Value-at-Risk Theory and Practice
The first advanced book on value-at-risk
Chapter 3, Page 123 Exercise 20
Consider a singular random vector X with mean vector and covariance matrix
[3.54]
Transform X into an equivalent two-dimensional random vector Y with mean vector 0 and covariance matrix I:
[3.55]
Solution
We first solve for k. By our multidimensional identity [3.31]:
[s1]
so we seek a factorization . Applying a Cholesky factorization, we obtain:
[s2]
Solving next for d, by our multidimensional identity [3.30]:
[s3]
Accordingly, our transformation is:
[s7]
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.
Applying a Cholesky factorization, we obtain:
