Multivariate elliptical truncated moments

Multivariate elliptical truncated moments

In this study, we derived analytic expressions for the elliptical truncated moment generating function (MGF), the zeroth-, first-, and second-order moments of quadratic forms of the multivariate normal, Student’s t, and generalised hyperbolic distributions. The resulting formulae were tested in a numerical application to calculate an analytic expression of the expected shortfall of quadratic po...

Multivariate truncated moments

1 We derive formulae for the higher order tail moments of the lower truncated multivariate standard normal (MVSN), Student’s t, lognormal and a finite-mixture of multivariate normal distributions (FMVN). For the MVSN we propose a recursive formula for moments of arbitrary order as a generalization of previous research. For the Student’s t-distribution, the recursive formula is an extension of t...

Truncated moments of parton distributions

We derive evolution equations satisfied by moments of parton distributions when the integration over the Bjorken variable is restricted to a subset (x0 ≤ x ≤ 1) of the allowed kinematical range 0 ≤ x ≤ 1. The corresponding anomalous dimensions turn out to be given by a triangular matrix which couples the N–th truncated moment with all (N+K)–th truncated moments with integer K ≥ 0. We show that ...

Directions for computing truncated multivariate Taylor series

Efficient recurrence relations for computing arbitrary-order Taylor coefficients for any univariate function can be directly applied to a function of n variables by fixing a direction in Rn. After a sequence of directions, the multivariate Taylor coefficients or partial derivatives can be reconstructed or “interpolated”. The sequence of univariate calculations is more efficient than multivariat...

Fourth moments of multivariate GARCH pro esses