No abstract

Recent interest in modern regression modelling has focused on extending available (mean) regression models by describing more general properties of the response distribution. An alternative approach is quantile regression where regression effects on the conditional quantile function of the response are assumed. While quantile regression can be seen as a generalization of median regression, expectiles as alternative are a generalized form of mean regression.Generally, quantiles provide a natural interpretation even beyond the 0.5 quantile, the median. A comparable simple interpretation is not available for expectiles beyond the 0.5 expectile, the mean. Nonetheless, expectiles have some interesting properties, some of which are discussed in this article. We contrast the two approaches and show how to get quantiles from a fine grid of expectiles. We compare such quantiles from expectiles with direct quantile estimates regarding efficiency. We also look at regression problems where both quantile and expectile curves have the undesirable property that neighbouring curves may cross each other. We propose a modified method to estimate non-crossing expectile curves based on splines. In an application, we look at the expected shortfall, a risk measure used in finance, which requires both expectiles and quantiles for estimation and which can be calculated easily with the proposed methods in the article.

Transcranial alternating current stimulation (tACS) sees increased use in neurosciences as a tool for the exploration of brain oscillations. It has been shown that tACS stimulation in specific frequency bands can result in aftereffects of modulated oscillatory brain activity that persist after the stimulation has ended. The general relationship between persistency of the effect and duration of stimulation is sparsely investigated but previous research has shown that the occurrence of tACS aftereffects depends on the brain state before and during stimulation. Early alpha neurofeedback research suggests that particularly in the alpha band the responsiveness to a manipulation depends on the ambient illumination during measurement. Therefore, in the present study we assessed the brain’s susceptibility to tACS at the individual alpha frequency during darkness compared to ambient illumination. We measured alpha power after 10 min of stimulation in 30 participants while they continuously performed a visual vigilance task. Our results show that immediately after stimulation, the alpha power in the illumination condition for both the stimulated and sham group has increased by only about 7%, compared to about 20% in both groups in the ‘dark’ condition. For the group that did not receive stimulation, the power in darkness remained stable after stimulation, whereas the power in light increased by an additional 10% during the next 30 min. For the group that did receive stimulation, alpha power during these 30 min increased by another 11% in light and 22% in darkness. Since alpha power already increased by about 10% without stimulation, the effect of illumination does not seem to have interacted with the effect of stimulation. Instead, both effects seem to have added up linearly. Although our findings do not show that illumination-induced differences in oscillatory activity influence the susceptibility toward tACS, they stress the importance of controlling for factors like ambient light that might add an independent increase or decrease to the power of brain oscillations during periods, where possible persistent effects of stimulation are explored.

In regression scenarios there is a growing demand for information on the conditional distribution of the response beyond the mean. In this scenario quantile regression is an established method of tail analysis. It is well understood in terms of asymptotic properties and estimation quality. Another way to look at the tail of a distribution is via expectiles. They provide a valuable alternative since they come with a combination of preferable attributes. The easy weighted least squares estimation of expectiles and the quadratic penalties often used in flexible regression models are natural partners. Also, in a similar way as quantiles can be seen as a generalisation of median regression, expectiles offer a generalisation of mean regression. In addition to regression estimates, confidence intervals are essential for interpretational purposes and to assess the variability of the estimate, but there is a lack of knowledge regarding the asymptotic properties of a semiparametric expectile regression estimate. Therefore confidence intervals for expectiles based on an asymptotic normal distribution are introduced. Their properties are investigated by a simulation study and compared to a boostrap-based gold standard method. Finally the introduced confidence intervals help to evaluate a geoadditive expectile regression model on childhood malnutrition data from India.