It is a common practice to analyze complex longitudinal data using flexible nonlinear mixed-effects (NLME) models with normality assumption. However, a serious departure of normality may cause lack of robustness and subsequently lead to invalid inference and unreasonable estimates. Covariates are usually introduced in such models to partially explain inter-subject variations, but some covariates may be often measured with substantial errors. Moreover, the response observations may be subject to left-censoring due to a detection limit. Inferential procedures can be complicated dramatically when data with asymmetric (skewed) characteristics, left-censoring and measurement errors are observed. In the literature, there has been considerable interest in accommodating either skewness, censoring or covariate measurement errors in such models, but there is relatively little work concerning all of the three features simultaneously. In this article, we jointly investigate a skew-t NLME model for response (with left-censoring) process and a skew-t nonparametric mixed-effects model for covariate (with measurement errors) process. We propose a robust skew-t Bayesian modeling approach in a general form to analyze data in capturing the effects of skewness, censoring and measurement errors in covariates simultaneously. A real data example is offered to illustrate the methodologies. The proposed modeling alternative offers important advantages in the sense that the model can be easily fitted in freely available software and the computational effort for the model with a skew-t distribution is almost equivalent to that of the model with a standard normal distribution.

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A smoothing spline is used to propose a novel model for the time-varying quantile of the univariate time series using a state space approach. A correlation is further incorporated between the dependent variable and its one-step-ahead quantile. Using a Bayesian approach, an efficient Markov chain Monte Carlo algorithm is described where we use the multi-move sampler, which generates simultaneously latent time-varying quantiles. Numerical examples are provided to show its high sampling efficiency, in comparison with the simple algorithm that generates one latent quantile at a time given other latent quantiles. Furthermore, using Japanese inflation rate data, an empirical analysis is provided with model comparisons.

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We consider a fractional 2^m₁+m₂ factorial design derived from a simple partially balanced array (SPBA), and we assume that the non-negligible factorial effects are the general mean, all the main effects and the two-factor interactions between the m₁ factors and the m₂ ones, and m_k ≥ 2 (k = 1, 2). In this paper, we give a necessary and sufficient condition for an SPBA to be a partially balanced fractional 2^m₁+m₂ factorial design such that all the non-negligible factorial effects are estimable, whose design is said to be of resolution R({00, 10, 01, 11}). Such a design is concretely characterized by the suffixes of the indices of an SPBA.

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The method of probability generating functions is extended for obtaining exact distributions of the number of occurrences of a discrete pattern in undirected graphical models. General results for deriving the distributions are given with illustrative examples. Further, a device for reducing calculations is proposed. It works effectively when the graphical model is relatively simple. An algorithm for obtaining the distributions including the device is also given. In order to show the feasibility of our method, exact distributions of the number of occurrences of a “1”-run are derived in two undirected graphical models whose vertices are allocated on a sphere and a torus, respectively. As an application of our results, the exact reliabilities of consecutive-k-out-of-n:F systems corresponding to the undirected graphical models are obtained.

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This paper is concerned with the asymptotic theory for penalized spline estimators in additive models. The focus of this paper is on the penalized spline estimators obtained by the backfitting algorithm. The convergence of the algorithm as well as the uniqueness of its solution are shown. Asymptotic equivalence between the penalized spline estimators by the backfitting algorithm and the convenient estimators proposed by Marx and Eilers (1998) is addressed. Asymptotic normality of the estimators is also developed, by which an approximate confidence interval can be obtained. Some numerical experiments confirming theoretical results are provided.

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