We define a distribution on the unit sphere S d−1 called the elliptically symmetric angular Gaussian distribution. This distribution, which to our knowledge has not been studied before, is a subfamily of the angular Gaussian distribution closely analogous to the Kent subfamily of the general Fisher-Bingham distribution. Like the Kent distribution, it has ellipse-like contours, enabling modelling of rotational asymmetry about the mean direction, but it has the additional advantages of being simple and fast to simulate from, and having a density and hence likelihood that is easy and very quick to compute exactly. These advantages are especially beneficial for computationally intensive statistical methods, one example of which is a parametric bootstrap procedure for inference for the directional mean that we describe.

Existing parametric regression models in the literature for response data on the unit sphere assume that the covariates have particularly simple structure, for example that they are either scalar or are themselves on the unit sphere, and/or that the error distribution is isotropic. In many practical situations, such models are too inflexible. Here, we develop richer parametric spherical regression models in which the covariates can have quite general structure (for example, they may be on the unit sphere, in Euclidean space, categorical or some combination of these) and in which the errors are anisotropic. We consider two anisotropic error distributions-the Kent distribution and the elliptically symmetric angular Gaussian distribution-and two parametrisations of each which enable distinct ways to model how the response depends on the covariates. Various hypotheses of interest, such as the significance of particular covariates, or anisotropy of the errors, are easy to test, for example by classical likelihood ratio tests. We also introduce new model-based residuals for evaluating the fitted models. In the examples we consider, the hypothesis tests indicate strong evidence to favour the novel models over simpler existing ones.

The study of action selection in humans can present challenges of task design since our actions are usually defined by many degrees of freedom and therefore occupy a large action-space. While saccadic eye-movement offers a more constrained paradigm for investigating action selection, the study of reach-and-grasp in upper limbs has often been defined by more complex scenarios, not easily interpretable in terms of such selection. Here we present a novel motor behaviour task which addresses this by limiting the action space to a single degree of freedom in which subjects have to track (using a stylus) a vertical coloured target line displayed on a tablet computer, whilst ignoring a similarly oriented distractor line in a different colour. We ran this task with 55 subjects and showed that, in agreement with previous studies, the presence of the distractor generally increases the movement latency and directional error rate. Further, we used two distractor conditions according to whether the distractor’s location changes asynchronously or synchronously with the location of the target. We found that the asynchronous distractor yielded poorer performance than its synchronous counterpart, with significantly higher movement latencies and higher error rates. We interpret these results in an action selection framework with two actions (move left or right) and competing ‘action requests’ offered by the target and distractor. As such, the results provide insights into action selection performance in humans and supply data for directly constraining future computational models therein.

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We consider a one-dimensional stochastic differential equation that is observed on a fine grid of equally spaced time points. A novel approach for approximating the transition density of the stochastic differential equation is presented, which is based on an Itô-Taylor expansion of the sample path, combined with an application of the so-called 𝜖-expansion. The resulting approximation is economical with respect to the number of terms needed to achieve a given level of accuracy in a high-frequency sampling framework. This method of density approximation leads to a closed-form approximate likelihood function from which an approximate maximum likelihood estimator may be calculated numerically. A detailed theoretical analysis of the proposed estimator is provided and it is shown that it compares favorably to the Gaussian likelihood-based estimator and does an excellent job of approximating the exact, but usually intractable, maximum likelihood estimator. Numerical simulations indicate that the exact and our approximate maximum likelihood estimator tend to be close, and the latter performs very well relative to other approximate