On the Breslow–Holubkov estimator

Breslow and Holubkov (J Roy Stat Soc B 59:447–461 1997a) developed semiparametric maximum likelihood estimation for two-phase studies with a case–control first phase under a logistic regression model and noted that, apart for the overall intercept term, it was the same as the semiparametric estimator for two-phase studies with a prospective first phase developed in Scott and Wild (Biometrica 84:57–71 1997). In this paper we extend the Breslow–Holubkov result to general binary regression models and show that it has a very simple relationship with its prospective first-phase counterpart. We also explore why the design of the first phase only affects the intercept of a logistic model, simplify the calculation of standard errors, establish the semiparametric efficiency of the Breslow–Holubkov estimator and derive its asymptotic distribution in the general case.     

ELICITING A DIRECTED ACYCLIC GRAPH FOR A MULTIVARIATE TIME SERIES OF VEHICLE COUNTS IN A TRAFFIC NETWORK

The problem of modelling multivariate time series of vehicle counts in traffic networks is considered. It is proposed to use a model called the linear multiregression dynamic model (LMDM). The LMDM is a multivariate Bayesian dynamic model which uses any conditional independence and causal structure across the time series to break down the complex multivariate model into simpler univariate dynamic linear models. The conditional independence and causal structure in the time series can be represented by a directed acyclic graph (DAG). The DAG not only gives a useful pictorial representation of the multivariate structure, but it is also used to build the LMDM. Therefore, eliciting a DAG which gives a realistic representation of the series is a crucial part of the modelling process. A DAG is elicited for the multivariate time series of hourly vehicle counts at the junction of three major roads in the UK. A flow diagram is introduced to give a pictorial representation of the possible vehicle routes through the network. It is shown how this flow diagram, together with a map of the network, can suggest a DAG for the time series suitable for use with an LMDM.     

Bayesian analysis of human movement curves

Summary.  We consider the Bayesian analysis of human movement data, where the subjects perform various reaching tasks. A set of markers is placed on each subject and a system of cameras records the three-dimensional Cartesian co-ordinates of the markers during the reaching movement. It is of interest to describe the mean and variability of the curves that are traced by the markers during one reaching movement, and to identify any differences due to covariates. We propose a methodology based on a hierarchical Bayesian model for the curves. An important part of the method is to obtain identifiable features of the movement so that different curves can be compared after temporal warping. We consider four landmarks and a set of equally spaced pseudolandmarks are located in between. We demonstrate that the algorithm works well in locating the landmarks, and shape analysis techniques are used to describe the posterior distribution of the mean curve. A feature of this type of data is that some parts of the movement data may be missing—the Bayesian methodology is easily adapted to cope with this situation.     

Model-based clustering for social networks

Summary.  Network models are widely used to represent relations between interacting units or actors. Network data often exhibit transitivity, meaning that two actors that have ties to a third actor are more likely to be tied than actors that do not, homophily by attributes of the actors or dyads, and clustering. Interest often focuses on finding clusters of actors or ties, and the number of groups in the data is typically unknown. We propose a new model, the latent position cluster model , under which the probability of a tie between two actors depends on the distance between them in an unobserved Euclidean 'social space', and the actors' locations in the latent social space arise from a mixture of distributions, each corresponding to a cluster. We propose two estimation methods: a two-stage maximum likelihood method and a fully Bayesian method that uses Markov chain Monte Carlo sampling. The former is quicker and simpler, but the latter performs better. We also propose a Bayesian way of determining the number of clusters that are present by using approximate conditional Bayes factors. Our model represents transitivity, homophily by attributes and clustering simultaneously and does not require the number of clusters to be known. The model makes it easy to simulate realistic networks with clustering, which are potentially useful as inputs to models of more complex systems of which the network is part, such as epidemic models of infectious disease. We apply the model to two networks of social relations. A free software package in the R statistical language, latentnet, is available to analyse data by using the model.     

Easy-to-construct confidence bands for comparing two simple linear regression lines

This paper shows how to construct confidence bands for the difference between two simple linear regression lines. These confidence bands provide directly the information on the magnitude of the difference between the regression lines over an interval of interest and, as a by-product, can be used as a formal test of the difference between the two regression lines. Various different shapes of confidence bands are illustrated, and particular attention is paid towards confidence bands whose construction only involves critical points from standard distributions so that they are consequently easy to construct.     

Generalized inferences on the common mean vector of several multivariate normal populations

The hypothesis testing and confidence region are considered for the common mean vector of several multivariate normal populations when the covariance matrices are unknown and possibly unequal. A generalized confidence region is derived using the concepts of generalized method based on the generalized p$p$-value. The generalized confidence region is illustrated with two numerical examples. The merits of the proposed method are numerically compared with those of existing methods with respect to their expected area or expected d-dimensional volumes and coverage probabilities under different scenarios.     

Generalized model for IGC growth in aluminum alloys

A generalized brick wall model is developed to describe intergranular corrosion in Equi-axed AA7178-T6 and Wingskin AA7178-T6 aluminum alloys. The intergranular corrosion rate is highly related to grain size and shape. High strength aluminum alloys are often elongated and anisotropic, with the fastest nominal IGC growth rate in the longitudinal direction (L) or long transverse direction (T) and the slowest in the short transverse direction (S). We propose a three-way intersection model and use it to simulate the corrosion kinetics for each direction. With a proper combination of model parameters, the generalized IGC model provides a good fit to experimental data developed by the foil penetration technique.     

A simulation study of a discrete-time multiserver retrial queue with finite population

In this paper, we deal with a discrete-time multiserver retrial queue with finite population. Firstly, we study the Markov chain at the epochs immediately after slot boundaries making emphasis on the computation of its steady-state distribution. Then, the main performance measures are investigated. Besides, we simulate the waiting time of a customer in the retrial group under three different queueing policies. Some numerical examples are given to illustrate the analysis.     

Some theory and practical uses of trimmed L-moments

Trimmed L-moments, defined by Elamir and Seheult [2003. Trimmed L-moments. Comput. Statist. Data Anal. 43, 299–314], summarize the shape of probability distributions or data samples in a way that remains viable for heavy-tailed distributions, even those for which the mean may not exist. We derive some further theoretical results concerning trimmed L-moments: a relation with the expansion of the quantile function as a weighted sum of Jacobi polynomials; the bounds that must be satisfied by trimmed L-moments; recurrences between trimmed L-moments with different degrees of trimming; and the asymptotic distributions of sample estimators of trimmed L-moments. We also give examples of how trimmed L-moments can be used, analogously to L-moments, in the analysis of heavy-tailed data. Examples include identification of distributions using a trimmed L-moment ratio diagram, shape parameter estimation for the generalized Pareto distribution, and fitting generalized Pareto distributions to a heavy-tailed data sample of computer network traffic.