Abstract
Expectiles are M-quantiles which are defined as the minimizers of asymmetric quadratic expected loss functions. Increasing attention has been devoted to expectiles because of their desirable properties as coherent, law-invariant, and elicitable financial risk measures. In this dissertation, univariate expectiles for random variables are extended into set-valued expectiles for random vectors in the higher dimensional spaces. The set-valued expectiles consid ered here include expectile regions in the literature, which describe the centrality of multivariate distributions like other depth regions; and the newly proposed cone expectiles–representing the tails of distribution depending on a vector preorder generated by a convex cone. The dual representations for both the expectile regions and the cone expectiles are derived in Chap ter 2, together with the fundamental properties of the cone expectiles. In addition, set-valued sublinear risk measures can be constructed from cone expectile sets in the same way as the univariate one. For applications, cone expectile ranking functions and cone expectile orders are also proposed for ordered data analysis. The second contribution of this thesis concerns the empirical set-valued expectiles and the empirical expectile depth constructed from expectile regions. The main focus is placed on the computation of these empirical sets and depth functions using different optimization techniques. This is possible due to the same dual information shared in both notions. In particular, in their dual forms, empirical set-valued expectiles can be seen as upper images of vector linear programming problems, whereas the empirical expectile depth can be computed using linear programming and bisection search. The proposed methods are applicable in the higher di mensional settings beyond Rd, although the curse of dimensionality still exists. Finally, the convergence rate of empirical expectile sets is established for certain families of distributions of random vectors. The final chapter proposes a variant of the expectile depth, termed the non-zero or stretched expectile depth, that assigns a strictly positive depth value to outlying observations outside the support of the multivariate distribution. This is done by replacing the distribution of the (scalarized) random variable with a mixture of the original distribution and a re-scaled version centered around the data point of interest. The stretched expectile trimmed region is stated as a consequence; the properties of both new depth and region are studied. The computation of the empirical stretched expectile depth can be done using the same algorithm as the original empirical expectile depth with appropriate rescaling of the data. The multivariate non-zero expectile depth may find applications in risk management, for instance, in approximating the probability of rare yet potentially catastrophic extreme events happening in multidimensional settings.