviation (SCAD) penalty to select the relevant variables in the PLSIM. The resulting SCAD estimators are shown to be asymptotically normal and have the oracle property. Performance of our estimation procedure is illustrated through numerous simulations. The approach is further applied to a real data example.     

Orthogonalized Residuals for Estimation of Marginally Specified Association Parameters in Multivariate Binary Data

Abstract. This paper focuses on marginal regression models for correlated binary responses when estimation of the association structure is of primary interest. A new estimating function approach based on orthogonalized residuals is proposed. A special case of the proposed procedure allows a new representation of the alternating logistic regressions method through marginal residuals. The connections between second‐order generalized estimating equations, alternating logistic regressions, pseudo‐likelihood and other methods are explored. Efficiency comparisons are presented, with emphasis on variable cluster size and on the role of higher‐order assumptions. The new method is illustrated with an analysis of data on impaired pulmonary function.     

Simulation of Constrained Variables in Engineering Risk Analyses

The problem of sampling random variables with overlapping pdfs subject to inequality constraints is addressed. Often, the values of physical variables in an engineering model are interrelated. This mutual dependence imposes inequality constraints on the random variables representing these parameters. Ignoring the interdependencies and sampling the variables independently can lead to inconsistency/bias. We propose an algorithm to generate samples of constrained random variables that are characterized by typical continuous probability distributions and are subject to different kinds of inequality constraints. The sampling procedure is illustrated for various representative cases and one realistic application to simulation of structural natural frequencies.     

A clustering approach to interpretable principal components

A new method for constructing interpretable principal components is proposed. The method first clusters the variables, and then interpretable (sparse) components are constructed from the correlation matrices of the clustered variables. For the first step of the method, a new weighted-variances method for clustering variables is proposed. It reflects the nature of the problem that the interpretable components should maximize the explained variance and thus provide sparse dimension reduction. An important feature of the new clustering procedure is that the optimal number of clusters (and components) can be determined in a non-subjective manner. The new method is illustrated using well-known simulated and real data sets. It clearly outperforms many existing methods for sparse principal component analysis in terms of both explained variance and sparseness.     

Approximate composite marginal likelihood inference in spatial generalized linear mixed models

Non-Gaussian spatial responses are usually modeled using spatial generalized linear mixed model with spatial random effects. The likelihood function of this model cannot usually be given in a closed form, thus the maximum likelihood approach is very challenging. There are numerical ways to maximize the likelihood function, such as Monte Carlo Expectation Maximization and Quadrature Pairwise Expectation Maximization algorithms. They can be applied but may in such cases be computationally very slow or even prohibitive. Gauss–Hermite quadrature approximation only suitable for low-dimensional latent variables and its accuracy depends on the number of quadrature points. Here, we propose a new approximate pairwise maximum likelihood method to the inference of the spatial generalized linear mixed model. This approximate method is fast and deterministic, using no sampling-based strategies. The performance of the proposed method is illustrated through two simulation examples and practical aspects are investigated through a case study on a rainfall data set.     

On a Unified Generalized Quasi–likelihood Approach for Familial–Longitudinal Non‐Stationary Count Data

Abstract. In this paper, conditional on random family effects, we consider an auto‐regression model for repeated count data and their corresponding time‐dependent covariates, collected from the members of a large number of independent families. The count responses, in such a set up, unconditionally exhibit a non‐stationary familial–longitudinal correlation structure. We then take this two‐way correlation structure into account, and develop a generalized quasilikelihood (GQL) approach for the estimation of the regression effects and the familial correlation index parameter, whereas the longitudinal correlation parameter is estimated by using the well‐known method of moments. The performance of the proposed estimation approach is examined through a simulation study. Some model mis‐specification effects are also studied. The estimation methodology is illustrated by analysing real life healthcare utilization count data collected from 36 families of size four over a period of 4 years.     

Analyzing multivariate longitudinal binary data: A generalized estimating equations approach

The authors consider regression analysis for binary data collected repeatedly over time on members of numerous small clusters of individuals sharing a common random effect that induces dependence among them. They propose a mixed model that can accommodate both these structural and longitudinal dependencies. They estimate the parameters of the model consistently and efficiently using generalized estimating equations. They show through simulations that their approach yields significant gains in mean squared error when estimating the random effects variance and the longitudinal correlations, while providing estimates of the fixed effects that are just as precise as under a generalized penalized quasi‐likelihood approach. Their method is illustrated using smoking prevention data.     

A Survey of Weak Instruments and Weak Identification in Generalized Method of Moments

Weak instruments arise when the instruments in linear instrumental variables (IV) regression are weakly correlated with the included endogenous variables. In generalized method of moments (GMM), more generally, weak instruments correspond to weak identification of some or all of the unknown parameters. Weak identification leads to GMM statistics with nonnormal distributions, even in large samples, so that conventional IV or GMM inferences are misleading. Fortunately, various procedures are now available for detecting and handling weak instruments in the linear IV model and, to a lesser degree, in nonlinear GMM.     

A consistent simulation-based estimator in generalized linear mixed models

We propose a strongly root-n consistent simulation-based estimator for the generalized linear mixed models. This estimator is constructed based on the first two marginal moments of the response variables, and it allows the random effects to have any parametric distribution (not necessarily normal). Consistency and asymptotic normality for the proposed estimator are derived under fairly general regularity conditions. We also demonstrate that this estimator has a bounded influence function and that it is robust against data outliers. A bias correction technique is proposed to reduce the finite sample bias in the estimation of variance components. The methodology is illustrated through an application to the famed seizure count data and some simulation studies.     

