Estimation and inference of the joint conditional distribution for multivariate longitudinal data using nonparametric copulas

In this paper we study estimating the joint conditional distributions of multivariate longitudinal outcomes using regression models and copulas. For the estimation of marginal models, we consider a class of time-varying transformation models and combine the two marginal models using nonparametric empirical copulas. Our models and estimation method can be applied in many situations where the conditional mean-based models are not good enough. Empirical copulas combined with time-varying transformation models may allow quite flexible modelling for the joint conditional distributions for multivariate longitudinal data. We derive the asymptotic properties for the copula-based estimators of the joint conditional distribution functions. For illustration we apply our estimation method to an epidemiological study of childhood growth and blood pressure.     

Uniform in bandwidth consistency for various kernel estimators involving functional data

The paper investigates various nonparametric models including regression, conditional distribution, conditional density and conditional hazard function, when the covariates are infinite dimensional. The main contribution is to prove uniform in bandwidth asymptotic results for kernel estimators of these functional operators. Then, the application issues, involving data-driven bandwidth selection, are discussed.     

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.     

Spatio‐temporal Ornstein–Uhlenbeck Processes: Theory,Simulation and Statistical Inference

Spatio‐temporal modelling is an increasingly popular topic in Statistics. Our paper contributes to this line of research by developing the theory, simulation and inference for a spatio‐temporal Ornstein–Uhlenbeck process. We conduct detailed simulation studies and demonstrate the practical relevance of these processes in an empirical study of radiation anomaly data. Finally, we describe how predictions can be carried out in the Gaussian setting.     

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.     

On the complex dynamics of functional-coefficients nonlinear autoregressive time series models

In this article, a new chaotic functional-coefficient nonlinear autoregressive time series model is formulated. Asymptotic stability of the equilibria of the skeleton is studied. Complex dynamics of the proposed model are investigated by means of the bifurcations, time series diagrams, and phase portraits. The effects of noise intensity on its dynamics and the intermittency phenomenon are also discussed via simulation. Two chaotic indicators, namely, the fractal dimension and the Lyapunov exponent methods are investigated for the model.     

A note on efficient simulation of multidimensional spatial autoregressive processes

In applications of spatial statistics, it is necessary to compute the product of some matrix W of spatial weights and a vector y of observations. The weighting matrix often needs to be adapted to the specific problems, such that the computation of Wy cannot necessarily be done with available R-packages. Hence, this article suggests one possibility treating such issues. The proposed technique avoids the computation of the matrix product by calculating each entry of Wy separately. Initially, a specific spatial autoregressive process is introduced. The performance of the proposed program is briefly compared to a basic program using the matrix multiplication.     

Mixture inverse Gaussian for unobserved heterogeneity in the autoregressive conditional duration model

In this paper, we assume that the duration of a process has two different intrinsic components or phases which are independent. The first is the time it takes for a trade to be initiated in the market (for example, the time during which agents obtain knowledge about the market in which they are operating and accumulate information, which is coherent with Brownian motion) and the second is the subsequent time required for the trade to develop into a complete duration. Of course, if the first time is zero then the trade is initiated immediately and no initial knowledge is required. If we assume a specific compound Bernoulli distribution for the first time and an inverse Gaussian distribution for the second, the resulting convolution model has a mixture of an inverse Gaussian distribution with its reciprocal, which allows us to specify and test the unobserved heterogeneity in the autoregressive conditional duration (ACD) model.

Our proposals make it possible not only to capture various density shapes of the durations but also easily to accommodate the behaviour of the tail of the distribution and the non monotonic hazard function. The proposed model is easy to fit and characterizes the behaviour of the conditional durations reasonably well in terms of statistical criteria based on point and density forecasts.     

Generalized Kerridge’s inaccuracy measure for conditionally specified models

Recently, conditional Renyi’s divergence of order α and Kerridge’s inaccuracy measures are studied by Navarro et al. (2014 Navarro, J., Sunoj, S.M., Linu, M.N. (2014). Characterizations of bivariate models using some dynamic conditional information divergence measures. Commun. Stat. Theory Methods 43:1939–1948.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]). In the present article, a generalized dynamic conditional Kerridge’s inaccuracy measure is introduced, which can be represented as the sum of conditional Renyi’s divergence and Renyi’s entropy. Some useful bounds are obtained using the concept of likelihood ratio order. The results are extended to weighted distributions. Sufficient conditions are obtained for the monotonicity of the proposed measure. Characterizations for bivariate exponential conditional distribution are presented based on the proposed measure.