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Understanding how people make decisions from risky choices has attracted increasing attention of researchers in economics, psychology and neuroscience. While economists try to evaluate individual’s risk preference through mathematical modeling, neuroscientists answer the question by exploring the neural activities of the brain. We propose a model-free method, 3-dimensional image functional principal component analysis (3DIF), to provide a connection between active risk related brain region detection and individual’s risk preference. The 3DIF methodology is directly applicable to 3-dimensional image data without artificial vectorization or mapping and simultaneously guarantees the contiguity of risk related brain regions rather than discrete voxels. Simulation study evidences an accurate and reasonable region detection using the 3DIF method. In real data analysis, five important risk related brain regions are detected, including parietal cortex (PC), ventrolateral prefrontal cortex (VLPFC), lateral orbifrontal cortex (lOFC), anterior insula (aINS) and dorsolateral prefrontal cortex (DLPFC), while the alternative methods only identify limited risk related regions. Moreover, the 3DIF method is useful for extraction of subjective specific signature scores that carry explanatory power for individual’s risk attitude. In particular, the 3DIF method perfectly classifies both strongly and weakly risk averse subjects for in-sample analysis. In out-of-sample experiment, it achieves 73 -88  overall accuracy, among which 90 -100  strongly risk averse subjects and 49 -71  weakly risk averse subjects are correctly classified with leave- k -out cross validations.

Multivariate expectiles, a new family of vector-valued risk measures, were recently introduced in the literature. [22]. Here we investigate the asymptotic behavior of these measures in a multivariate regular variation context. For models with equivalent tails, we propose an estimator of extreme multivariate expectiles in the Fréchet domain of attraction case with asymptotic independence, or for comonotonic marginal distributions.

In this paper we derive a semiparametric efficient adaptive estimator for the GJR-GARCH(1,1){(1,1)} model. We first show that the quasi-maximum likelihood estimator is consistent and asymptotically normal for the model used in analysis, and we secondly derive a semiparametric estimator that is more efficient than the quasi-maximum likelihood estimator. Through Monte Carlo simulations, we show that the semiparametric estimator is adaptive for the parameters included in the conditional variance of the GJR-GARCH(1,1){(1,1)} model with respect to the unknown distribution of the innovation.