A Spline‐Based Semiparametric Maximum Likelihood Estimation Method for the Cox Model with Interval‐Censored Data

Abstract. We propose a spline‐based semiparametric maximum likelihood approach to analysing the Cox model with interval‐censored data. With this approach, the baseline cumulative hazard function is approximated by a monotone B‐spline function. We extend the generalized Rosen algorithm to compute the maximum likelihood estimate. We show that the estimator of the regression parameter is asymptotically normal and semiparametrically efficient, although the estimator of the baseline cumulative hazard function converges at a rate slower than root‐n. We also develop an easy‐to‐implement method for consistently estimating the standard error of the estimated regression parameter, which facilitates the proposed inference procedure for the Cox model with interval‐censored data. The proposed method is evaluated by simulation studies regarding its finite sample performance and is illustrated using data from a breast cosmesis study.     

ROBUST INFERENCE IN PARAMETRIC MODELS USING THE FAMILY OF GENERALIZED NEGATIVE EXPONENTIAL DISPARITIES

We examine robust estimators and tests using the family of generalized negative exponential disparities, which contains the Pearson's chi‐square and the ordinary negative exponential disparity as special cases. The influence function and α‐influence function of the proposed estimators are discussed and their breakdown points derived. Under the model, the estimators are asymptotically efficient, and are shown to have an asymptotic breakdown point of 50%. The proposed tests are shown to be equivalent to the likelihood ratio test under the null hypothesis, and their breakdown points are obtained. The competitive performance of the proposed estimators and tests relative to those based on the Hellinger distance is illustrated through examples and simulation results. Unlike the Hellinger distance, several members of this family of generalized negative exponential disparities generate estimators which also possess excellent inlier‐controlling capability. The corresponding tests of hypothesis are shown to have better power breakdown than the Hellinger deviance test in the cases examined.     

Double filter instrumental variable estimation of panel data models with weakly exogenous variables

In this article, we propose instrumental variables (IV) and generalized method of moments (GMM) estimators for panel data models with weakly exogenous variables. The model is allowed to include heterogeneous time trends besides the standard fixed effects (FE). The proposed IV and GMM estimators are obtained by applying a forward filter to the model and a backward filter to the instruments in order to remove FE, thereby called the double filter IV and GMM estimators. We derive the asymptotic properties of the proposed estimators under fixed T and large N, and large T and large N asymptotics where N and T denote the dimensions of cross section and time series, respectively. It is shown that the proposed IV estimator has the same asymptotic distribution as the bias corrected FE estimator when both N and T are large. Monte Carlo simulation results reveal that the proposed estimator performs well in finite samples and outperforms the conventional IV/GMM estimators using instruments in levels in many cases.     

A class of correlated weighted Poisson processes

In this paper, we propose new classes of correlated Poisson processes and correlated weighted Poisson processes on the interval [0,1], which generalize the class of weighted Poisson processes defined by Balakrishnan and Kozubowski (2008), by incorporating a dependence structure between the standard uniform variables used in the construction. In this manner, we obtain another process that we refer to as correlated weighted Poisson process. Various properties of this process such as marginal and joint distributions, stationarity of the increments, moments, and the covariance function, are studied. The results are then illustrated through some examples, which include processes with length-biased Poisson, exponentially weighted Poisson, negative binomial, and COM-Poisson distributions.     

Estimators for a simultaneous equation model with lagged endogenous variables and autocorrelated errors

Two full information estimators and a limited information estimator for the simultaneous equation model with autocorrelated errors are studies by the Monte Carlo method.The estimators share features of the two-step Gauss-Newton procedure and of Aitken generalized least squares.One full information method generates the estimated endogenous variables used in the later stages of computation from the unrestricted reduced form while the other uses a restricted reduced form.The observed small sample behavior was close to that suggested by asymptotic theory.     

Complete convergence for arrays of row-wise ND random variables under sub-linear expectations

In this paper, complete convergence for arrays of row-wise ND random variables under sub-linear expectations is studied. As applications, the complete convergence theorems of weighted sums for negatively dependent random variables have been generalized to the sub-linear expectation space context. We extend some complete convergence theorems from the traditional probability space to the sub-linear expectation space and our results generalize corresponding results obtained by Ko.     

Unit root test for short panels with serially correlated errors

This paper proposes a unit root test for short panels with serially correlated errors. The proposed test is based on the instrumental variables (IVs) and the generalized method of moments (GMM) estimators. An advantage of the new test over other tests is that it allows for an ARMA-type serial correlation. A Monte Carlo simulation shows that the new test has good finite sample properties. Several methods to estimate the lag orders of the ARMA structure are briefly discussed.     

A Family of Near-Exact Distributions Based on Truncations of the Exact Distribution for the Generalized Wilks Lambda Statistic

For the case where at least two sets have an odd number of variables we do not have the exact distribution of the generalized Wilks Lambda statistic in a manageable form, adequate for manipulation. In this article, we develop a family of very accurate near-exact distributions for this statistic for the case where two or three sets have an odd number of variables. We first express the exact characteristic function of the logarithm of the statistic in the form of the characteristic function of an infinite mixture of Generalized Integer Gamma distributions. Then, based on truncations of this exact characteristic function, we obtain a family of near-exact distributions, which, by construction, match the first two exact moments. These near-exact distributions display an asymptotic behaviour for increasing number of variables involved. The corresponding cumulative distribution functions are obtained in a concise and manageable form, relatively easy to implement computationally, allowing for the computation of virtually exact quantiles. We undertake a comparative study for small sample sizes, using two proximity measures based on the Berry-Esseen bounds, to assess the performance of the near-exact distributions for different numbers of sets of variables and different numbers of variables in each set.