Principal component analysis (PCA) is a widely used dimension reduction tool in the analysis of many kind of high-dimensional data. It is used in signal processing, mechanical engineering, psychometrics, and other fields under different names. It still bears the same mathematical idea: the decomposition of variation of a high dimensional object into uncorrelated factors or components. However, in many of the above applications, one is interested in capturing the tail variables of the data rather than variation around the mean. Such applications include weather related event curves, expected shortfalls, and speeding analysis among others. These are all high dimensional tail objects which one would like to study in a PCA fashion. The tail character though requires to do the dimension reduction in an asymmetric norm rather than the classical L 2 -type orthogonal projection. We develop an analogue of PCA in an asymmetric norm. These norms cover both quantiles and expectiles, another tail event measure. The difficulty is that there is no natural basis, no 'principal components', to the k-dimensional subspace found. We propose two definitions of principal components and provide algorithms based on iterative least squares. We prove upper bounds on their convergence times, and compare their performances in a simulation study. We apply the algorithms to a Chinese weather dataset with a view to weather derivative pricing.

No abstract

No abstract

No abstract

High-frequency data can provide us with a quantity of information for forecasting, help to calculate and prevent the future risk based on extremes. This tail behaviour is very often driven by exogenous components and may be modelled conditional on other variables. However, many of these phenomena are observed over time, exhibiting non-trivial dynamics and dependencies. We propose a functional dynamic factor model to study the dynamics of expectile curves. The complexity of the model and the number of dependent variables are reduced by lasso penalization. The functional factors serve as a low-dimensional representation of the conditional tail event, while the time-variation is captured by factor loadings. We illustrate the model with an application to climatology, where daily data over years on temperature, rainfalls or strength of wind are available.